CANTOR'S ARGUMENT
4 The Assumptions of Set Theory
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Cantor's argument is short and lucid. It has been around now for over a hundred years. Probably every professional mathematician alive today has studied it and found no fallacy in it. So there is every temptation to imagine that anybody who writes a paper attacking it must be of dangerously unsound mind. One should resist this temptation; the facts don't support it. (Wilfred Hodges [N0]).
It is the most extraordinary result in mathematics. In a paper published in 1891, the German mathematician Georg Cantor showed that there is more than one kind of infinity. Beyond the so-called natural numbers, the infinite sequence 1,2,3,4 … that continues forever and endlessly, there exists another realm of transfinite numbers. Cantor showed that since we cannot list out every possible combination of alphabetic characters, even if the list were infinitely long, the number of possible combinations must lie beyond every finite number that exists [N1]. For suppose each item E on the list is an infinite sequence of characters: m (mann) and w (weib):
E = ( m, m, w, w, w, m, w, … )
and suppose we could list every combination there was, in the form of a table with numbered rows. Then we could construct such a combination by taking the first letter of the first row, changing it so that 'm' turns into 'w', or conversely, then taking the second letter of the second row and changing it in the same way, changing the nth letter in the nth row of the infinite table to its opposite, and so on forever. This forms a so-called diagonal combination of letters. Now ask, was the diagonal already listed in the original table? It can’t have been on the first row, because the first letter of the diagonal was deliberately changed to be different from the first letter of that row. It can’t have been the second row, because the second letter of the second row is different from the second letter of the diagonal. Altogether, there is no row n in the original table that matches the diagonal combination, because the nth letter of this row was deliberately changed to be different from the nth row of the diagonal combination. The diagonal is not on the table, and we supposed this was any such table. Since the rows of any such table can be numbered, it follows that there is no way of putting all the possible combinations of letters into a one to one correspondence with the numbers 1, 2, 3 … and so on. The number of possible combinations must be greater than any finite number.
The argument is wonderfully brief, and contrary to all intuition. If there are infinitely many natural numbers, they will never run out, so, surely we could use them to count everything (including themselves) that there is. Cantor's argument may be valid for finite combinations of integers or letters, you would think. Though we cannot use a finite set of tiles to count the finite number of possible arrangements of the same tiles - and we don’t need anything so strong as Cantor's argument to prove this - given an unlimited number of tiles, why can we not count every possible one of the (now unlimited) arrangements of the tiles? Naïvely, there is just one kind of infinity; there is nothing between any finite set of things, and the unimaginable, unlimited extent of everything that is and could be, except other finite sets. Cantor's argument yields the astonishing and incredible idea that for any infinite set of things, there is a larger one.
Cantor's ideas aroused hostility at the time he was writing, and afterwards. Poincare supposedly declared that later generations would regard the Mengenlehre 'as a disease from which one has recovered.' [N2]. Wittgenstein was implacably hostile to it, and also compared it to a malady from which mankind would be delivered 'as the result of some development or other' [N3]. For a short time they brought about a crisis in mathematics, since in their original form they seemed to involve a contradiction. Zermelo showed that they did not, by axiomatising them as set theory in a way that showed they were not obviously inconsistent. But even today, there remains a reasonably widespread view that Zermelo's axiomatisation is an ad hoc attempt to salvage as much of Cantor's original ideas as possible, and that it is, in some sense, an unhappy compromise, a botched job. [N4] Most would agree with Hilbert that Cantor had created a 'paradise' for mathematicians, but some entertain doubts [N5].
The purpose of this essay is an elucidation of the philosophical ideas underlying Cantor's argument. In the hundred years since it appeared it has been much discussed as a piece of mathematics. Yet most discussions betray an almost uncritical acceptance of the philosophy behind his ideas. Mathematicians see no obvious inconsistency arising from the assumptions of standard set theory, and tend to view the results as unproblematic. Philosophers defer to received opinions in mathematics and, almost without exception present the argument as uncontroversially correct, and limit themselves to the consequences for human rationality, given that most intelligent people who have no set-theoretical background find the argument difficult to grasp or believe [N6].
Yet Cantor writes as a philosopher. His defence of his theory of the infinite engages both with the mathematical philosophy of the ancients and of his contemporaries. His writing is for the most part a stylish German, replete with references in Latin, French, Italian, to writers of the tradition of Locke, Descartes, Berkeley and Leibniz. His work is a defence and a demonstration of philosophical ideas that goes beyond what is simply mathematics. These ideas, I shall argue, are not uncontroversially correct. Some of them are patently absurd. I shall elucidate Cantor's argument and to address that important question: on what grounds, if any, can we say its conclusion is true?
Cantor says he will prove that there is an infinite set or 'manifold' (Mannigfalltigkeit) which cannot be correlated in a reciprocal one-to-one way with the set or 'totality' (Gesamtheit) of all finite integers. His argument involves three main assumptions [N7]. First, that the notion of a set, or totality can be made coherent. Second, the assumption now known as 'Cantor's Theorem', that not all subsets of a set M can be 'correlated in a reciprocal one-to-one way' with the elements of M. Third, that the natural numbers form a set. It follows by logic that the set of natural numbers exists, and cannot be correlated 1-1 with the set of all its subsets. Fundamental to these is the idea of a set. The diagonal sequence of characters is such a set, the proof depends on showing that its existence, and therefore the uncountability of the set to which it belongs, is a consequence of this idea.
The first difficulty, for those who seek to make the argument coherent, is to explain what a set is. It is not entirely clear in what sense it is a thing at all. Cantor gives two definitions. The first is that a set (Menge [N7a]) is not many things, but a single thing formed in some way out of many things, a 'gathering-together' (Zusammenfassung) into a whole of 'determined well-distinguished objects … of our intuition' [N7b]. The second is that it is not one thing, but many things brought together into some sort of unity by the mind, 'a many which can be thought of as one' (jedes Viele, welches sich als Eines denken laesst).
These definitions reflect two old and conflicting ideas about plurality. The first is that a set is one thing. It is the obvious idea suggested by the fact that nouns like 'set', 'collection', 'class' and so on appear in the singular, qualified by the singular definite article 'a', and by the fact that we count a set as one thing. We say a set of dishes, one pair of shoes. We say 'here is a', rather than 'here are many'. Here is one group of people at the door (rather than two). We talk about 'another', i.e. a second group, and of course we can order things into groups: the first, the second, the third, and so on, or we can refer to many groups by their number, as in 'divide the applicants into ten groups'. It is also suggested by the grammatical structure of a sentence like 'Socrates is a man', which was thought by early set theorists some to express a relation (expressed by 'is a') between the referent of 'Socrates', and the referent of 'man' [N8].
The second view is that a set is fundamentally many things. Nothing is really created when a number of objects are collected together, or when we refer to them as a collection. Being aggregated is just 'a manner of existence' of the aggregated objects, as Leibniz says [N9]. Why should a number of rings, interlaced to form a chain, constitute a genuine substance any more than if they had openings in them through which they could be separated? Aggregation is nothing more than relations between different objects.
These opposing views reflect a deep puzzle about the nature of groups, collections, and wholes. Many things can't in fact be one thing (at least, not in the sense that a thing is whatever is fit to be the referent of a singular noun phrase). For if many things, such as Peter and Paul, were identical with one thing, whatever was predicated of the many would be predicated of the one also, and the plural predicate 'were apostles', which is asserted of Peter and Paul, would also be asserted of some singular Peter-and-Paul entity which is neither Peter not Paul. Which is absurd, for a sentence like 'Peter and Paul were apostles' [N10] is obviously equivalent to the two sentences 'Peter was an apostle' and 'Paul was an apostle'. The two sentences assert the singular predicate 'is an apostle' of just two things [N11]. If they do not refer to a single Peter-and-Paul object, neither does any sentence to which they are equivalent [N12]. A sentence using plural expressions does not say anything about any individual apart from those referred to by the expressions. What refers to many things cannot simultaneously refer to one thing, and what refers to one thing cannot refer to many. 'One' and 'many' are fundamentally opposed: illud quod est unum non est plura [N14].
But if many things cannot be one, how can we apparently refer to them as one, using singular nouns like 'collection', 'aggregation', 'set' and so on? That is why some philosophers have proposed that sets are 'one' only in some fictitious or imaginary sense. These many things can't actually be one, so it must be that we think of them as one. Leibniz says that a combination of two diamonds may be called 'a pair' of diamonds, but that this is only an 'entity of reason'. Such things 'have their unity in our mind only'. Hume argues that we can think of the whole universe as a unity, but that this is 'a fictitious denomination, which the mind may apply to any quantity of objects it collects together'. Dedekind defines a set as a number of things comprehended under a common point of view, in the mind [N15]. Wang says that we can view a number of objects both separately and 'as a unity', and that viewing them together 'suggests a loose link which ties the objects together in our intuition' [N16]. Indeed, the term 'universe' itself comes from the Latin unus plus versus, meaning literally 'turned (or combined) into one'.
But how can we even think that many things are one thing? If this is a contradiction, we are thinking a contradiction. In any case, as Bolzano forcefully argued [N17], a plurality is not fictitious or imaginary. It is not that we think of an plurality or collection as one plurality or one collection, for the nine of diamonds and the nine of hearts actually are one pair of cards (not many pairs), the set of dishes is actually a set of dishes (not many sets). If 'a many' denotes a single thing, as the singular article implies, then there is no need to think of it as one, for it is already so. If it denotes many things, how can those things be thought of, except by mistake, as one [N17a]?
I shall argue that the puzzle arises from an elementary misunderstanding about the grammar of collective nouns [N18]. It is a mistake about the meaning of 'one' in the noun 'one couple', and about why a collective noun of the form 'an X of Y's' is singular, even when the genitive 'of Y's' is plural. A collective noun phrase is qualified by 'one' because it represents a single instantiation of the corresponding predicate. For example, an instance of the one-place predicate 'x is a person', is when the value of the variable x is Alice, another is when the value of x is Bob, another when it is Carol, and so on. This is indicated by the articles 'a' or 'one' attached to the noun formed from the predicate, thus: Alice is a person, Bob is one person, and so on. But an instance of the two-place predicate 'x is married to y' is when both of the variables are assigned values; for example when the value of x is Alice and y is Bob, or when the value of x is Carol, and y is David. This is also indicated by the articles 'a' or 'one' attached to the noun 'married couple', thus Alice and Bob are a married couple. Each is an example of one married couple, even though it is an instance of two people, because 'x is a person and y is a person and x is married to y' has two argument places, whereas 'x is a person' has only one. This is simply a way of referring to two individuals, not to a singular entity. Or rather, the expression 'singular entity' means whatever is fitted to be the subject of a distributive predicate like person(x) or shoe(x), rather than a non-distributive predicate such as 'couple', which applies to a number of things together. Nouns like 'set', 'collection', 'combination', and so on, are like 'couple' or 'pair', except they do not signify exactly how many individuals form the set. They are, as Van Inwagen says [N19], 'variably multigrade': they can apply to any number of entities. A winning combination of cards can be three queens, or four queens, or three queens and two tens, or any number of cards whatsoever. Whichever cards the combination is of, it is a single assignment of values to the predicate 'combination(x, y,)'. The combination is many, in the sense that they are many cards, it is one, in the sense that it is one combination. Thus, 'an X of Y's exists' means
(E x, y, z, …) X(x, y, z, …) and Y(x) and Y(y) and Y(z) ….
This also explains how we count a set as one. Expressions like 'a set' or 'a lot' are counting sortals used to count many things together, rather than just one thing at a time. Imagine a language where 'one shoe' was represented as 'a single of shoes', where we would contrast those things which are single, with those which are pairs, triples, dozens, or combinations or groups or classes. A single set of things is not a single thing, any more than a dozen pairs of things, i.e. 24 things, is 12 things. 'There is a single set of things' does not imply 'There is a single thing that is a set'.
Thus, it is a confusion to suppose a set is many things thought of as one, because a set is not one thing at all, though it is one set. We do not think of a combination as one object but as one combination of objects, and we are correct, because the combination actually is, in reality, a combination. It is a confusion to suppose that collective nouns like 'a pair of' or 'a set of', 'a combination of' and the like, represent one-place predicates. They do not and cannot. It is essential to the logical grammar of 'a pair of shoes', that 'pair' counts as one instance of a two-place predicate, and 'shoe' counts as two instances of a one-place predicate. Wrongly supposing that 'collection' is a one-place predicate, and not finding the object that satisfies it among the objects that form the set, we imagine the missing object must be an 'entity of reason', or a 'fictitious denomination of the mind'. The apparently singular nature of the collective noun is, as Hartley Slater has said [N20] a 'grammatical mirage'.
The confusion is prevalent throughout the literature on sets and classes [N21]. Leibniz says that an aggregate is not really and truly an entity, because it is not an entity. It is not truly singular, it is not a 'true unity' because we cannot truly quantify it with the article 'a' or the number word 'one'. Yet he uses the singular article 'an' in the expression 'an aggregate', indeed, he has no other way of expressing his point. The noun 'aggregate' is not a monadic predicate. And so, for that reason, it has to occur in the singular when there is only a single assignment of its variables. An aggregate of things is 'really and truly' an aggregate. Russell asks whether a class of many objects is to be regarded as itself one or many?
Taking the class as equivalent simply to the numerical conjunction 'A and B and C and etc.,' it seems plain that it is many; yet it is quite necessary that we should be able to count classes as one each, and we do habitually speak of a class. Thus classes would seem to be one in one sense and many in another [N22].
He is correct that the singular article 'a' means 'one', and so that 'a class' means 'one class'. But he supposes that one class of things is 'in a sense' one thing, which is not correct. A class of things is one class of things, and it is not a single thing precisely because it is a class of things. It is not that 'one' and 'many' have different senses, as he suggests. It is that they qualify different nouns. 'One' qualifies 'class of things', 'many' qualifies 'things'. A class is one in the sense that it is one class, it is many in the sense that it is many things, just as a pair of shoes is a pair of things, and it is not a single thing because it is a pair of things [N23].
In summary, the word 'set' is linguistically a collective noun 'set of things', signifying those very things of which the set is a set. A collective noun phrase such as 'that X of Y's' is a way of referring to what 'those Y's' refers to, and a combination or plurality is identical with the things which it is a combination or plurality of. Cantor's idea, that a set is many things thought of as one thing, is a mistake. If a set is one set of many things, it is wrong to think of them as both many and one. Rather, an set is many things referred to as one set of things.
Cantor also says that a set is in some sense 'formed' from many things. This idea is essential to the second part of his argument - known as 'Cantor's Theorem' - that any list of subsets must always omit some "diagonal" subset. It has an elementary set-theoretical formulation. Let M be any set of things, and let f be any function that correlates elements of M with subsets of M. We can easily prove there is a set D – the "diagonal" set – that does not correspond to any element of M. Take each element m of M in turn, and ask whether m is in the subset f(m) to which it corresponds. If not, put it in D. Thus, we separate off a set of things D, consisting of all elements of M that are not in the subset to which they correspond. But of course D is also a subset of M, and there cannot be an m such that f(m) = D. Or suppose there is. Then m is in f(m), or it is not. If it is not, then it satisfies the description 'n is not in f(n)'. But any element that satisfies this description is, by definition, in the set D, and, since D = f(m), is in f(m). But then it satisfies 'n is in f(n)', and so cannot be in D, since D contains only those elements that satisfy 'n is not in f(n)'. If it isn't in D it is, if it is, it isn't. The assumption that there is an m such that f(m) = D, leads to a contradiction. But f was any function that correlates elements of M with subsets of M, and so for any such f, there is always a subset of M that is 'out of range'; and we assumed M was any set. So, no set can be correlated with its subsets in this way [N24].
The argument, as Hodges says, is short and lucid, or it seems so. The existence of the diagonal set D seems to follow naturally from the idea that we can 'sort' through the elements of the set M, selecting only those which are not in the subset given by the correspondence, and "form" the diagonal set. This principle, that we can form a subset of any set using by segregating those elements that possess some property, is known as comprehension. It is absorbed into the fabric of modern logic and mathematics, and is fundamental to it. It is formulated as Axiom III, the Aussonderungs axiom, in the standard system of set theory set out by Zermelo in 1908, where Cantor's theory is reduced to seven independent axioms.
Axiom III: Whenever the propositional function -- (x) is definite for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which -- (x) is true [N25].
The idea of forming a set by sorting is essential to this axiom. Indeed, though it is usually translated as the 'Axiom of Separation', the German word Aussonderung means 'sorting', so it could equally have been called the 'sorting' axiom. It is considered the most clear and self-evident of the set-theoretical axioms. Yet it is not clear or self-evident. The difficulty is that if a set is identical with its elements, the axiom is false, for there may be no elements 'for which -- (x) is true'. There are no things, hence no set that the subset M' could be identical to. Moreover, even if there are some elements for which -- (x) is true, it is not obvious that the expression 'the elements for which -- (x) is true' meaning all such elements, has a reference, but I shall discuss this later.
The justification by sorting or separating out is blatantly circular. We are supposed to separate out every element of M which satisfies -- (x), into some pile, perhaps. So every element of M either satisfies or does not satisfy the definite property 'x lies in the pile of elements for which -- (x) is true'. But then we need to show that there is a set M' containing as elements for which this definite property is true! That was where we began: with a set of objects M, and a definite property, and the need to show how the property defines a set M. It ends with exactly the same things: the objects in M, and another definite property of lying in some pile. What justifies the conclusion that the second definite property defines a set of objects within M, given that the first definite property does not?
Underlying this, of course, is an important principle. If there are objects signified by the definite noun phrase 'the F', and if any of these are G, then the definite description 'the F which are G' also signifies a definite number of objects. If M is all the horses that ever ran in the Derby, and F be 'ran a race in 1998'. Then 'the M that are F' refers to a definite subclass of the M, namely those animals that ran in the 1998 Derby. But this principle cannot itself be justified by the sorting explanation, which is merely a circuitous way of defining an indefinite noun phrase, 'an object sorted into a set containing objects belonging to the M that satisfy the property F', from the simpler noun phrase 'an F belonging to the M'. We still have to justify the existence of a set of objects picked out by the longer noun phrase, and so would have to go through M again, sorting out those objects which were in the pile, from those not in the pile, so creating another pile, and another definite property, and so on, and so on. We are dealing with a principle of language that is absolutely fundamental, that cannot be justified by any argument or explanation, and which we have to assume in all our reasoning.
Note also that if sets are identical with their elements the requirement for a "definite" or "well-defined" property is otiose. It is sufficient that the elements satisfying the property exist. If they do not, no set exists, even if the property is well-defined. There is no set-of-unicorns, for example. If they do, then there always exists the perfectly well-defined property of being identical with one of those elements. Cantor says, for example, that a well-defined set exists if it is "internally determined" on the basis of its definition whether any arbitrary object in a domain belongs to the set or not [N26]. But the condition is irrelevant: whatever set of objects we consider, if they exist all, it is already "determined" whether or not any arbitrary object belongs to the set, since the object is either one of those objects, otherwise it is not (assuming, as he says, the principle of "excluded middle"). Or suppose we discovered some set of objects which have no "basis of definition", and for which it is not "internally determined" whether an arbitrary object is one of those objects. Then which objects are we talking about? Wouldn't they also constitute a set?
Naturally, if a property is not "well-defined", it is difficult to sort out objects that have the property from objects that do not, and thus to sort the former into a pile. But this confuses an epistemological question with a logical one. The question is always: do any objects satisfy the property? If there are, we can speak of 'those objects', and we can ask of any object whether it is identical with any of them, or not.
It could be objected that a set is "formed" by separation in the same way that a pile of objects is formed by the sorting process, or in the way that a crowd is formed as it gathers together. Suppes says, for example, that the axiom of separation permits us to 'separate off the elements of a given set which satisfy some property and form the set consisting of just those elements [N27]'. The difficulty with this argument is that in set theory a set exists so long as its elements exist, even when they have not been sorted into a pile, or gathered into a crowd. The existence of the set is guaranteed by the existence of individuals individually possessing the right distributive properties. The diamonds form a set, so long as they are diamonds. The crowd of agitators form a set, so long as they are agitators. It is true, under all circumstances and unconditionally that 'those elements x of M for which -- (x) is true' form a set. By contrast, being a crowd, or a pile, is a non-distributive property of all the things that from the crowd or the pile, together, and does not depend on any distributive properties that apply to them. The cards forming a single suit are all mixed randomly in the pack of cards until they have been sorted into a separate pile. The people were having breakfast, or sleeping, before they met together as a crowd, and before they were a crowd. The crowd, or the pile, may not exist even though the elements do.
Given any X's, and some property F, there is no reason to suppose any collection of objects comes into existence as a result of "sorting" the objects among the X's into a pile of those things that are F, and a pile of those things that are not. Either none of the X's are F, in which case there is no collection at all, or there are, but these are just all those objects among the X's that are F, and these were not created by the sorting process. The set is the objects. It is them; its creation is their creation. Thus the process of separation cannot "form" anything.
This is particularly obvious in the case of the so-called empty set and singleton set. The empty set, denoted by {}, is supposedly a set with no elements: for example, the set of unicorns, which is guaranteed to exist by the axiom of Separation because of the existence of the set of horses {x: x is a horse}, and because the predicate 'x has a horn' then defines a set of horses with horns {x in {x: x is a horse}: x has a horn}. There is no such set in ordinary language, because 'those horses that have horns', if it referred, would refer to nothing but horses with horns, and there are no such things [N28]. Note the absurdity of any appeal to sorting, in order to justify the existence of the empty set. We sort through the set of all horses, we find nothing with a horn, so we have "formed" the empty set! Ordinarily, we do not think like this. 'There are no horses with horns' does not assert the existence of an object which does not contain horses [N29].
Similarly, if a set consists of nothing but its elements, there cannot be a 'singleton set' containing exactly one object. A set consisting of one element would be that element. Hence, the absurdity of the appeal to sorting through the set of horses in order to "form" a singleton set. For example, the predicate 'won the Derby in 1780', is satisfied by only one horse, Diomed. If we "sort" through the set of horses to find one which won the Derby in 1780, there is just Diomed. The Axiom then "allows" us to infer the existence of some other object than Diomed. This is not intuitive. The definite description 'the horse that won the Derby in 1780' refers to Diomed.
In the case of sets with more than one element, there is still the problem of what exactly is "formed". If the set is identical with its elements, nothing is formed at all: there simply are those elements for which -- (x) is true. If it is not, then something exists for which -- (x) is not true, namely the set itself. Given the existence of things which do satisfy a certain property, is it self-evident that anything exists which does not satisfy that property? Some philosophers have argued that a set is a mereological sum or fusion of the objects of which it is composed, and that plural terms signify such a sum [N30]. On this view, the word 'and' is a way of forming a new proper name out of individual proper names, and the expression 'Peter and Paul' is a compound name for such a fusion, a Peter-and-Paul entity formed out of but separate from its individual elements, rather as water is a separate substance formed out of the separate elements hydrogen and oxygen. This is fundamentally a mistake, as I have argued above. Sentences containing 'Peter and Paul' as a subject can be analysed in terms of distributive or of non-distributive predication of the relevant individuals. There is nothing signified by 'Peter and Paul' except what is signified by 'Peter' and what is signified by 'Paul'. 'Those people' refers to nothing but: those people.
Some philosophers have argued that sets are required in order to express certain scientific, mathematical or logical theories. Sets are convenient fictions, abstract entities invoked to explain what could not be explained otherwise. Quine argued that they are required even to explain a sentence as apparently innocent as 'a is an ancestor of b'. The formal representation of this sentence,
(X) [ b in X and (y) (z) [ y in X & parent(z,y) -> z in X ] -> a in X ]
can be paraphrased as 'a is in every set which contains b and every parent of every member'. The capitalised variable X clearly cannot stand for one of my ancestors and there is a standard proof that it cannot be eliminated [N32]. Since it occurs in the existential statement '(E X)', it seems we are "ontologically committed" to the sort of thing that this so-called second order variables range over [N33]. Quine says that the price paid for the expressive power of language is "ontological": we must presuppose objects of a special and abstract kind, namely sets or "second order entities". The existential '(E X)' says that the variable X takes a value and, as he says, to be assumed as an entity is to be assumed as a value of a variable [N34]. The sentence 'a is an ancestor of b' is not ontologically innocent.
Yet it is easy to paraphrase such sentences in a way that avoids collective nouns in favour of plurals, and of a relation which connects singulars and plurals. We can say, for example 'There is a group of people such that y is one of them, and any person among them is either the same person as x, or is such that their parent is among them'. The relation expressed by the English 'one of' and 'among' can be defined in terms of the connective 'and' of 'Peter and Paul'. When we say 'The Earth is one of the planets', we are not connecting the Earth to some singular object signified by 'the planets'. We are saying that the planets are identical with the Earth and the other planets. A single thing x is one many things Y, if and only if there are some objects Z (i.e. the 'rest of' Y), such that x together with those objects constitute Y. Formally:
x is one of Y iff (E Z) Y = x & Z
The only "ontological commitment" is to whatever the value of the quantifier 'some people' takes, namely whatever 'those people' refers to [N35].
A counter-argument (by Rouilhan and others) is that words like 'among' or 'one of', and plural terms are the way that natural language deals with mathematical sets. He says that Boolos' argument begs the question, since it relies on the assumption that plural existential quantification manifestly, obviously enjoys ontological innocence. We must distinguish between sentences like 'I have some ancestors' which can be represented as the first-order sentence 'I have at least one ancestor', and sentences like Quine's, which cannot represented without the use of English plurals, and cannot be formalised without the use of 'capitalised variables'. This is fundamentally how ordinary language distinguishes between first and second-order quantification. In 'some critics are honest' the plural is distributive: each critic one by one has the property of being honest. In 'some critics admire only each other', it is collective: there is some property that some critics have 'all together', and thus it affirms the existence of a collection of critics. How do we show that the kind of innocence that legitimately belongs to the first sentence, also belongs to the second? Rouilhan argues that the value of such a second-order variable is what he calls a multiplicity.
This is a mistake, for reasons that should already be clear. I have argued that we can paraphrase sentences with plural referring terms by sentences with singular terms, and so the plural terms imply the existence of no more than what the use of the singular terms commits us to. The sentence 'Peter and Paul were apostles' is exactly equivalent to 'Peter was an apostle and Paul was an apostle'. Though we cannot paraphrase away the use of plural variables, as Quine's examples show, that means nothing, for it is to the existence of possible values of the variables that we are committed, and the existence of the plural values (Peter and Paul) – involves no more than the existence of the singular values (Peter and Paul). The plural quantifier '(E X)' asserts the existence of some things. But it does not assert the existence of anything that is not one of those things [N36]. The objection that it refers to some thing that is not one of them, is fundamentally mistaken. As for the idea of 'a' multiplicity, I have already argued that we cannot do without the 'one' in 'one pair of things'. Nor can we do without the 'a' in 'a multiplicity'.
In summary, the Axiom of Separation has no intuitive justification. Set theory appeals to our intuitions about sorting into piles or groups, and regard it as the most natural and self–evident of the set-theoretical axioms, but the appeal is inherently circular. If we cannot prove, without such a principle, that 'those F's which are G' has a reference, we cannot prove that 'those F's which have been sorted into a pile containing only objects that are G' has a reference either. Nor is anything "formed" by this sorting process. The same objects that existed before, acquire a non-distributive property, such as being a crowd, or a pile. No new object is formed, although the same objects acquire a new property. And no singular object is referred to by expressions that have a plural reference. While plural variables cannot be eliminated, it is to the existence of possible values of the variables that we are committed, and this amounts to no more than the existence of singular values.
The final assumption underlying the diagonal argument, and the key to Cantor's philosophical view of the infinite, is that the series of finite numbers is in some sense "finished" or complete, that the numbers form a set, N. Then the existence of the diagonal set follows from the Separation Axiom. But is there a set of all numbers? Early in life we learn to count in sequence, or at least to parrot a sequence of numerals in the right order. Later, we learn the fundamental techniques of addition, of manipulation of the decimal number system, and of constructing numbers as large as we like. As part of this, we learn that any number of things that we can represent in this way, can always be increased by the addition of another. It is fundamental to any number that some other number comes after it, that everything that counts as a number has a successor. This suggests there cannot be a set that includes all numbers. Such a set must include the successor of every number. Any set of numbers there is, must itself have a number, and there must be some number that follows that. No set of numbers can possibly include all the numbers there are. In effect, all numbers are finite. This natural conception of the infinite as "potential" was the standard view from Aristotle until Cantor.
… it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number [N37].
It is Cantor's main target. He argues [N38] that every finite number is followed by another finite number. So a set v of finite numbers which itself has a finite number, cannot include all finite numbers. Why should there not be a set w of all such numbers v? Such a set will not itself have a finite number – being the set of all finite numbers, so its number must be non-finite, or "infinite". Let us simply say that there such a set. This is formalised by Axiom VII of Zermelo's system, the Axiom of Infinity.
Axiom VII. Axiom of infinity (Axiom des Unendlichen) There exists in the domain at least one set Z that contains the null set as an element and is so constituted that that to each of its elements a there corresponds a further element of the form { a }, in other words, that with each of its elements a it also contains the corresponding set { a } as element.
It states that there is a set Z (a so-called inductive set) that includes an initial 'zero' element and every successor of every member. While there is no "largest" set in Z (for the axiom states that to any such set there corresponds a "larger" one), there can nonetheless exist a set of such sets, larger than all of them. The philosophical objection, that there can be no number so large that we cannot imagine a larger one, is true only of finite numbers. The set including all numbers that have the property of finitude cannot itself have the property of finitude [N39]. There is nothing contradictory in supposing this.
… however contradictory it might be to speak of a greatest number [groessten Zahl] of the [finite] class, there is nevertheless nothing offensive in thinking of a new number [eine neue Zahl] which we shall call w, and which [my emphasis] will be the expression for the idea that the entire assemblage [der ganze Inbegriff] of [the class of finite numbers] is given in its natural, orderly succession [N40].
The conception of infinity as "potential" assumes that nothing other than finite numbers exist, and any attempt to prove otherwise involves a petitio principii. When the finitist is asked how he knows that there are no infinite numbers, his reason, according to Cantor, is that 'all numbers are finite,' which, of course, is the very point at issue.
All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on general grounds as well. It is their prwton yeudox [initial falsehood] that from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices [N41].
Cantor claimed to have discovered this petitio in the philosophical arguments of Aristotle, Spinoza, Leibniz, Locke, Descartes and others [N42]. All finitist arguments assume the very thing they are trying to prove.
The argument is simple, and is accepted in all standard mathematical accounts of the natural numbers. But it depends on the idea I have called into question, that a set is a single thing that, because it is single, cannot be identical with the many things of which it is a set. Otherwise there cannot be an infinite plurality, as is shown by an argument that is, in essence, due to Leibniz [N43], and which depends (a) on what I shall call Malezieu's principle: if a set or plurality is its members, any set that lies outside the domain of quantification must include at least one element that lies outside the domain. As I have argued, the existence of a plurality amounts to nothing more than the existence of the individuals which constitute it. A plurality is not a single thing (though, as I have argued, it is a single plurality). If a plurality of elements exist, it is not the case that anything has to exist apart from those elements. To be sure, the plurality is not identical with any of its elements. But that is because it is identical with all of them. It is many things, and for that very reason we should not suppose it is a single thing, distinct from them.
A plural variable such as 'any critics' therefore ranges over exactly the same things that a singular variable (critic) range over, for there is nothing else for there to be. A singular object v is in the domain of a plural variable X if and only if for some plural value V of X, v is one of V. Similarly, a plurality V is in the domain of a plural variable X if and only if all v in V are in the range of X. For example, the reference of 'those critics' lies within the domain of 'any people who live in the street' only if each of those critics lives in the street. Thus Malezieu's principle says that any plural value V lying outside some domain D includes some object that itself lies outside D.
(By contrast, a mathematical set is a single thing distinct from any of its elements. If the set exists, then something exists which is not one of its elements, for the set is one thing, it is not identical with any of its elements, and is therefore distinct from all of them. Thus the set itself, being a singular object, can lie outside the domain consisting of singular objects such as its elements. The set of all critics is something other than a critic).
But (b) in order for there to be infinitely many things, it is necessary that there be some domain that fails to contain a set of all such things. For example, the domain corresponding to the range of 'every finite set', which contains every finite set, but which cannot contain the set N of all elements in the domain; for that, by definition, is an infinite set.
Then (c) if all sets are identical with their elements, Malezieu's principle says that at least one element of N lies outside D, which is contradictory. Hence there cannot be an infinite set, if a set is many things. If there are not so many X's that there are not more of them [N44], there cannot be a set or plurality consisting of all X's.
Now Cantor's objection was to the assumption that the variable X ranges over finite pluralities only. If there is some infinite domain of finite sets D, why can't there be some wider domain D' that contains the set Z of all the elements in D? Which is correct, if a set is conceived as being one thing rather than many things. But when we quantify using a plural variable X, each of whose values represents many things, a value-of-things, there can be no universe where there is an X consisting of every object in D. If such a plurality X existed outside D, by Malezieu's principle at least one such object in X would be excluded from D. Each value of a plural variable is identical with a number of objects, so any value outside the narrow domain D includes at least one individual object outside D as well. (If, for example, V is the value consisting of the numbers 1 and 2 taken together, and V lies outside the domain, either of the individual numbers 1 or 2 (or both), lie outside it also). But we supposed that V includes just objects in D. Hence there is no value V corresponding to every object in D, either within D (because that is what we supposed), or in any other domain (because excluding V would entail excluding some object among D). There can be no value "outside the range" corresponding to all objects "in the range". Cantor's complaint - that the finitist position involves a petitio principii – is valid only when a set is a singular thing, different from all its elements, and thus lying outside the domain that contains only its elements.
There are a number of objections to consider. (i) Set theory asserts the existence of infinite sets without contradiction. Why can't we also assert the existence of an infinite plurality? I repeat the argument above. Mathematical sets are singular objects that are not identical with their elements. When X ranges over mathematical sets, there certainly can be another object, a singular object lying outside the range of our quantifiers, that "contains" the set of all objects in the narrow domain D. We may exclude this singular object from D without excluding any of its elements. (Indeed, there is a version of set theory where exactly this happens). But this is possible only because a singular set is a single object that contains elements, whose criterion of identity is different from the criterion of identity of its elements, and whose existence can be denied without denying the existence of the elements. We can exclude a singular set V, whose elements are all the counting numbers, because this does not involve excluding any elements of V. The excluded object is V itself [N46]. But if V is identical with its elements we cannot exclude it, because V is not itself a thing, but many things, and to exclude it would mean excluding some of them.
(ii) Could there be a value of X that corresponds to "all objects" in the domain, even if there actually isn't? This is impossible for the same reason. If the universe changes in such a way that some set-of-objects V comes into existence, then, because V is identical with those objects, some individual object must have come into existence. Which is impossible, for V was assumed to be the value consisting of all elements in the domain that existed before V comes into existence. Every individual object that exists after V was created, existed before V was created. Therefore, if a set-of-all-objects in the world does not exist in some possible world D, it fails to exist in every possible world that contains every object in D.
(iii) Isn't it contradictory to deny the existence of a plurality that includes every object within the range of the quantifier? Every one of those things exists, so all of them exist. This is argued, in effect, by Cantor [N47], who says that the domain of any variable quantity cannot itself be something variable. Any x that is an element of the domain at one time must be an element of it at any other time, otherwise there cannot be a proof "for all x". The domain or range of a variable x is 'a definite, actually infinite set of values'. Any potential infinite presupposes an actual infinite. But it is this which begs the question. It presumes that the existence of a domain that is the range of 'every x', implies a reference for 'all the x's'. It presumes there is a possible value of x representing all the objects that x can range over. Yet we specifically state that this is not the case, using a statement like the axiom of infinity - that every value X omits some value, and thus no value "includes" every value. The universe where X is unrestricted still contains infinitely many objects. Any possible value excluded from this universe is either a plurality of individuals, in which case at least one individual in the universe has been excluded from it, which is impossible, or it is a Cantorean set, in which case the value does not represent all the objects at all, but a single object different from any of them.
(iv) But if a set is identical with its elements, and if the set of all the elements does not exist, surely at least one of the individuals does not exist? Isn’t the argument contradictory? But it is not contradictory, for Malezieu's principle states that there is no set whose elements lie in the domain, which is not itself in the domain. It does not state that there is always a set that contains every element in the domain. If there is such a set, it lies in the domain. Otherwise, there is no such set at all. Not existing within the domain, and not existing at all, are separate concepts. To repeat the argument above, the existence of a domain that is the range of 'every x', does not imply a reference for 'all the x's'. Malezieu's principle says only that if there is a reference for 'all the x's', and if every one of those x's is in the domain, the reference of 'all the x's' is also in the domain.
(v) What then stops us referring to "all objects"? I reply: what stops us is the truth of the assertion that there are infinitely many. What fails to stop it in the finite case is the assertion that there are only finitely many. It's exactly as though one had asked, 'What stops there being any American aristocrats, when we truly assert there are no American aristocrats', or 'What stops it in that case, but not in the case that there are a very small number of American aristocrats'. It is not that we lack a mental ability, or anything like that. It's not that we are prevented from thinking of infinitely many objects by some failure to collect "them" together in thought, as Hume argued. That, as Bolzano argued, is wrong [N48]. There is no breakdown for very large finite numbers. The breakdown happens, as I have argued, in virtue of what "infinitely many" means, namely that every plural value X, without restriction, fails to include at least one object. If there are infinitely many things, it is because there are not so many of them, that there are not more of them besides.
Thus, Cantor's conception of an actual infinite depends on the assumption that a set is a singular thing different from any of its elements. On the assumption that a set is identical with its elements, it is invalid. It is only possible to assert the existence of a domain containing infinitely many things by denying the existence of a set containing every object in the domain.
#4 The Assumptions of Set Theory
It may still be objected that there are such things as sets, in some set-theoretical sense. I have argued for a particular sense of the word 'set'. In this sense, sets are identical with their elements, and so there cannot be an empty set or a singleton set, and so there cannot, in particular, be any infinite sets. But it could reasonably be objected that the set-theoretical axioms both define and assert the existence of an entirely different kind of object. If there are sets of this kind, this justifies Cantor's argument for an actual infinite, and the existence of higher infinities beyond. Why shouldn't there be such things?
Yet there needs to be a reason for supposing that there are such things. The problem is that all the reasons commonly given for the existence of sets, are no more than arguments for the indispensability of plural terms, rather than for the existence of singular objects over and above the referents of such terms.
I have already mentioned two such reasons. One is that that 'being an ancestor of b' cannot be expressed without invoking sets. But it can, since it is sufficient to quantify over plural nouns quantifiers or collective nouns, which I have argued can be reduced to plurals, and which do not involve "sets". The other is that a collective noun is singular: a set of things is one set of things. This involves the confusion that collective nouns like 'a pair of' or 'a set of', 'a combination of' and the like, represent one-place predicates. Note that it is difficult even to explain the idea of a set without the use of synonyms such as 'collection' or 'plurality'. Shoenfield says that 'the notion of a set is so natural that it would call for investigation'. And then he says that a set is a collection of objects! To analyse this notion any further requires plurals. For example, 'a set may be described by words, for example: A = the first three natural numbers greater than zero, B = the colors red, white, blue, and green'. Or 'a set C is the union of two sets A and B if its members are exactly those objects that are either members of A or members of B'. All of this presumes we understand plural referring expressions like 'those objects that are either members of A or members of B'.
Another reason sometimes given is our intuition that a set is determined by its elements, captured by Zermelo's axiom of Extensionality. The axiom states that any sets which have the same members are themselves the same. But this merely captures intuitions about plural terms, namely that if every one of the A's is one of the B's, then the A's are identical with those B's which are A, hence – if every of the B's is one of the A's – the A's are identical with the B's. If each of the people at the door is one of the people who live at number 14, then the people at the door are equivalent to those people at number 14 who are at the door, hence – if every person at number 14 is one of the people at the door - are equivalent the people who live at number 14. This relies completely upon our understanding of the relation 'one of', which is essentially of part to whole, upon the meaning of definite descriptions like 'the people at the door', and upon the plural identity expressed by the word 'are' [N49]. Elementary set-theoretical reasoning – for example, the reason underlying the distributive principle - is essentially reasoning with plural terms [N50].
Indeed, extensionality is a good reason for not believing that a set is anything different from its elements. If the set is different from its elements, as with a crowd, why suppose it is determined by its elements? A crowd of individuals is not identical with those individuals, for people can leave the crowd, and people join it, without it being a different crowd. 'This crowd' does not refer to the same things as 'the individuals in this crowd'. But then we have to suppose the axiom of extensionality fails to apply, that the crowd is not "determined by its elements". To establish that the "criterion of identity" for being the same crowd is different from the criterion of identity for being the same individuals, we must imagine a case where there is the same crowd, but different individuals, that is, establish a case or situation or example where the axiom of extensionality fails to apply. In order to suppose a different criterion of identity, we have to suppose that extensionality is false. As Russell says, classes which have the same members are not necessarily the same class. 'In order to perceive that man and featherless biped are not identical, it is quite unnecessary to take a hen and deprive the poor bird of its feathers'. A man is not the same kind of thing as a featherless biped. [N57]. And the objects which fall into the same kind are perpetually changing. Think of the class of Londoners, the Buckner family. A class is not made, as Mill says, by drawing a line around a particular set of individuals [N58]. The axiom of extensionality is self-evidently true only if we read the epsilon as signifying 'one of', and if we interpret the sign for a set as a plural term referring rigidly to the same collection of individuals. But then a set of things is not "formed" from those things. It is them.
Another reason for supposing there are sets is that every common noun signifies a property, and so implies the existence of a class of things that happen possess the property. But, as I have argued, it is wrong to suppose that the plural definite description formed from the noun designates any things at all. 'All the unicorns that there are' refers to nothing. It is also a mistake to imagine that the noun refers to anything else. An early argument for sets (by Peano and others) is that sentences like 'Socrates is a man' assert a relation signified, as it were, by 'is a' between the subject and the set or class signified by the word man. In early set theory 'Socrates e man' means 'Socrates is a man'. This idea is historically connected with theories of the line of Boole, Peano and Frege. But, as Russell discovered in 1902, it leads to contradiction. It is possible to assert or deny that a class is a member of itself, i.e. we can assert or deny 'x e x'. But 'x e x' is a predicate, which is true of any class which belongs to itself, and false of any class that does not belong to itself, and which (apparently) divides all classes into the class F of those which belong to themselves, and the class F* of those which do not. But to which of these classes does F* belong? If to F, F* is one of the classes that belong to themselves, and so F* belongs to F*, not to F. Yet if it belongs to F*, it is one of those classes which do not belong to themselves. So it does not belong to itself, and so does not belong to F*. If it belongs to itself, it doesn’t, and if it doesn’t, it does. This contradiction was a disaster for the early "class theory" of Frege and Peano [N51].
The contradiction was avoided by the idea that sets are objects that are somehow formed from their elements, as in Zermelo's modified comprehension axiom, the Aussonderungs Axiom. This prevents any straightforward contradiction, by allowing the formation of sets only as subsets of existing sets, and permits the building of sets in stages, so that sets are formed only from elements formed at earlier stages. This is sometimes called the iterative conception: 'Here are some things. Now we bind them up into a whole. Now we have a set.' [N52]. There can be sets of non-self-membered sets, but no set of all non-self-membered sets, and thus certain classes – for example, the class of all classes that are not members of themselves – are "too large" to be themselves members of any class or set. They do not "form" sets. Such classes were later called proper classes [N53]. This arbitrary solution allowed for the subsequent development of set theory, and the term 'set' gradually displaced the word 'class', to make room for the distinction between 'set' and 'class'. But it did not address the underlying difficulty, of what the words 'set' and 'class' actually mean. If a statement of the form 'x is an F' really asserts the relation e between x and a class F, as Peano asserted, then the statement 'the class F* is a non self-membered class', being a proposition of that form, asserts the relation e between F* and the class non self-membered class. But F* is that class! And so it really is a member of itself, and so the paradox follows. We can escape it only by supposing that a sentence 'x is an F' does not state a relation between two things. But then there is no reason to suppose there are such things as classes, or such a relation as e, in the first place. We prevent the paradox only by supposing that 'x is an F' is not really of the form 'a e F'. But then what exactly is the relation signified by the Greek epsilon? If it does not represent the relation of subject to predicate, or if it does not signify the relation between singular and plural expressed by 'one of', what does it mean? The truth of the axioms depends on this fundamental relation e of set theory. But it is not explained or defined within the theory.
A standard view is that the paradox is also a difficulty for so-called "naïve" set theory. Naïve set theory assumes so-called "unrestricted" comprehension, that a set can be formed from any predicate, rather than by "forming" it from an already-existing set, as the Axiom of Separation stipulates. It is supposedly embedded in our ordinary instincts or intuitions about plurality, and so (by implication) is not a difficulty intrinsic to set theory itself. It is our logical intuitions that are contradictory, because we assume, as Goedel says, that 'every propositional function exists 'as a separate entity' [N55]'.
But this is just propaganda by set theorists, who like to portray their theory as a victory over common prejudice. Our ordinary ideas about sets do not lead to contradiction. Goedel's presumption of some logical intuition that 'for every propositional function there exists the class of objects satisfying it', i.e. that any definite description has a referent, is extraordinary. In naïve set theory itself there may be such an assumption, but that is because it is not properly "naïve". There is a considerable amount of theory built into the idea of a "propositional function", which requires first the idea of a mathematical function, in which objects are "mapped onto" objects, and then transferring this idea, somewhat awkwardly, and dubiously, into the domain of linguistic analysis. We do not ordinarily suppose that the sentence 'Socrates is bald' asserts a relation between two entities, Socrates and some object named by '--is bald'. Like set theory, ordinary language allows us to form a referring phrase from a predicate. We tack the word 'the' onto a noun, to form singular or plural referring terms: 'the X' or 'the X's'. This is the ordinary language analogue of a class, and 'x is one of the X's' the analogue of the membership relation. But grammar prevents this leading to paradox, since there is no direct analogue of the predicate 'x is not a member of x'. If x is a single object, it must "one of" many objects, not a single one. If x is many objects, on the other hand, it cannot be "one" of anything. Ordinary language also has the relation "among". We can say 'the X's are among the Y's', where both X and Y are plural terms, meaning that any x that is one of the X's is also one of the Y's. But then, if the X's=the Y's, it follows that for any X's, it is always true that the X's are among the X's, and false that the X's are not among the X's. Hence 'the X's such that the X's are not among the X's' is a term that purports to refer, but in fact refers to nothing. Russell's contradiction only follows when suppose there is some object of reference. For it would refer to those things not among themselves, about which things we would then wonder if they were among themselves, in which case they are wouldn't be, and vice versa. But we have no such intuition. There is no presumption in ordinary language that there is a reference for the description formed by tagging 'the' onto a common noun, hence no presumption there is a corresponding object or set-of-objects. We have no "common-sense assumption" that any definite description refers to anything simply in virtue of its meaning. Of course a definite description has a meaning, but that, as Russell himself showed, that is not the same as having a reference. He also showed, as we know, that the sentences 'the F is G' and 'the F is not G' can both be false [N56].
Finally, note the difficulty that some collections of objects cannot form mathematical sets anyway. There cannot be a set of all sets, for example. According to the iterative conception, we must collect together the elements of the set, before the set itself is formed. But if we had collected together all sets, before the set of all sets is formed, we hadn't collected all sets after all. Cantor already saw this. If there is a set W of all numbers, arranged in order, it would have a number d. But a set of ordinals containing every predecessor of each of its members has an ordinal number that is greater than any of its members. Since W is the set of all numbers, it would contain d, and then d would be greater than d, per impossible. He concluded that the ordinals are an 'absolutely infinite' or 'inconsistent' multiplicity (Veilheit): a multiplicity whose elements cannot be assumed to 'be together' without contradiction, or such that it is impossible to think of the multiplicity as a unity (Einheit).
I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, 'transfinite' or 'suprafinite') if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements.
… In contrast, infinite sets such that the totality of their elements cannot be thought of as "existing together" or as a "thing for itself" or an ajwrismenon, and that therefore also in this totality are absolutely not an object of further mathematical contemplation, I call 'absolutely infinite sets' [N59].
This would now be called a 'proper' class, a class that is "too big" to form a set. But if some collections of objects do not form sets, what reason is there to suppose that any of them do? The whole idea that sets exist arises from the strong pre-theoretical intuition that any generalizations about things 'of a certain kind', involves grouping 'those things' together in some way. Mirimanoff asserts that 'the existence of individuals must imply the existence of sets of them' [N60], Boolos asks 'How could there not be a collection, or set, of just those things to which any given predicate applies?' (seemingly ignoring the case where there are no such things). This leaves us with the apparent paradox, that (A) the existence of individuals implies the existence of sets of them, yet (B) a set of individuals need not exist, even though the individuals do, since there is no set of all the sets that there are.
This is all the result of combining two contradictory meanings of the word 'set'. Our intuition (A) that the existence of individuals implies the existence of a set of them, follows only on the assumption that a set is identical with its elements, and by the same argument that if Tully is Cicero, and Cicero exists, then Tully exists. Indeed, in saying that the existence of individuals implies the existence of a set of them, it is necessary to invoke this identity: 'them' and 'set of them' must refer back to the same individuals. But the idea (B) that a set of individuals need not exist, even though the individuals do, depends on the contrary assumption that a set is a single thing that is not one of its elements. Then, if there were a set of all things, it would not itself be one of those things, and so not be a set of all things. There is no real contradiction, since the argument depends on contradictory readings of the term 'set'. The assumption that a 'set of things' means the set of elements, implies that if any things exist, so does the set-of-them (in this plural sense of the word 'set'). The assumption that a 'set' means something different from a 'set of things', means that a set need not exist, even if its elements do.
This leads to the strangest parts of Cantor's thought. He assumes without question that we can refer to many things as one [N61]. Thus many objects are 'a multiplicity', 'a whole', 'a totality', the system of all ordinal numbers is 'a' system, 'an' inconsistent, absolutely infinite multiplicity, and so on. But then an inconsistent multiplicity is one thing which is not one thing at all, or which, bizarrely, cannot be 'thought of' as one thing. Or which can be thought of as one thing – since he refers to it as 'a set', but which cannot actually exist 'with the totality of its elements', or which, just as confusingly, is an infinite set such that the totality of [its] elements cannot be thought of as 'existing together'. This makes no sense. What does 'thinking as one' mean? If we can refer to the system or multiplicity as one thing, how is it we can refer to it as one, but not "think of it" as one? What is the force of 'together' as in 'thought of as existing together', or 'actually existing with the totality of elements'? What does it mean [N62]? It is the unavoidable by-product of the idea that a set of things is both a multiplicity 'thought of as one', and a single thing as well.
But, as I have argued, it is impossible to develop infinitary set theory by plural reasoning, for if a set is identical with its elements, there can be no empty set, no singleton set, and – most importantly – no infinite set. Set theory begins with an appeal to our standard intuitions about plurality, but goes on to develop ideas and propositions which are inconsistent with them. We cannot explain the word 'set' in terms of a meaning it does not have.
In overall summary. Cantor's argument for the existence of transfinite numbers depends upon the assumption that there can be a set consisting of infinitely many elements. I have argued that we have no appropriate conception of a set that allows for this.
If a set or plurality is not one thing but many things, Malezieu's principle applies. There is no plurality whose elements are in the domain, which is not itself in the domain. Hence an infinite plurality, in this sense, is impossible. If there are infinitely many things, there is some domain D in which there no largest plurality exists: every collection of elements of the domain, omits some element. But then no set V of all elements exists at all. By definition, none exists inside the domain; by Malezieu's principle, none exists outside the domain. A plurality is identical with its elements, and any value outside the narrow domain D therefore includes at least one individual object that is outside D as well. If V existed at all, at least one such object among V would be excluded from D. We can exclude a singular or mathematical set V, because this does not involve excluding any elements of V. The excluded object is V itself. But we cannot exclude a plurality, because a plurality is not a thing, but many things, and to exclude it would mean excluding one of those things.
It is difficult to make sense of his argument by appealing to other ideas about sets. The idea that it is many things "thought of as one" is a mistake, based on a confusion about the grammar of collective nouns. The number-word attached to such a noun tells us how many times it applies, not how many things it applies to. The 'one' in 'one pair of shoes' tells us that the word 'pair' applies once, but to two shoes. The idea that a set is one thing "formed out of many things" is based on a confusion about what is achieved by sorting things, and about the reference of plural referring terms. The appeal to the notion of sorting that is commonly used in support of Zermelo's separation axiom is blatantly circular. The idea of a class leads directly to a contradiction, as is well known. In any case, the logic of our everyday talk about pluralities depends on reference to more than one thing, and to the relations of identity signified by 'are', 'one of' and 'among'. There is no intuitive justification for set theory. What self-evidence it has, is derived from a misreading of statements that are intuitive and justified.
Is set theory wrong? Obviously not in the sense that its theorems do not follow from its axioms. Of course they do. Everything that is said to follow from the assumptions made by Cantor, and which are embedded in the axioms of standard set theory, follows logically from those assumptions. So far, no contradiction has been proved to result from them. I have questioned the assumptions themselves. Are they really true? Are they axioms in the standard sense, that is, in the dictionary sense of propositions that are self-evident, assented to as soon as understood? I have argued that they are not.
Edward Buckner
London, January 2006
[N0] Wilfred Hodges (1998).
[N1] He proves, in fact, 'that there are infinite manifolds (Mannigfalltigkeiten) which cannot be correlated in a reciprocal one-to-one way with the totality (Gesamtheit) of all (aller) finite integers 1, 2, 3,…, v, …', i.e. that such manifolds have a larger number or 'power' (Machtigkeit) than the sequence of finite integers. However, as he observes, it can be proved from this, that it is impossible to present in sequential form (in der Reihenform) all real numbers (reelen Zahlen) of an arbitrary interval.
[N2] Quoted in Morris Kline's Mathematical Thought, p1003.
[N3] 'Think of the use of the motor-car producing or encouraging certain sicknesses, and mankind being plagued by such sickness until, from some cause or other, as the result of some development or other, it abandons the habit of driving.' Remarks on the Foundations of Mathematics II.23.
[N4] Mayberry 2000, p. xv.
[N5] Wittgenstein: 'If one person can see it as a paradise of mathematicians, why should not another see it as a joke?') Remarks on the Foundations of Mathematics, V. 7., p. 264. See Hilbert 1926, in Putnam & Benacerraf p.191).
[N6] Thus, Peter Suber argues that we can only conceive of infinity by 'unlearning' the intuitions we have cultivated in our experience of the finite.
The truth on this question [of infinity] will violate at least one of our intuitions. For my purposes here, this lesson is at least as important as the mathematical details of the correct answer, for it implies that we should not trust our intuitions in this domain, nor should we expect to confirm mathematical results about infinity with our intuitions ('Infinite Reflections').
The difficulty in unlearning these intuitions is clearly demonstrated by those eccentric and willful attempts to refute the diagonal argument, submitted to scholarly journals from time to time. Wilfrid Hodges dedicated one of his essays to the two-dozen-odd people whose (unpublishable) refutations of Cantor's diagonal argument came to him as referee or as editor in the last twenty years or so. (Hodges 1998).
[N7] This does not include the assumption that exists the set P(N) of all subsets of the set N of natural numbers exists. What I am calling 'Cantor's argument' is the argument that the set of natural numbers exists, and cannot be correlated 1-1 with the set of all its subsets (if such a set exists). Note also, that what I am calling Cantor's argument should not be confused with Cantor's Theorem, which states that no set M can be put into 1-1 correspondence with the set P(M) of all its subsets. 'Cantor's Theorem' was first so-called by Zermelo (1908). This uses a slightly different method of proof than the 1891 paper.
[N7a] Bolzano seems to have been the first to use this word (meaning a crowd, rabble, or disorganised mass) in preference to the word 'class', the nineteenth century term of art. 'Menge' connotes an arbitrary collection, as with a crowd, for example dorfmenge (village crowd) or 'the man in the crowd' (Der Mann in der Menge mit den vielfarbigen Spiegeln auf seinen Nagelschuhe). The semantic focus is on the crowd as made up of individual people, a sense in which it had a negative connotation, as of things that are turbulent, confused or chaotic, incapable of self-control, a rabble, the opposite of what is signified by leader or ruler. The crowd is opposed to the individual, who is little (wenig), yet somehow constituted of them: something halfway between an army and a mob (Bosanquet). Leibniz says that a mob (correspondence with Arnauld, p. 81) has no unity at all.
By contrast, the word 'class' (from the Latin classis meaning military rank, hence a division of the people) connotes a division made according to some rule or method. Hence Boole, Russell and even Quine (in the 1950's) calling set theory 'class theory', and hence the survival of mathematical terms such as 'equivalence class'.
Note that, in Latin, 'class' had no philosophical, metaphysical or logical connotation. It acquired this meaning after being imported from the Latin in the 17C. The corresponding Latin words would have been a genus, meaning a kind of thing, or species, meaning part of a genus, or universal, many things together "turned into" one thing. Some medieval philosophers believed that, besides individual men such as Socrates, Plato, Aristotle, there is another sort of object, signified by the name 'man', a substance common to many things, but different from any of them, and connected with them in virtue of their all being men. It was called in a universal, or genus or species. This idea fell into disrepute in the 17C, and the word 'class' may have been introduced to avoid these unwelcome philosophical connotations – see Locke, Essay, Book III.
It is ironic that Peano, who decided to write his mathematical papers in Latin, used the word 'class', in a sense that was not really Latin at all.
[N7b] 'Beiträge zur Begründung der transfiniten Mengenlehre', 1895 Mathematische Annalen 46, GG p. 112
[N8] Indeed, the epsilon sign e was originally introduced (by Peano in 1888) to denote this relation. In early set theory 'Socrates e man' means 'Socrates is a man'. According to this view ordinary sentences of subject-predicate form, like 'Socrates is a man' assert a relation signified, as it were, by 'is a' between the subject and the 'class' signified by the word man. Set theories of this line lead to paradox, as I shall discuss later.
[N9] Leibniz (everyman) p. 79
[N10] Mill says that 'Peter and Paul preached at Jerusalem and Galilee' are equivalent to the four sentences 'Peter preached at Jerusalem', 'Paul preached at Jerusalem', 'Peter preached at Galilee ' and 'Paul preached at Galilee'. This makes it obvious that 'X preached at Jerusalem and Galilee' does not assert some relation between X and some Jerusalem-and-Galilee object, but is elliptical for 'X preached at Jerusalem and [preached at] Galilee'. Bolzano 1851, §3, says that the simplest way of referring to an aggregate is by combining singular terms with the conjunction 'and', as in 'the sun, the moon and the earth'. Russell also mentions it (P of M § 71) in connection with his distinction between a class as many – an enumeration of individuals, and a class as one. Unlike Mill, he distinguishes between distributive and non-distributive use of plural predication.
When we say 'A and B are yellow,' we can replace the proposition by 'A is yellow' and 'B is yellow'; but this cannot be done for 'A and B are two'; on the contrary, A is one and B is one (§ 71).
[N11] A predicate may also be 'non-distributive', as in 'Peter and Paul were two apostles'. Here, the proposition expresses a relation (an irreducible one) between singulars. See McKay for a detailed account of distributive and non-distributive predication. Whereas a distributive predicate applies to each of a number things of individually, a non-distributive predicate applies to all of them together, or collectively. A standard test used in grammar is whether a sentence in the plural can be converted to a sentence in the singular by adding the quantifier 'each' to the subject. The intended meaning of 'London commuters travel thousands of miles a year' is that each London commuter travels thousands of miles a year, so 'travel thousands of miles a year' is distributive. By contrast, in the sense in which it is intended, the sentence 'London commuters travel millions of miles a year' does not entail the corresponding singular sentence with 'each' (since it is saying that the commuters as a whole travel millions of miles a year).
[N12] Oystein Linnebo (2005) mentions an argument that the two sentences are not equivalent. Though the semantic value of a plural expression like 'Peter and Paul' is not an object in the Fregean sense (if with Frege we mean by 'object' the referent of a singular first-order term) nonetheless it is something. A plural expression like 'Peter and Paul' refers to a plural entity, and this plural entity (his words) is not an object. A plurally referring noun phrase thus represents 'an ontological commitment over and above the individual objects referred to [such as Peter and Paul]'. However, the argument clearly depends on referring to the semantic value in a singular way. To say that 'Peter and Paul' means Peter and Paul does not obviously imply that 'Peter and Paul' means any thing. One must use a singular first-order term like 'this plural entity'. But then a plural entity is an object in the Fregean sense, and we still have the difficulty of this one object (the meaning of 'Peter and Paul') being identical with many objects (Peter and Paul).
Of course, we say that 'Peter and Paul' has "a" meaning, and we talk of the meaning (not "the meanings") of a complex expression like a sentence. We say 'this semantic value'. A semantic value is an object, in the Fregean sense. But it does not follow that a plural sentence is "ontologically committed" to such values. 'Peter and Paul exist' is not the same as 'the meaning of "Peter and Paul" exists'.
According to McKay, Link (1998) endorses the view that one thing can be many things. Dionysius says that 'there is no kind of multitude that is not in a way one.', though he adds 'But what are many in their parts, are one in their whole'.
[N13] orphan
[N14] Ockham (Summa Logicae, I. 45, cf. 1.14, Aquinas Summa Theologia I. Q11. 2, Plato, The Sophist Pt IV).
[N15] Leibniz: letter to Arnauld, Gottingen April 1687. Hume: Treatise I. II. ii 'Of the infinite divisibility of Space and Time' (see also Frasca-Spada Space and the Self in Hume’s Treatise). Dedekind 1888 §1, Berkeley 'An Essay towards a New Theory of Vision' § 109.
[N16] My emphasis. This psychologism is surprisingly widespread among mathematicians and mathematical logicians. Suber describes an actual or completed infinity as the bundling together of objects to form a set of some definite cardinality. Simpson says that 'an actual infinity is something like an infinite set regarded as a completed totality'. Hilbert says that we have an actual infinity when 'we consider the totality of the numbers 1, 2, 3, 4, ... itself as a completed entity, or when we regard the points of a line segment as a totality of objects that is actually given and complete' ('On the Infinite'). On this view, it is in some sense essential to infinite sets that we think of them in the right way. The obvious objection is that an appeal to psychology is hardly a satisfactory proof of existence. If a set cannot exist as a completed unity unless considered or regarded or thought of as so, in what sense do infinite sets really exist at all? Even the psychology is not clear. What is meant by the phrases 'consider the totality ... itself as a completed entity', or 'an infinite set regarded as a completed totality'? They read almost like arbitrary re-arrangements of the words 'complete', 'totality', 'entity', and so on. It is unclear in what sense the objects that form a set are "comprehended together", or form a collective whole. How does regarding or contemplating a number of objects unite them into a set in question? And it is odd, as Frege famously remarked, that the most exact of sciences should seek support from psychology, a science that, at the time he was writing, as well as today, is feeling its way none too surely (Grundlagen § 27, cf. § 27, Grundgesetze, Geach and Black p.129).
[N17] Bolzano argues that events at the poles of the earth (then unexplored) take place even though even though no humans or other thinking beings are present to observe them. But then the propositions ('absolute propositions and truths') which record these events would have to mention 'wholes and sets', since 'every physical body is a whole, and produces very many of its effects only through the multitude of parts of which it is composed'. If these sets are not objectively present, how could such propositions be true? What content could they have, if the presence of an observer were a necessary condition of their truth?
Another objection to this idea is the difficulty of explaining the 'as' in 'we refer to the pair as one thing. If we are referring to the pair "as" one thing, are we actually referring to one thing? Is it the pair we are referring to? But that is two things.
[N17a] As Laycock (2006, 2.1) notes,
[typically] use of the expressions ‘set’ and especially ‘class’ seems to have no ontological import, and to constitute no more than a (potentially misleading) facon de parler, serving in fact the purely syntactic function of transforming a straightforwardly plural expression or sentence into a collective but singular expression or sentence–‘The integers include prime numbers’, for instance, to ‘The class of integers includes prime numbers’, or again, ‘All men are mortal’ to ‘The class of men is included in the class of mortals’.
[N18] It is remarkable that collective nouns occur so frequently in the philosophical literature on set theory, without having attracted attention from philosophers or mathematicians. Cantor talks about 'a unification of entities into a whole' (eine Vereinigung der Einheiten zu einem Ganzen), 'a totality of definite elements' (jeden Inbegriff bestimmter Elemente), eine endliche Anzahl der gesetzten Einheiten (a finite number of posited units). and so on. Bolzano says ' a well-defined aggregate of objects', and introduced Menge ('crowd') as the standard term for a set. Other words are Vielheit ('manyness' or multiplicity); Allheit ('allness' or totality); Inbegriff (aggregate); Mannigfaltigkeit (manifold).
[N19] For example, the plural predicate 'X carried a coffin' is variably multigrade: it may combine with a plural referring expression referring to four people, or six, or eight (Inwagen). This view is sometimes known as plural quantification.
It is commonly associated with George Boolos, though it is older, and is related to the view of classes defended by Mill and other nominalist philosophers. Mill argued that a universal or 'class' is not a peculiar kind of thing, having an objective existence distinct from the individual objects that fall under them, but 'is neither more nor less than the individual things in the class'. (Mill 1904, II. ii. 2, see also I. iv. 3). Interest revived in plurals with work in linguistics in the 1970's by Scha, Link, Landman, Schwarzschild and Lasersohn and others, who developed ideas for a semantics of plurals.
Boolos' work in logic is seminal, however, and was the first to connect ideas from semantics and linguistics with mathematical logic. He argued that we need not construe second-order quantifiers as ranging over anything other than the objects over which first-order quantifiers range. 'Ontological commitment is carried by our first-order quantifiers; a second-order quantifier needn't be taken to be a kind of first-order quantifier in disguise, having items of a special kind, collections, in its range. It is not as though there were two sorts of things in the world, individuals, and collections of them, which our first- and second-order variables, respectively, range over and which our singular and plural forms, respectively, denote. There are, rather, two (at least) different ways of referring to the same things'.
[N20] Slater 'Grammar and Sets' 20XXX. That it is a mirage is especially clear in the case of collective nouns such as 'a couple', 'a dozen', 'a hundred' and the like. The noun 'couple' in the expression 'a couple of shoes' does not refer to some third entity different from either of the two shoes. It simply expresses the fact that there are two shoes, a fact that can be expressed in first order logic using just two individual variables:
(Ex, y) shoe(x) and shoe(y) and x<>y
Even if we call the two a 'a pair', this signifies only that the two shoes match each other in some way. Being a pair of shoes is just a "manner of existence" of the two shoes.
[N21] Russell says (Principles of Mathematics § 71 ff) that 'A and B is what is denoted by the concept of a class of which A and B are the only members.'. This is obviously wrong: because 'A and B' does not denote A or B, it does not follow that it denotes something else, rather than, as is obvious, A and B. Michael Resnik says that plural expressions 'refer to collections quite explicitly. How else are we to understand the phrase "one of them" other than as referring to some collection and as saying that the referent of "one" belongs to it?' (1988, p. 77). Certainly, a plural expression may well refer to a single collection of things. But Resnik implies that it refers to one thing, rather than one collection of things. It is a feature of language that is at the same time elementary, yet interesting enough to mislead the cleverest people.
To be sure, the word 'thing' has a general and vague sense, in which it may substitute for practically any linguistic item whatsoever. For instance 'there is something I forgot to pack, namely that pair of shoes'. But this is mere bad grammar, of which ordinary language is famously tolerant. In any case, it is not being used as a counting sortal. If counting is sortal-relative, as I have argued, and if 'thing' is a counting sortal, as it seems to be, and if a pair of things is really a thing, we have the problem that many things can be one thing. Counting pairs of things is not the same as counting things.
Note also that the grammatical distinction between singular and plural in English and Romance languages is a matter of inflection only, and can be philosophically misleading. The sentence 'that couple is Alice and Bob' expresses an identity, and we can reverse the terms without loss of truth value, viz. 'Alice and Bob are that couple". But note the transition from 'is' to 'are', even though (if the sentence is true) the logical subject does not alter. This is surely a grammatical idiosyncrasy. In languages that have no plural inflection, such as Cantonese or Japanese, the difficulty would not exist. There are numberless examples of how it can be misleading. The passage above quoted from Russell (Principles of Mathematics § 74, passim) is an example. Jonathan Lowe argues that
… 'the planets', refers plurally to Mercury, Venus, Mars, Earth, Jupiter, Saturn, Uranus, Neptune and Pluto, and the predicate attributes the property of being nine to that plurality. We should not be misled here by the fact that the expression 'that plurality' is grammatically singular in form. This is a mere idiosyncrasy of idiom and does not signify that there is some further thing, 'the plurality of the planets', in addition to the planets themselves.
Certainly, the plurality of the planets is not a thing in addition to the nine planets. However, the grammatically singular form of 'that plurality' is not mere idiosyncrasy. The singular 'a' in 'a pair of shoes' not idiosyncratic, since it means "one" pair of shoes, as I have argued. The idiosyncrasy is in the way European languages have a redundant plural form in the verb, as well as the noun.
[N22] Principles of Mathematics § 74.
[N23] Set theorists are in effect trying to define the relation between what is signified by the singular noun phrase, and what is signified by the embedded plural, in terms of an expression that presumes that relation. Cantor defines a set as a collection or gathering together of the elements, as though the relation were the gathering itself. But of course the phrase 'a gathering together of elements' is itself a collective noun phrase of the form 'an X of Y's', and so presumes the existence of the very relation that was to be explained.
[N24] Zermelo's paper was notable for containing the first mention of "Cantor's theorem" explicitly and by name:
Cantor's Theorem: If M is an arbitrary set, then always M < P(M) [the power set of M]. Every set is of lower cardinality than the set of its subsets.
which he proves using the argument given in the main text. The argument is closely related to, and is a generalised form of, the diagonal argument. Note the close resemblance of this proof, to the way he disposed of Russell's Paradox. If there is such an R, then if R is in R, it satisfies the predicate 'x is not in x', and so is not in R. but if it is not in R, it does not satisfy the predicate, and so is in R, which contradiction. It is easily seen that Russell's set R is simply Cantor's set {x: x not in f(x)} with f set as the identity relation x = f(x).
[N25] This would now be written f(x), but I have reproduced Zermelo's notation. The dashes '--' are schematic for a formula containing x.
[N26] Cantor is trying to justify a principle, which it is natural to justify by employing, by taking any property whatsoever, and showing that we can refer to things that have this property. Mill makes a similar assumption in A System of Logic (I. vii. 1) when he says that as soon as we use a general term to signify attributes, then 'the things which happen to possess those attributes are constituted ipso facto a class'. But this is circular, since it invokes the very principle that is to be explained. We cannot prove the existence of any set of things just by constructing an expression that seems to refer to them, and using it. The expression 'the things which happen to possess those attributes' cannot be assumed to have a reference, without using the principle in question.
[N27] My emphasis. Shoenfield says that a set is formed by gathering together certain objects to form a single object (Shoenfield 1967 p.238). According to Hao Wang, a set is a single object formed by collecting the members together (Wang 1974, in Benacerraf & Putnam, p. 530). Suppes says that the axiom of separation permits us 'to separate off the elements of a given set which satisfy some property and form the set consisting of just those elements' Suppes p. 2. (my emphasis).
The idea that sets are formed is also essential to mereology (from the Greek mereos, 'part'): the theory that "collections" of objects are formed by objects combining in a particular way, analogous to how water comes into existence by combining hydrogen and oxygen. A collection is not its elements, any more than water is simply hydrogen and oxygen. It is a singular thing, because it is a sum or fusion of its elements, a thing brought into existence by the right kind of combination: just as water is a combination of two gases, but not a mixture of gases, just as a person is, in a way that we still do not quite understand, a fusion of bodily parts arranged in the right way, just as an aeroplane, as the old air force joke goes, is more than '100,000 small parts flying in close formation'.
Most versions of mereology misrepresent the logic of plurals and collective noun phrases. Even to express the idea of a compound entity or mereological sum, we need to say something like 'those two people form a mereological sum', which contains use the referring phrase 'those two people'. This cannot refer to the sum, otherwise we could not say that those people do not form a mereological sum. The plural cannot refer to the compound entity, any more than 'those two gases' – referring to hydrogen and oxygen – can refer to water. Furthermore, according to mereologists, the parts must compose an ordinary object of some kind, not any summand composed of 'queer or scattered objects', such as, in Varzi's words
… you and I, my cat and your umbrella, or Chisholm's left foot and the top of the Empire State Building -- not to mention categorially distinct summands such as Chisholm's left foot and Sebastian's stroll, your life and my favorite Chinese restaurant, or the color red and the number 2. Such “sums” fail to display any degree of integrity whatsoever and there appears to be no grounds for treating them as unified wholes (Varzi).
By contrast, we can multiply refer to categorially distinct objects, as Varzi inadvertently shows. He says (my emphasis) that 'such sums fail to display any degree of integrity whatsoever and there appears to be no grounds for treating them as unified wholes.' He refers to them as sums, as them, or as 'scattered or otherwise ill-assorted summands'. We can quantify over any such ill-assorted objects - as any such objects - and we can refer to them as those ill-assorted objects. We can even use collective noun phrases to pick out such objects. For an illuminating discussion on the subject, which does not fall into these traps, see Megan Wallace's paper 'On Composition as Identity'.
[N28] Cantor and Dedekind both avoided the null set. As late as 1930, Zermelo chose in an important paper to axiomatise set theory without the null set (Zermelo 1930).
Using a logic of plurals, the empty set is not necessary at all. We assume e.g. that there exist some A's, call these 'the A's'. If it turns out that no x is an A, then, for no X, is it the case that X=the A's, and so no individual x can be one of the A's. This is like what happens with 'the empty set', except we end up with reference failure of a plural term, rather than a reference for a singular term {x: x is A} that fails to "contain" anything.
For example, the standard proof that in any well-ordered integral domain, there are no elements between 0 and 1, is to suppose that the set S of all such elements between 0 and 1 is non-empty. If S has any elements at all, it must have a smallest element x, but then x2 < x, and so x is not the smallest element in S. Thus S has no elements. But this proof can be restated without any reference to sets, by showing that the supposition that there are integers between 0 and 1, and thus that there is a smallest integer between 0 and 1, leads to a contradiction. Any reference to an "empty set" is otiose.
Gowers mentions the question: if there is no empty set, what is the intersection of the sets {1,2} and {3,4}? But this can be rephrased as the question: What numbers are included both among the numbers 1 and 2, and among the numbers 3 and 4? Answer: none. The plural referring expression 'the numbers included both among the numbers 1 and 2, and among the numbers 3 and 4' does not have a reference.
John Mayberry has argued that 'without empty sets and singletons, we should have to make special provision for the cases in which a predicate F satisfied by no objects', in order that {x e S: F(x)} defines a set (Mayberry 2000, p. 74). Thus he define two special elements a b contained in every set that 'do not count as objects' for the purposes of his theory, so that {a, b} is the empty set. This is entirely unnecessary. In such a case, we need only suppose the set-symbol behaves like the English 'the F's', i.e. perfectly well-formed, but has no reference, so that no object that is one of 'them'.
[N29] See also Kanamori 2003, for the history of the null set idea. The original versions of set theory in the Bolzano-Dedekind-Cantor line contained contradictions as a result of permitting the empty set. As Frege pointed out, if a set is a mere aggregate, it consists of objects, and 'must vanish when these objects vanish. If we burn down all the trees of a wood, we thereby burn down the wood'. (Cf Frege, 1893 Geach and Black p.130).This was one of the difficulties in Peano's class theory. Russell (Russell 1903, § 69) points out that if a class is identical with its elements, there can be no such thing as the null-class, though there are null class-concepts, and that a class having only one member is to be identified 'contrary to Peano's usage' with that one term. We cannot define such a thing into existence, for it is not possible that it exists at all.
[N30] Uli Sauerland (1998 p.3) argues that the referent of a plural noun phrase is a 'plural entity', formed by the mereological union operation '+'. 'John and Mary' refers to the plural entity John+Mary. But this contrast between "singular entities" and "plural entities" is bogus. A pair of entities is not a single entity that somehow has the property of being plural, it is just two things.
[N32] Boolos – on the standard proof why it can't be eliminated xxx.
[N33] We characterise a first-order language by its adherence to distinct syntactic categories. One category is the formal equivalent of proper names: symbols standing in for individual objects. The other category are predicates: fragments of language analogous to sentences from which one, two or more proper names have been removed, and which can be used to symbolise one, two and n-place relations. A first order language keeps these categories distinct by allowing quantification, as it were, only over the individual objects. It is tempting to say that standard first order logic in some way prohibits us from referring to more than object, but that is not quite right, for it has no way of stating this prohibition. It can only express the idea of plurality of any individual objects such as x and y at all, by stating that x and y are different objects, i.e. by saying x is not the same thing as y, ~x=y. It cannot express the fact that x is or 'are' in fact more than one object. The idea of a grammatical plural lies beyond the limits of any simple first order language. Such a thought is expressed in ordinary language by means of an identity statement where one term is uncompounded, and the other is a concatenated plural, a series of proper names connected by '&'. This sign, and what it expresses by joining proper names, is not available in any standard first order language.
[N34] See Methods of Logic (2nd edition, 1952) p. 228. Quine says that in order to express the principles of mathematical theory, we have to go beyond first order logic. 'But the price paid for this increased expressive power is ontological: objects of a special and abstract kind, viz. classes [i.e. sets], are now presupposed. Formally it is precisely in allowing quantification over class variables "alpha", "beta" &c, that we assume a range of values for these variables to refer [sic] to. To be assumed as an entity is to be assumed as a value of a variable'. (Methods of Logic p. 228).
Another well-known example that cannot be interpreted in first order logic is the so-called Geach-Kaplan sentence: 'some critics admire only each other'. See Boolos 1894.
[N35] Which is the joke behind the title of Boolos 1984.
[N36] Tom McKay has given the name of singularism to the prejudice that 'every plural predication is at its root a singular predication'. As McKay argues, there are certain kinds of noun phrase that are plural: they refer to many things. That is no reason to suppose they must refer to one thing.
Rouilhan says that 'E X' asserts the existence of 'a multiplicity', and rejects Boolos' assertion that, by itself, a multiplicity 'adds nothing to ontology'. He says that 'multiplicities do not enjoy the same type of being that units do, but they have their own type of being'. But what is their own type of being? He says that just as individuals exist in the sense of the first-order existential quantifiers, 'E x', 'E x', multiplicities exist in the sense of second-order existential quantifiers, 'E X', ' EY', and so on. What a painful confusion. 'E X' does not assert the existence of anything that is not one of the things asserted to exist. If I say 'some people here speak only to one another', I assert the existence of some people. And if those people happen to be identical with Alice and Bob, then those people are the only ones we are talking about. Indeed, the expression 'those people', which seems to refer to a multiplicity of people, in this case a couple, does not refer to anyone apart from Alice and Bob. See de Rouilhan (2000). The problem of "commitment" is one of those problems which are caused, rather than resolved by, the use of formal methods which are no more than a superficial interpretation of everyday language.
[N37] Aristotle Physics 207b8, my emphasis.
[N38] In § 11 of the Grundlagen as follows. A number (Anzahl) of units (Einheiten) that are separately asserted to exist may become a set by being gathered together, literally a being-gathered together (Zusammengefasstwerden) into 'one thing' (einem Ding). Any finite number-symbol v (e.g. '3') signifies both three different individual units, and their "unification" into a whole, a single set (Menge).
the number v is the expression for a finite number [Anzahl] of such positings following one another in a sequence; it is also the expression for the unification [Vereinigung] of the posited units into a whole [zu einem Ganzen].
To any such number (Anzahl) of things we can always add another unit, creating a further number of things. These are united into a new whole number (Zahl) v', the next in the sequence. This continues indefinitely. The process is endless in the sense that there is no last number that can be formed in this way. Adding one to a finite number always yields another number which is finite, there is no 'greatest finite number' (es gibt unter ihnen keine groesste).
Since there are infinitely many finite numbers, these may themselves be unified into a whole, just as for any finite number of things. We can form the idea of all the finite numbers, of a multiplicity formed out of every finite number. This number cannot be finite, since it cannot be realised by the addition of a single thing to a finite multiplicity. If it could we could add something else to it, and it would not be a multiplicity of all the things that could be thus added to.
[N39] Note that the set Z asserted to exist by the Axiom of Infinity, though it contains all finite numbers, does not necessarily contain just finite numbers. Z is an inductive set, a set that contains the initial element zero, and which contains the successor of every one of its elements element it contains. I.e.
inductive (X) iff 0 e X & (x) [ x e X -> {x} e X ]
The Axiom of Infinity postulates the existence of at least one inductive set Z, from which it is trivial to prove the existence of a minimal or "smallest" inductive set w, i.e. the set containing every inductive set in Z. This is guaranteed to exist by the Axiom of Separation.
w = {x in Z: ~ ($ A) [ inductive(A) & ~ x e A] }
It is intuitively the "smallest" inductive set, the set {1, 2, 3, 4, …} containing every finite number, and no other.
Frege's definition is similar (although his version of the comprehension principle was undermined by Russell's paradox). He defines infinity as the number belonging to the concept of member of the natural number series beginning with 0, i.e. the concept finite Number. Every concept has a number attached to it. So the concept '- is a finite number' must have a number, which cannot of course itself be a finite number.
[N40] Grundlagen § 11.
[N41] Letter to Gustac Enestrom, quoted in Dauben p. 125.
[N42] Ewald p. 890 GL.5
[N43] Via Hume, who argues (after the French geometer Malezieu in Book, section II of the Treatise 'Of the infinite divisibility of Space and Time') that existence only belongs to a number of things on account of the individual things of which they are composed. 'Twenty men may be said to exist; but it is only because one, two, three, four, &c. are existent' (my emphasis). Malezieu's principle is essentially the idea I have invoked here, reading 'exist' as 'exist in the domain'. Thus, if a plural value lies outside the domain, at least one of its elements lies outside the domain also. Thus no plurality is divisible ad infinitum, or it would not exist at all, lacking the true existence given to it by the indivisible unities of which it is composed.
To the objection that anything may be considered to be a unity, Hume replies that such a unity is a 'fictitious denomination', applied by the mind. Though he attributes the argument to Malezieu, the same distinction between real and fictitious unities is found in Leibniz (for example, in the letter to Arnauld above). All entities by aggregation are really "fictitious", so that they are not unities at all, but really many things. We must distinguish unum per se (that which is one of itself) from unum per accidens (that which is one per accidens). The idea that there is a "fictitious" unity is confused, as I have said, but it is essentially correct. "An" aggregate is not one thing, but many. Leibniz argues that the existence of an aggregate presupposes the existence of things that have a true unity 'for [the aggregate] only takes its reality from the reality of those of which it is composed, so that it will not have any at all, if each entity of which it is composed is itself an entity by aggregation '. (Morris p. 78). 'The plural presupposes the singular, and where there is not a thing still less can there be several things'.
See also Spinoza, Ethics I Prop. VIII. Leibniz. Aquinas says that the fundamental conception of unity consists in indivisibility (ratio unius consistit in indivisibilitate) Summa Q11.2. Note Leibniz' point can be rewritten salva veritate as 'Where there are not single entities still less can there be several entities', which fails to express the point so well.
[N44] This was the medieval definition of infinity. Ockham writes, for example, that the parts of the continuum are infinite because there are not so many parts that there are not more of them (partes continui sunt infinitae quia non tot quin plures - Exp. phys. III, 10). I.e. any parts that we take (where 'any' has unrestricited range) are such that there are more parts besides.
[N46] A similar argument is made by Richard Arthur (2004), who locates it in Leibniz (Theodicy 249, G.vi.232.), and who paraphrases Leibniz (in an imaginary dialogue) as follows:
I say that the collection of all substantial unities or the accumulation of an infinite number of substances, is, properly speaking not a whole any more than infinite number itself. And I will prove it to you. For we both agree that there is an actual infinity of true unities, and these can be aggregated, at least in thought. Now suppose the aggregate of these unities is itself a unity; let us call this unity U. Now according to the principle of the genetic formation of sets that you mentioned above, this unity is not included in U itself. Yet if it is the aggregate of all unities, it must be included in itself. Therefore U is not a true unity.
[N47]
… in order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite. ['Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen' (Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar 11 (19), 1-10 (1886), p. 9; cf. Hallett, p. 25.
Thanks to Richard Arthur for this reference.
[N48]
[This is] a mistake engendered by the false opinion that a whole consisting of certain objects a, b, c, d… cannot be constructed in thought unless one first forms separate mental representations [Vorstellungen] of its separate component objects. This is by no means true. I can think of the set, of the aggregate, or if you prefer it, the totality, of the inhabitants of Prague or of Peking without forming a separate representation of each separate inhabitant.
[N49] It could be objected that plural definite descriptions pick out objects by means of descriptive properties, and that reference to them is not rigid in the sense required by the axiom. 'The people who live in Spencer Walk' may have a different reference at different times, if people move in or move out of the street. And this is true, I concede, if the definite descriptions are used in the same sense at different times. Otherwise the same argument would prove that 'John Smith' is not a rigid designator. As Mark Sainsbury has argued, definite descriptions appear to be used rigidly when they occur in the subject position of a sentence. Contrast 'Napoleon was the greatest French soldier, but might not have been the greatest French soldier ', where the predicate 'the greatest French soldier' is non-rigid, with 'the greatest French soldier (referring to Napoleon) was Corsican but might not have been Corsican', where 'the greatest French soldier' is clearly rigid. Kripke argues that the "natural" reading of
Someone other than the US President in 1970 might have been the US President in 1970. (Kripke 1972/1980: 48)
requires the first occurrence of the definite description to be rigid and the second non-rigid. This seems also to be true of plural definite descriptions. For example
The people who live in Spencer Walk might not have been the people who live in Spencer Walk
where 'the people who live in Spencer Walk' is rigid in the first (subject) occurrence, but non-rigid in the second. That is to say, our intuitions that sets are completely determined by their elements is the result of intuitions developed about rigidly referring plural terms. But the logic of such terms does not support Cantor's argument for, as I have argued, for they presuppose that a set of elements is identical with those elements, and thus can be neither empty nor infinite.
[N50] A intersect (B union C) = (A intersect B) union (A intersect C). Read 'A union B' as 'those things which are A's OR B's' and 'A intersect B' as 'those A's which are B's'. Thus we prove distribution as follows.
1. Suppose there is an x which is one of the A's which are B's, or one of the A's which are C's.
2. Then in each case x is an A.
3. In the one case it is a C, in the other case it is a B
4. Thus x is one of those A's which are among the B's or C's
5. Conversely, suppose x is not one of the B's which are A, NOR one of the B's which are C.
6. If x is an A, it is not B or C.
7. Thus it is not one of those A's which are B's or C's
[N51] Russell is usually attributed with the discovery of the paradox (hence 'Russell's Paradox' – see Principles § 101. But it is reasonably well known that Zermelo independently discovered the paradox some years earlier. Some argue that Cantor was aware of it. Tait cites the passage already mentioned above as Cantor's version of the Separation Axiom.
I call a manifold (an aggregate [Inbegriff], a set) of elements which belong to any conceptual sphere, well-defined, if on the basis of its definition and in consequence of the logical principle of excluded middle, it must be recognised that it is internally determined whether an arbitrary object of this conceptual sphere belongs to the manifold or not, and also, whether two objects in the set, in spite of formal differences in the manner in which they are given, are equal or not. ('Uber unendliche, lineare Punktmannigfaltigkeiten', 2, Mathematische Annalen 20, 1882, pp 113-121. Quoted in Tait 'Cantor's Grundlagen and the paradoxes of Set Theory')
However, this depends on reading 'conceptual sphere' as a set of elements. If read as meaning the domain or range of a variable, however, as the quotation below suggests, the paradoxes of naïve set theory follow. E.g. let the range of x be 'any set'. The description 'x is not a member of x' yields a well-defined set, in Cantor's sense.
Certainly, Frege did not spot it. Russell wrote to Frege about the paradox in June 1902, just as Frege was preparing the second volume of his Grundgesetze der Arithmetik. 'I conclude that under certain circumstances a definable collection [Menge] does not form a totality.' Frege hastily prepared an appendix in response to the paradox, but this later proved unsatisfactory. It is generally thought that this led Frege completely to abandon his life's work on the logic of classes. 'Not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic, seem to vanish'.
[N52] Boolos p.18.
[N53] Von Neumann, J. (1925). 'Eine Axiomatisierung der Mengenlehre'. J. für Math. 154:219--240. translated as 'An Axiomatization of Set Theory', in van Heijenoort.
[N55] 'Russell's mathematical logic', Benacerraf / Putnam pp 447-69
[N56] Reference for 'On Denoting'.
[N57] Russell, Principles of Mathematics § 69
[N58] System of Logic, System p. 61, I. V. Frege says that the class defined by 'inhabitant of Germany' has a different 'range' (Werthvelauf) at different times.]
[N59] Cantor to Hilbert 2 October 1897
[N60]
One believes and it appears evident, that the existence of individuals must imply the existence of sets of them; but Burali-Forti and Russell have shown by different examples that a set of individuals need not exist, even though the individuals exist. As we cannot accept this new fact, we are obliged to conclude that the proposition which appears evident to us [i.e. that the existence of individuals must imply the existence of sets of them] and which we believe to be always true is inexact, or rather that it is only true under certain conditions. Mirimanoff D. 1917a: 38) 'Les Antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles at les antinomies cantoriennes', L'Enseignement mathématique, 1st ser. vol. 19
A.W. Moore says that we have a pre-theoretical intuition 'that, if we are able to make generalizations about things of a certain kind, then those things can be grouped together into determinate totality'.
[N61] The oddity of the classical idea of a set is that its definition already contains, and presupposes a term referring to all the objects that are to be members of the set. Each contains a grammatically plural term referring to a number of things. Cantor talks about 'objects of our intuition', Dedekind of the 'various things a, b, c, …', Wang and Shoenfield of 'the objects which belong' to a set. There is already a way of thinking, conveyed by grammatical plurals, of more than one thing, that does not require the idea of a set of them, and which is presumed in the very definition of a set. Each goes on to postulate the existence of an object referred to by a grammatically singular term such as 'a whole of determined well-distinguished objects' (Cantor), 'a system S' (Dedekind), 'the set' (Wang and Shoenfield).
It is strange, if the purpose of set theory is to allow talk about more than one object, that it presupposes, and entirely fails, to explain the use of plurals, which are the natural way of doing this, in order to postulate the existence of something we cannot refer to as a plural, and which clearly deficient to talk about more than one thing.
[N62] Understandably, Dedekind replied that he could make no sense of it all:
A visit from you will always be welcome to me and my sister, but I am not at all ripe for a discussion of your communication: it would for the time being be quite fruitless! You will certainly sympathise with me if I frankly confess that, although I have read through your letter of 3 August many times [my emphasis], I am utterly unclear [my emphasis] about your distinction of totalities [inbegriffe] into consistent and inconsistent; I do not know what you mean by the 'co-existence of all elements of a multiplicity', and what you mean by its opposite [my emphasis]. I do not doubt that with a more thorough study of your letter a light will go on for me; for I have great trust in your deep and perceptive research. But until now … I have not had the time or the necessary mental energy to immerse myself in these things.
See also Frege's Grundlagen (§ 28), where he argues that the expressions 'multitude' (Vielheit), 'set' (Menge), 'plurality' (Mehrheit), due to their vagueness, are unsuitable for use in defining number (Zahl). They all carry the connotation of being stuck together, or of being together in the same physical space (raumliches Zusammensein). He observes that a man may have a horse in America and a horse in Germany, and if they are not brought together, they will not be a physical phenomenon in the sense of being physically together. Yet they will still be two horses.
[N49] Russell 1907, p. 29.