The Formalisation of Set Theory

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The Formalisation of Set TheoryAuthor(s): John TuckerSource: Mind, New Series, Vol. 72, No. 288 (Oct., 1963), pp. 500-518Published by: Oxford University Press on behalf of the Mind AssociationStable URL: http://www.jstor.org/stable/2251865 .

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III.-THE FORMALISATION OF SET THEORY

BY JOHN TUCKER

IN this paper I attack formalism and put forward the view that the foundations of mathematics are its indispensable informal concepts. By formalism I mean, not the project which became obsolete in 1931, but the current view that in logic and mathematics it is essential to make a fresh start with a formalised language; the view that nothing less than complete formalisation will do. Against this I argue that there are many features of informal language which cannot be thrown away, that in the development of formalised concepts the most that can be done is to extend some informal concepts but not all.

I deal with the concepts of set theory because each known branch of mathematics can be derived from set theory. It is essential to have a correct account of these concepts since an erron- eous one will be applied to the whole of mathematics. I show that the paradoxes of set theory can be explained only by an examination of the rules of informal language. For these paradoxes arise when a rule of informal language is broken. They can be legitimately avoided only when they are understood and they can be understood only when the rule in question has been identified. But there are two obstacles to this simple view. One is the wholesale rejection of informal language by formalists. The other is inattention to the rules of informal language by those who are not formalists. Both of these obstacles can be removed.

The distinctions between natural and artificial language, and between ordinary and technical language do not affect the arguments which follow. The only contrast which is relevant is that drawn by formalists themselves between formal and informal language. For while the first two of these contrasts mark a difference in degree, the contrast between formal and informal is intended to mark a difference in kind. For according to formalists the sentences of an informal language hang together in such a way that the language can only be interpreted as a whole, while the sentences of an informal language are separately meaningful. The fact that the sentences of the informal language may be a mixture of ordinary and technical, natural and artificial words or signs has no bearing on this distinction. Nor has the fact that the rules for the use of the ordinary words may be implicit, while those of the technical terms are explicit. The Russell paradox, for example, is just as much a paradox when it is obtained in English by deviating from one of the so-called ' customs ' of

500



THE FORMALISATION OF SET THEORY 501

that language as when it is obtained within a system of technical terms.

Formalists stonewall any explanation of the paradoxes of set theory which is offered by appealing to the distinction between formal and informal. The formalist objection to any informal treatment of the paradoxes is that it applies to them only as they appear in informal language and not to their counterparts in formal systems. Against this I argue that to understand the paradoxes as they occur in informal language is necessarily to understand them as they appear in formalsystems. Foraparadox, in the sense used in set theory, is the appearance of either one or two contradictions and a contradiction can appear only by means of a negation. But since, as will be shown, all formalised negation derives from and remains dependent on informal negation there is no such thing as a purely formal paradox.

If informal language is rejected, so are treatments of the cause and cure of contradictions given in terms of the rules of informal language. It is essential, therefore, to prove first that informal language cannot be totally rejected; to show that formalism is untenable. For if formalism is untenable, the distinction between formal and informal paradoxes cannot be maintained.

In putting forward the view that the foundations of mathematics are its indispensable informal concepts I reject: (i) the view that the foundations of mathematics are or should be a formal system from which the rest of mathematics can be formally derived. This is not to say that no such unification is possible, but only that it would not be in any sense a foundation. For there is no one privileged way of effecting this unification and no compulsory point of departure. I reject: (ii) the view that the foundation of mathematics is mathematical logic. For mathematical logic is more mathematics. It may, for example, be treated as a branch of recursive number theory. The line between it and the rest of mathematics can only be arbitrarily drawn so as to make the view trivially correct. I reject: (iii) the distinction made by Russell and G6del between the philosophical and the mathematical aspects of foundational problems.' I illustrate the harmful effects of this distinction on treatments of the paradoxes. It encourages the view that it is more impor- tant to know that a formalism works than to know how it works. It discourages any attempt to understand how the paradoxes arise. I replace it by the distinction between clarification and formal development. I reject: (iv) the view that mathematical

1 B. Russell, Logic and Knowledge (ed. Marsh), p. 102. K. G6del in The Philosophy of Bertrand Russell (ed. Schilpp), p. 146.



502 J. TUCKER:

logic is a clarification of mathematical concepts.' For if it is a clarification, how are we to describe the activity of removing the conceptual confusions into which mathematical logicians frequently fall? I indicate the need for such clarification in the case of Skolem's view that the concepts of set theory are relative and in the case of proposals to accept or reject the actual infinite.

1. The refutation offormalism

Formalists take up stronger or weaker positions. Some, like Church and Tarski, demand that informal language be replaced entirely.2 Others, like Quine, think that informal language is necessary to start with but that it is left behind completely once the basic symbols have been defined in terms of it.3 Others, like Curry, admit that informal language is never entirely left behind but think that formalisation must result in greater precision, and further, that this precision can be indefinitely in- creased.4 Each of these positions is untenable. It can be proved that complete formalisation is in principle impossible. It can be proved that the search for greater precision in respect of the indispensable features is senseless. In particular, it can be shown that Church's demand for a consistency which reveals itself structurally is unobtainable.

(i) Indispensable informal concepts There are some informal concepts without which it is im-

possible to operate a formal system and others without which it is impossible to give such systems an interpretation.

(a) It is not possible to operate a formal system without making use of the informal notion of a rule. The proof of this is straightforward. For a formal system is produced by giving a set of rules. This cannot be done without making use of the informal notion of a rule in the first instance; and any subse- quent attempt to formalise the notion of a rule must make use of the informal notion for the following reason. Any new rule for the concept of rule would have to be laid down by means of the old concept of rule and every time the new version was applied, the old version would necessarily be used in applying it. Therefore every use of the new concept would involve the use of

I R. L. Goodstein, Mathematical Logic, p. 1. 2 A. Church, Mathematical Logic, throughout. A. Tarski, Logic, Semantics.

Metamathematics, throughout. 3 W. V. Quine, 'Mr. Strawson on Logical Theory', MIND (1953), p. 445.

4H. B. Curry and R. Feys, Combinatory Logic, p. 25.




the old concept. It might be thought that the new concept of rule could be introduced with the aid of the old concept and that the old concept could then be discarded. But this is impossible, for by discarding the old rule is meant " not making any use of it " and not making any use of it would preclude any use of the new rule. The rules for the use of a formalised concept of 'rule' must be the informal rules with which we start. It follows that ' rule ' if formalised would necessarily involve the use of informal rules. This also shows that conventionalism is untenable. The conventionalist position is that all the rules of language can be changed and the argument provides an example of a rule which cannot in principle be changed.

If the formalist premiss that all the usage which occurs in informal language is unclear is correct, then we are condemned by that premiss to perpetual unclarity. For if the informal notion of a rule is unclear and if any formalisation of the notion of a rule involves the use of that informal notion, then any formalisation of the notion of a rule must be unclear.

(b) Any proposal to formalise the notion of substitution is a proposal to produce a new definition of substitution. But a definition is itself a rule for the substitution of one expression for another. In order to understand what the new definition is, it is necessary to understand that it is a proposal to substitute a new for an old expression. So in order to introduce any new definition of substitution, whether formal or informal, it is necessary to make use of the informal notion of substitution. If there is something unclear about the informal notion, the unclarity is bound to be transmitted to the new version, since in moving from the old to the new concept of substitution, the old concept is used. The faulty concept has to be used in introducing any new version of it. The new version can only be understood with the aid of the faulty concept. So the fault cannot be eradicated by this means. If a set of rules which are the formal equivalent of the informal notion of substitution is provided, it is necessary to understand which of the symbols of the formal system may be substituted for the symbols used in stating the formalisation of substitution. In other words, it is necessary to make use of the informal notion of substitution in order to be able to apply the rules which state the formalised version.

(c) Informal negation is necessary for any formal system. It is needed both in operating the system and in talking about it. It is needed in operating the system, since the operations consist in applying rules and in applying a rule it is essential to know what the ruLle proscribes as well as what it prescribes. It is also



504 J. TUCKER:

needed in specifying the conditions for the consistency of the system. A given formal system might or might not contain a negation sign. If it does contain at least one such sign, then its consistency can be characterised by saying that a formula and its negation should not both be provable within the system. If it contains no negation sign, then Post's requirement that not all formulae of the given system be provable can be applied. The usual notion of consistency applies in those systems which contain negation signs. They are inconsistent if a formula and its negation are both provable. But this requirement can only be satisfied if the negation sign or signs can be recognised and a negation sign cannot be recognised by purely formal means. It cannot, for example, be recognised by a purely formal appeal to truth tables, since in order to understand a truth table it is necessary to understand that the different values may not be possessed by a given formula in the same context, and this is an informal negation. Post's definition of consistency is equally dependent on informal negation. For in order to understand that a formalism is consistent in accordance with Post's requirement, it has to be understood that some formulae cannot be proved. Which again is an informal negation.

Consistency proofs for the propositional and restricted predi- cate calculus consist in showing that all well-formed formulae have one of the two values and not the other. The negation sign of the formal system is defined in terms of these two values. If any formula has both values, then there is an informal contradiction, for the ruling is that a given formula cannot have both values. It follows that these proofs make essential use of informal negation.

(d) The informal notion of a group of things is indispensable and is used unchanged in set theory. In P.M. (I. *20), for example ' class' is defined by means of ' all', the use of which involves the informal notion of a group. ' Relation' is then defined in terms of ' class' (I.*21). But the notion of a class involves the relation of a group to its members. So relations also can only be understood by making use of the informal notion of a group of things. ' E ' is then defined as a relation. But a relation is a class and ' class' involves the notion of class membership. But since ' class' itself and therefore the notion of class membership are introduced by means of the informal notion of a group of things, it follows that this definition of ' E ' also depends on the informal notion. So does the notion fundamental to set theory, of one-one relations, since this involves the use of 'relation' defined as above.




(e) In order to be applied a language, formal or informal, must provide the means for identifying particulars in a common space and time. It follows that the most that can be effected by the formalisation of demonstrative reference is to provide a shorthand way of linking up informal demonstrative reference with a formal system. The informal method of referring is a necessary feature of any language and must therefore be in- corporated in any applicable formalism.

This is not meant to be an exhaustive list of indispensable informal concepts. It suffices to show that complete formalisa- ation is impossible.

(ii) The impossibility of complete formalisation Church argues that unless logic can be completely formalised,

it cannot be precisely defined, where by a precise definition he means one which is completely formalised. In his view, without such a formal definition there can be no such thing as logic.' But this argument is itself informal and can easily be seen to be unformalisable. For it consists in the assertion that only formalised arguments are to be accepted. If this informal premiss is itself questioned, if a justification of it is demanded, then further informal details are given. Support is found for it by an appeal to greater precision, greater rigour and so on. Neither the premiss nor the justification can be formalised. For how could the recommendation that only fully formalised arguments are to be accepted itself be fully formalised? The pure formalism would have to admit of an interpretation which read ' only fully formalised arguments are to be accepted'. Now, of course, we could make up a code which would decipher into these words just as we can make up codes which decipher into sentences such as 'this sentence is unprovable'. The mere arrangement of such words can be codified and deciphered as we please, and in this trivial sense the sentence ' only fully formal arguments are to be accepted' could be formalised. But such a proceeding would be trivial, since it would not of itself give us any reason for accepting or rejecting the action recommended by the interpreted formula. No amount of codification could do this. The grounds for recommending complete formalisation must themselves be informal. However much formalisation has been carried out, there must always remain one informal argument, namely, the argument by means of which it is established that only formal arguments are to be accepted. But if only fully formalised arguments

1 A. Church, 'The Richard Paradox', American Mathematical Monthly, xli (1934), 360.



506 J. TUCKER:

are to be accepted and the argument by means of which this is established is unformalisable, then this argument ought not to be accepted. A thorough-going formalist cannot claim that the informal reasons which he gives in support of his policy are acceptable. Why then should we take any notice of what he says?

Church's own practice is inconsistent with his demand for complete formalisation. His explicit aim in trying to establish 'Church's thesis' was that of producing a formal definition of effective calculability which is equivalent to the corresponding informal notion.1 It is not therefore a formal theorem capable of formal proof or disproof.2 It is a definition which can be judged informally to be more or less in conformity with the informal notion. Since Church's thesis cannot be fully formalised, it follows that on Church's own view Church's thesis is and must remain unsatisfactory.

Church says that his aim is to reduce semantics to syntax, where by syntax is meant the non-referring components of the formalism and by semantics is meant its referring symbols.3 But if this were achieved, there would be no referring symbols left and the formalism would be inapplicable and therefore useless. For in applying a formalism, formalists are not allowed to trade on the informal usage which they have officially discarded. Now, in order to apply a formalism it is necessary to assign powers of reference to some symbols and, as already pointed out, this can be done only by making use of informal demonstrative reference or its trivial equivalent.

(iii) These unformalisable concepts are already as precise as they can be.

Curry takes up a more restricted formalist position.4 He says that the language from which we start cannot be transcended but that this language is inherently vague. He also says that any degree of precision can be obtained by successive approximations.5 But he does not specify the standard to which these approximations are made. What, for example, could be meant by saying that a given formalisation of substitution is of a higher degree of precision? This presupposes some standard by which

1 A. Church, 'An Unsolvable Problem of Elementary Number Theory', American Journal of Mathematics, lviii (1936), p. 346.

2 See L. Kalmar, 'An Argument against the Plausibility of Church's Thesis', p. 72 of Constructivity in Mathematics, ed. A. Heyting.

3 A. Church, Mathematical Logic, pp. 64-65. 4H. B. Curry, 'Outlines of a Formalist Philosophy of Mathematics'. 5 H. B. Curry and R. Feys, op. cit. p. 25.




the informal notion of substitution is known to be imprecise. But how can there be such a standard if the ' imprecise ' concept is required both in introducing and in using the 'precise' one? By the same argument it does not make sense to claim that any indispensable informal concept is vague and capable of greater precision.

(iv) Church's special requirement is unobtainable Church rejects all informal treatment of the paradoxes of set

theory.' It is not enough in his view to offer an explanation of the unsoundness of those arguments which lead to contradictions. What is required is a guarantee that the individual sentences and methods of argument wA-hich are used are sound. He demands a method of recognising sentences and sequences of sentences as not leading to contradiction. In his view, to give an explanation of the contradictions without having a formal guarantee of consistency is 'to abandon the notion of mathematical proof altogether '.2 According to Church a theorem can only be properly said to have been proved if there is a guarantee that the system within which it occurs is consistent. Without this guarantee, he says, the most that can be said of a putative proof is that it would be a proof if the guarantee were available. This guarantee must, of course, be a purely formal one.

One reason why Church's demand cannot be met is that there is no such thing as a purely formal guarantee of the consistency of a purely formal system. The proof given of the untenability of formalism is enough to show this, but there is a more specific objection. We should, according to Church, be able to recognise formal sentences as not leading to contradictions. It would not be enough merely to show that a given sentence is well formed. For it might, he thinks, be well formed in accordance with rules which lead to a contradiction. Church wants more than this. It must be possible to recognise a formal sentence not only as well formed but as never in any context leading to a contradiction. It is never made clear what sort of feature in the sentence we are to look for over and above the property of being well formed. Church assumes that a formal sentence could possess some property which we could inspect and which would tell us that no contradiction could come from that sentence. Now this is at

1 See Church's review of Finsler's ' Gibt es Unentscheidbare Satze ? ' J.S.L. (1946), p. 132.

Also Church's review of F. Moch's ' On peut eviter les antinomies classiques sans restreindre la notion d'ensemble', J.S.L. (1956), p. 322.

2 J.S.L. (1956), ibid.



508 J. TUCKER:

variance with the general formalist view that it is the formal system as a whole, not its parts, which have such properties as consistency. If a system has been proved to be consistent and if a sentence of that system is well formed, then it can be safely concluded that that sentence will not give rise to a contradiction. No appeal is made to any property of the sentences other than to their being well formed. But if Church abandons this tenet of formalism, it is not clear what sort of property he has in mind. He assumes that consistency is the direct outcome of some positive property of a sentence of a formal system. But whereas an inconsistency does appear in the form of a recognisable feature, namely a contradiction, it is fallacious to conclude from this that consistency could also appear in some such recognisable form. For consistency is not a positive feature at all, it is merely the absence of inconsistency. Church naively assumes that all sentences which conserve consistency might have some feature in common. This is like thinking that all things which are not red might have some recognisable common feature. All that consistency-conserving sentences have in common is the absence of inconsistency-generating features. There is no future in the search for the feature which all things which are not red have in common. The same applies to Church's search for recognisable signs of consistency.

(v) The 'formalisation of intuition' Those formalists who think of themselves as having discarded

the concepts of informal language cannot at the same time think that they are engaged in formalising some of those concepts. They are led, therefore, to give a fictitious account of what they are doing. They say that they are striving to formalise their intuitions. But the use to which this appeal to intuition is put is that of providing a spurious explanation and in so doing, preventing the search for genuine ones. G6del, for example, takes the Russell paradox to be the discovery that our intuitions are self-contradictory (op. cit. p. 131). Wang concludes from the LRwenheim-Skolem result that our intuitions of the integers cannot be formalised.1 When a contradiction is found, our intuitions have been shown to be contradictory. When an incompleteness theorem has been proved, our intuitions have been shown to be incompletely formalisable. When opium makes people sleepy, it is because of its virtus dormitiva. It is evident

1 Hao Wang, 'On Formalisation ',MrND, April 1955, pp. 232-233; 'On Denumerable Bases of Formal Systems ', in Mathematical Interpretation of Formal Systems, p. 78.




that this appeal to mathematical intuition, like all appeals to intuition, simply reduplicates in a redundant way what is already known. When, for example, Godel compares mathematical intuition with sensory intuition, he does so by stating that numbers are objects postulated to explain the range of mathematical intuition just as physical objects are postulated to explain sensory intuition (op. cit. p. 137). This has no effect other than that of introducing a categorial confusion between concrete and abstract. It tells us nothing since we can attribute to mathematical intuition only what we already attribute to numbers. Similarly, when Wang compares the difficulty of formalising the intuitions of the integers with the difficulty of describing a specific hue, he characterises both as attempts to formalise intuition.' The comparison involves him in two mistakes. The supposed difficulty of describing a specific hue rests on an elementary mis- understanding of what description is. For to describe is to list qualities and to ask for the list of qualities of a single quality is absurd. Further, the application of the term 'intuition' to both hues and integers could only be justified by abandoning the distinction between concrete and abstract which would also be absurd.

Mathematicians are not striving to formalise their intuitions. They are striving to extend their already extensive technical concepts.

2. Consistency-formal and informal

Next I show (i) that each paradox of set theory does in fact arise from the breaking of a rule of informal language; and (ii) that attempts to deal with them in terms of imnpredicative definition and the diagonal argument are hopelessly confused.

(i) The informal origin of the paradoxes of set theory It follows from the argument of the previous section that there

can be no such things as purely formal systems. It follows also that the conditions which prevail within a formal system are the same as those which prevail in informal language. That, for example, the Russell paradox is the same trouble with the same cause and the same cure whether it occurs in a formal or an informal context. That if a satisfactory account of a paradox is given in terms of informal language the same account applies within a formal system. For a formal system is an extension of informal language to which it remains attached. The formalist

1 MIND, April 1955, op. cit. p. 232.



510 J. TUCKER:

rejection of any such treatment is based on the following premisses, each of which will be shown to be false:

(a) That the rules of informal languages lead to contradictions and that no explanation of these contradictions is possible in terms of informal language.' Tarski, for example, derives his entire programme for formalised semantics from the premiss that paradoxes such as the Richard and the Epimenedes cannot be explained by the means available in the informal languages in which they occur (pp. 401-403).

(b) That in a formal system for which there is no formal guarantee of consistency a contradiction may turn up unex- pectedly anywhere. That there may be latent contradictions spread throughout such a system.

(a) In Tarski's view the contradictions show that informal language is inconsistent. He fails to notice that in each case it is by breaking a rule of informal language that the contradiction is obtained. For example, the rule which is followed in informal language is that ' this is false ' is applied only to statements which could be false. It is applied only if there is some way of establishing that a given statement is false. The liar paradox is obtained by ignoring this requirement and by pretending that the sentence 'this is false' taken by itelf could be found to be either false or true. Again, in informal language 'does not apply to itself' is applied only to those expressions of the language which do not apply to themselves and it can be applied only if there is some way of establishing that a given expression does not apply to itself. The heterological paradox arises when this requirement is ignored, when it is pretended that ' does not apply to itself ' either could or could not apply to itself. Similarly, the rule for applying' expression which does not apply to the number assigned to it' is that there must be some way of establishing that the given expression does not apply to the number assigned to it. The Richard paradox is obtained by breaking this rule. Russell's paradox is obtained by breaking the rule that we say of a class that it is not a member of itself only if there is some way of establishing that it is not a member of itself.

But the aversion of formalists to informal language not only prevents them from recognising these cases as cases of rule- breaking, it also prevents them from seeing that they themselves make illegitimate use of such rule breaking. There is an argument for the indenumerabilty of real numbers which is intended

I A. Tarski, Logic, Semantics, Metamathematics, pp. 164-165.




to show by a reductio ad absurdum that the set of all sets of positive integers is not denumerable.1 Each positive integer is assigned a set of positive integers. Each positive integer either belongs to the set assigned to it, or it does not. Now, all those positive integers which do not belong to the set assigned to them form a set. It is then asked of this set whether the positive integer correlated with it is a member of the set or not, and the familiar pattern of the Heterological and Russell paradoxes follows. It is concluded that there is a set of positive integers which cannot be correlated with any integer. But an argument which proceeds by breaking a rule is invalid, even though the conclusion reached happens to be correct and can be obtained by other valid arguments.2

The Burali-Forti paradox is entirely different in structure from the preceding ones and should never have been classified with them. An obvious difference is that a single contradiction is obtained, not two, as in each of the preceding cases. The Burali-Forti contradiction is a consequence of two theorems. One is to the effect that the series of all ordinals up to and including any given ordinal exceeds the given ordinal by one. It follows from this theorem that there is no greatest ordinal. The other states that the series of all ordinals has an ordinal number. It follows that there is a greatest ordinal, namely, the ordinal number of the series of all ordinals. The two theorems contradict each other.

Russell's proposed solution consists in forbidding the use of 'all ordinals '.3 The second theorem is amended to state that the series of all ordinals of a given type has an ordinal number; and since that ordinal number is of higher type it no longer follows that there is a greatest ordinal. But this is no solution. It is an ad hoc way of avoiding the contradiction, not an explanation of how it arose in the first place. Nothing less than such an explanation will do as a solution to the Burali-Forti. It is not enough to forbid altogether the use of the expression' all ordinals ' merely because one particular use of the expression gives rise to a contradiction. It is essential to show how this use of 'all ordinals' has this effect in this particular case. It is clear from the first theorem that the class of ordinals is a self-generating class. A self-generating class is a class such that whatever

1 Hao Wang, 'Formalisation of Mathematics ', J.S.L., xix (1954), 244-245.

2 The same fallacy is to be found in S. C. Kleene, Introduction to Meta- mathematics, pp. 14-15, and in R. L. Goodstein, op. cit. pp. 96-97.

3Principia Mathemttica, vol. 1, p. 63, vol. 3, p. 73. 34



512 J. TUCKER:

group of members is considered, it follows by the property of self-generation that there is yet another member. If we speak of the totality of such a class we cannot go on generating from that totality. The Burali-Forti is produced by trying to do so. Granted that the series of all ordinals has an ordinal number, then the self-generating process cannot be applied to this ordinal. It is a unique ordinal. It is the only ordinal to which self- generation does not apply. There is a simple reason for this. Self-generation can be applied to it only by the use of 'more than all' which is a clear case of breaking a rule of informal language. This is the sole source of the contradiction.

Russell at one stage concluded from the fact that a contradiction is obtained that there must be something wrong with self-generating classes as such.' But it is not the property of self-generation which gives rise to the contradiction. The contradiction arises from a misuse of ' all ', not from some defect inherent in self-generating classes. It is unsatisfactory, therefore, merely to put a ban on such classes.

The difficulty found by Burali-Forti arises when the condition that the class of all ordinals has an ordinal is brought in. For it then seems to follow that self-generation applies to this ordinal also. It is not difficult to see that the property of self-generation cannot apply to the class of all ordinals. This follows quite simply from the way we use 'all'. When we say 'all' we mean the whole lot, we mean that there are no more to come. So when we speak of the class of all ordinals we really mean all of them. We cannot apply self-generation to this class because to do so would break the rule for this use of ' all'. We cannot speak of 'more than all'. It is clear, then, that the trouble arises not from the property of self-generation but from the breaking of the informal rule for making use of ' all'.

There is a theorem to the effect that the cardinal of the set of all sub-sets of a given set is greater than the cardinal of the given set. So for any given cardinal there is a greater cardinal. The notion of a cardinal is self-generating. By applying this result to the set of all sets Cantor had already obtained a contradiction which is isomorphic with that obtained later by Burali-Forti. It is solved in the same way.

This demolishes the Peano-Ramsey division of the paradoxes into semantic and syntactical, which is unfortunately standard textbook practice. It is clear that it puts together cases which are unlike and separates cases which are alike. The Russell is

1 B. Russell, ' On Some Difficulties in the Theory of Transfinite Numbers', Proc. London Math. Soc. (1905-6), p. 37.




separated from the Heterological and grouped with the Burali- Forti. The ironic feature of this division is that it induced formalists to put aside those paradoxes which in fact provided the clue to an understanding of them all. For it induced them to dismiss the Heterological paradox as merely to do with meaning in informal languages and so not their concern.'

(b) The fear that a latent contradiction might spread throughout a formal system is a common feature of formalism.2 This fear of latent contradictions shows a lack of faith in our capacity to understand the causes of contradictions. It gives rise to wholesale bans, like that on the impredicative, which are indiscriminate and lack all explanatory value. It discourages the search for the rule which has been broken. A contradiction which is understood cannot spread. For it if is understood, then the precise point at which it arises is known, the fault has been located and cannot have unforeseeable effects elsewhere. In any case, the presence of a contradiction in a formal system does not render the entire system meaningless. The ground given by Hilbert and Ackermann for this assertion is that if A and not-A are both provable within a formal system, then any sentence is provable (ibid.). But again, this follows only if a defeatist attitude is taken up. Of course, if A and not-A are provable within a formal system and if there is no way of finding out how and where the contradiction is generated, then any sentence of the system is provable and the system as a whole is useless. But we are never in this hopeless position. For there is no such thing as a contradiction which has no cause, no such thing as a contradiction which cannot be tracked to its source. We do not conclude that a system in which, for example, the Burali- Forti can be derived is useless; we simply find out at precisely which point the contradiction is generated within the system. This is in fact what formalists do, in spite of their precepts. When the Burali-Forti was derived from Quine's first version of Mathematical Logic it was not concluded that the whole of Quine's system would have to be scrapped. On the contrary, Rosser and Wang repaired the fault at the place where it occurred.3 In the same way, we do not discard informal languages wholesale merely because games like the Liar, the Heterological, the Russell, can be played in them by breaking a rule. Nor does

1 The clue was provided by G. Ryle in ' Heterologicality', Analysis, 1951.

2 For example, Hilbert and Ackermann in Mathematical Logic, p. 38; Quine's Methods, p. 252; B. Rosser's Logic for Mathematicians, p. 207.

3B. Rosser, 'The Burali-Forti Paradox', J.S.L. (1942), p. 2.



514 J. TUCKER:

the Liar, for example, have effects which spread throughout the informal language in which it is expressed. The rule is broken at that point, but the rules are followed elsewhere, that is all. The idea that the breaking of a single rule somewhere has un- predictable effects elsewhere is unfounded.

(iii) A clarification of some of the confusion about impredicative definition and the diagonal argument

The diagonal argument, which is central to classical set theory, gave rise to concern with the impredicative and to the Poincare- Russell approach to the problem of the consistency of set theory. A definition of x is impredicative it if makes use of bound vari- ables one of the possible values of which is x itself. Such definitions are thought to be essential to mathematics and yet are, according to Poincare, the sole source of the paradoxes of set theory. This view was taken over by Russell and has become orthodox.' It gave rise to attempts like those of Russell and Weyl to derive mathematics from a fundamental formalism without the use of such definitions. The current view is that impredicative definition is not well understood. According to G6del its legitimacy is not yet established (op. cit. p. 135). Ac- cording to Wang it is a notion of which the human mind is at present incapable of forming a clear and distinct idea.2

But since the paradoxes do not arise from impredicative definitions there are no such difficulties about their legitimacy. If impredicative definitions are needed in mathematics, mathematicians can have as many of them as they like. For no such definition gives rise to a contradiction. The sole causes of the contradictions are those already mentioned.

Russell tries to show that each paradox results from an impredicative definition (vol. I, p. 60 ff.). The use of this blunt in- strument leads him to write off all statements about all propositions as meaningless on the ground that a proposition about all propositions is viciously circular. The circle leads, he says, to the Epimenides. Russell treats this case as if it were the same sort of case as the Burali-Forti. He assumes that in this case also self-generating properties are present. This is clear from his supposition that ' new propositions are created by statements about " all propositions'7 n (p. 37). This is false. For new propositions would only be created by statements about all propositions if the notion of proposition resembled that of ordinal

1 S. C. Kleene, op. cit. p. 42. 2 Hao Wang, ' On Denumerable Bases of Formal Systems ', in Mathe-

matical Interpretation of Formal Systems, p. 78.




in the relevant respect, that is, in being self-generative. But propositions have no such property. A proposition about all propositions simply is one of those propositions. 'All propositions are meaningful' is a proposition about all propositions including itself. It is not, as Russell claims, a new proposition which tries unsuccessfully to stand outside the totality of all propositions. It stands effortlessly within that totality.

It is part of the orthodox view that the diagonal argument makes use of an impredicative definition. For the Russell paradox is said to have been derived originally from the diagonal argument,' and the Richard paradox is usually formulated by means of the diagonal argument.2 But it can be shown that those versions of the diagonal argument which make use of impredicative definitions are invalid and that the argument in its valid form makes no such use. It can also be shown that the paradoxes are not related to the diagonal argument. Wang, for example, claims that Cantor's proofs of the indenumerability of real numbers make use of impredicative sets.3 In support of this claim he gives the following invalid account of the diagonal argument: Suppose that for every positive integer there was a set of positive integers which corresponded to it. Then the set of all such sets of positive integers would be denumerable. Wang tried to prove by reductio ad absurdum that the set is in- denumerable. For suppose that the set of all sets of positive integers is denumerable. Then each positive integer has its corresponding set and each positive integer either is or is not a member of its corresponding set. Consider the set N of all those positive integers which are not members of their corresponding sets. Is n the positive integer whose correlate is N a member of N or not? If it is a member of N then by the condition of membership of N it is not a member of N. If it is not a member of N, then by the same condition it is a member of N. Wang thinks that these contradictions prove that the premiss is false and concludes that there is no such one-one correlation.

Now this argument is isomorphic with the 'heterological' family of paradoxes. For N is the set of all those positive integers which are not members of their correlating sets, and in order to be a member of N a positive integer must already be correlated with a set. N is a second-order set which is parasitic for its members on other first order sets, just as 'heterological'

1 Hao Wang, J.S.L. (1954). op. cit. p. 246. 2 Principia Mathematica, vol. 1, p. 61. Hilbert and Bernays, Grundlagen

der Mathematik, vol. 2, p. 263; S. C. Kleene, op. cit. pp. 38-39. 3Hao Wang, J.S.L. (1954), op. cit. p. 244.



516 J. TUCKER:

is a second order adjective parasitic for its applicability on other first order adjectives; just as 'this is false' is a second order statement dependent on other first order statements for its applicability, and so on. 'Is n a member of N? ' could only be answered in the affirmative or in the negative if n were already assigned to some set other than N. But this would be contrary to the condition that it is assigned to N and only to N. The argument turns on the breaking of this rule and is therefore invalid.

But in fact Cantor's diagonal argument is valid and does not make use of an impredicative set. It is usually given as a proof that the unending decimals cannot be paired off with the integers. This is done by giving a rule for the construction of an unending decimal which differs from any given unending decimal at one place. The rule is that for any given unending decimal an integer other than that at the nth place is substituted for that at the nth place. A denumerable series of unending decimals is obtained by pairing off the unending decimals with the integers. But by the above rule an unending decimal can be produced which cannot appear in this denumerable series. It cannot appear because the rule for writing it down consists in making it differ from each unending decimal in the denumerable series. It differs from the first given unending decimal in the first place, from the second in the second place, and from the nth in the nth place. It follows that there are more unending decimals than there are integers.

Now this argument differs entirely from that given by Wang. For the decimal which differs from each decimal in the denumerable list is not defined in terms of a totality of which it is a member. It is not defined in terms of a totality at all. All that is laid down is that it differs from each given unending decimal at the nth place. It is not laid down that the totality of unending decimals is required prior to its construction. No totality is mentioned.

The following argument is described in P.M. (p. 61) as the Richard paradox: E is the class of all finitely definable decimals. E has xo members. N is a decimal which is finitely defined by the diagonal rule. N differs from each member of E. But N is finitely defined and so is and is not a member of E. But this is not a paradox at all. The diagonal method does not of itself generate contradictions. For from the fact that N is finitely defined by the diagonal rule, it follows that E has more than xo members. So the assertion that E has xo members is thereby proved to be false, that is all. The form of the argument is simply that A is asserted and not-A is shown to be the case.




3. The distinction between the formal development of concepts and their clarification

This account of the paradoxes of set theory illustrates the distinction between the formal development and the clarification of concepts. It shows that, for example, the formal development of theories of types does nothing to clarify the situation which they are designed to avoid. Finally, I give two further samples of the distinction. In these cases formal developments which do not themselves require clarification give rise to inter- pretations which do. It is clear that further formal development will not of itself remove any conceptual confusion which is introduced by the interpretation. If mathematical logic were a clarification of concepts these cases would not arise. But they do. The confusion in both cases turns on the use of pairs of contrasting terms, one of which is never given any significance. This deprives its companion term of the significance which the contrast is intended to confer on it. In the case of the view that the concepts of set theory are relative the contrasting terms are ' relative' and 'absolute'. In the case of the actual infinite, the contrast is between ' actual ' and ' potential ' as applied to infinite sets.

(i) It is claimed by Skolem and others that the L6wenheim- Skolem theorem shows that the concepts of set theory are relative and not absolute as previously supposed, and further, that it dis- poses of platonism in set theory and so in mathematics generally.' Fraenkel and Bar-Hillel, for example, interpret the theorem as presenting us with a choice between platonism and relativism. The platonic half of the choice is said to consist in taking the concept of the sub-set of a given set to be a platonic reality.2 But what is it to 'take a concept as a platonic reality'? We are never told. Since we are never given instructions on how to ' take ' a concept in this way, it follows that we have no idea what it is to refrain from doing so either. It follows that no meaning has been provided for this contrast between relative and absolute.

Now, whatever might be meant by ' taking a concept as a platonic reality ', it is clear that if one concept is ' taken ' in this way, so must all the others. Platonists cannot be fastidious about which ones they ' take '. It follows that no discovery about the relations or lack of them which hold between concepts can be

1T. Skolem, ' Une relativisation des notions mathe'mati- les fonda- mentales ', Colloques Int. Centre Nationale de la Recherche Scientifique (Paris, 1958).

2A. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory, p. 108.



518 J. TUCKER: THE FORMALISATION OF SET THEORY

for or against platonism. For the relations which hold or do not hold, hold ex hypothesi between 'platonic realities'. The claim that the concepts of set theory are relative can be simply stated. All that is meant by saying that non-enumerability is relative is that a set which is non-enumerable in one system may be enumerable in another system. All that is meant by saying that a set is absolutely non-enumerable is that no such system is available.

A relativism which contrasts with platonism is meaningless. A meaningful relativism cannot contrast with platonism. It follows that the L6wenheim-Skolem theorem cannot dispose of platonism.

(ii) The results of set theory are discussed in terms of the rejection or acceptance of the actual infinite. It is sometimes claimed that classical set theory is the development of the concept of the actual infinite.' It is sometimes claimed that con- structivists want to dispense with the actual infinite in order to work only with the potentially infinite.2 But these opposing views can make sense only if a satisfactory way of introducing the notion of the actual infinite is forthcoming. The actual infinite is contrasted with the potential infinite which is itself then defined as a permanent possibility of the repetition of a given operation.3 But this contrast would make no difference to any proof and is therefore vacuous. For what difference would it make, say, to the diagonal argument if it were given with either of the following cautions? (a) This infinite square array of infinite decimals is not to be thought of as growing one by one, but is to be thought of as presented in its entirety. (b) This infinite square array of infinite decimals is to be thought of as growing indefinitely by increments. In either case, the diagonal differs at each place in the required manner.

* * * * *

This is only a preliminary skirmish. There is a lot more to be done by way of clarification of the motley of mathematics.

University College, Bangor

1 A. A. Fraenkel, Abstract Set Theory, p. 3; E. W. Beth, The Foundations of Mathematics, p. 365.

2 A. Mostowski, 'On various Degrees of Constructivism ', in Construc- tivity in l thematics, ed. A. Heyting, p. 181.

3 J. Ladriere, Les Limitations Internes des Formalismes, p. 219.



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The Formalisation of Set Theory

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