Is the Axiom of Choice a Logical or Set-Theoretical Principle?

JAAKKO HINTIKKA*

Abstract A generalization of the axioms of choice says that all the Skolem functions of a true first-

order sentence exist. This generalization can be implemented on the first-order level by gener- alizing the rule of existential instantiation into a rule of functional instantiation. If this gener- alization is carried out in first-order axiomatic set theory (FAST), it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what the truth of S means in a combinational (model-theoretical) sense, in any model of FAST there are sentences which are set-theoretical “true” but false in the normal sense of the word. This shows that the assumptions on which the axiom of choice rests cannot be fully implemented in FAST. The axioln nf choice is not a set-theoretical principle.

This title question is not completely well formulated as long as the bor- derline between logic and set theory remain as unclear as it has generally been. However, the contrast between different answers is unmistakable. In his attempt “New Grounding of Mathematics” (1922, p. 157) Hilbert asserted that “it must be possible to formulate Zermelo’s postulate of choice in such a way that it becomes just as valid and reliable, in the same sense of ‘valid’ as the arithmetical proposition that 2+2 = 4.” In contrast, in our days that axiom of choice is considered in the context of axiomatic set theory which is thought of as being on a par with any other axiomatic theory studied in mathematics. The axiom of choice then becomes merely one possible axiom for set theory. You can take it or leave it as you can do with any special-purpose mathematical axiom, in view of the fact it has been proved independent of the other usual axioms of set theory.

A doubting Thomas might object that Hilbert does not in the quoted pas- sage claim that is possible to @rn the axiom of choice into a logical truth. This point is well taken, but it does not pre-empty the contrast. For one thing,

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Hilbert’s own later epsilon-technique may be viewed as an attempt to turn a version of the axiom of choice into a logical principle. For another thing the objection turns on the ambiguity of the term of “logic”. According to Hilbert, Dedekind treated set theory as “a chapter of pure logic”, and most of the found- ing fathers of contemporary foundational studies considered set theory as a set of general truths and principles of argumentation that apply in all parts of mathematics and perhaps literally everywhere. What they did not argue, or, rather, what they left for the likes of Frege and Russell to argue, is that the mathematical universality of set theory implied its reducibility to logic. In con- trast, most of those logicians like Peano who concentrated on the actual step- by-step logical arguments in mathematics did not see any way of accommo- dating all of set theory, including the axiom of choice, let alone their deductive systems. At best, the axiom of choice was considered as an additional intu- itionist’s assumption with a dubious pedigree in pure logic. The import of Hilbert’s claim is thus no less clear than if he had used the magical phrase “logical truth”.

This early twentieth-century ambivalence is an anticipation of the present- day uncertainty concerning the status of higher-order logic. The axiom of choice is arguably a truth of higher-order logic. It can be thought of (at least tentatively) as being by the second-order axiom schema

(1) (vx)(~YY)s[x,Yl 3 (3f)(v’x)S[x,f(x)l But is second-order logic really logic, or merely “set theory in sheep’s cloth- ing”? The preferred way of thinking in our time and age considers set theory as being merely another mathematical theory that employs logic but does not reduce to it. Such a way of thinking is predicated on a rejection of the status of higher-order logic as genuine, basic logic. As a corollary, it seems to fol- low that the axiom of choice is a set-theoretical rather than logical principle.

What such a view nevertheless fails to acknowledge is a possibility which ought to be fairly obvious but which to the best of my knowledge has not been examined or implemented before. In a sense, the ideas (“intuitions”, as some folks would say) codified in the axiom of choice can be fully implemented on the level of first-order logic. In order to see this, consider the usual rule of exis- tential instantiation that leads us from a sentence of the form (3 x) S [XI to a formula of the form S[b] where b is a new individual constant. What under- lies this rule is the obvious meaning of the quantifier (3 x) in (3 x) S [XI. What it says is that there exists a “witness individual”, call it b, such that S [b] .

What is involved here is nothing stranger than the procedure of a judge who does not know the identity of an alleged perpetrator or does not want to divulge the name of a litigant and decides to refer to her or to him as “Jane Doe” or

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“Richard Roe”. The instantiating terms like our “b” are nothing but logicians’ “Jane Does” and “Richard Roes”.

It is amusing and instructive that according to some sources Vibte’s intro- duction of special algebraic symbols for unknowns in mathematical problems was modeled on lawyer’s use of special names like our “Jane Does” to desig- nate unknown or indefinite defendants. (See Vibte 1983, p. 13.) Since alge- braic and logical symbols are pretty much on a par, history has apparently reproduced logic here.

(2) ( V X W Y)S[X,Yl

By the same token, the sentence

is true if and only if there is a way of choosing, given any individual a, an indi- vidual b such that S[a,b]. Since here b depends on a, we are mathematically speaking saying that there is a function of a, say g(a), that produces a witness individual that verifies

(3) (3 y)S[a,yl. In other words, what (2) asserts is

(4) (3 f)(Vxx)S[x,f(x)l. But if so, if there are functions like that, then we can call one of them “Jane Doe” - well not really, perhaps we should call one of them more prosaically “g(x)” - and go on and argue from

( 5 ) (~xX)S[x,g(x)l. Again, in the same way we can motivate a general rule of inference of first- order logic that is formulated as follows:

Assume that we are given a sentence S=So[(3 x)Sl[x]] which contains (3 x)S , [XI as its subsentence and which is in the negation normal form. Then we can move from S to

(6) sois I[f(YI,YZ,. . .)I1 where f is a new function symbol and (Vyl), (Vyy2), ... are all the universal quantifiers in the scope of which (3 x) occurs in So.

This rule will be called the iule of functional instantiation. It is a first-order rule, for no higher-order quantifiers occur anywhere in its applications. In the usual first-order logic, it is a valid derived rule. However, it can be extended to independence-friendly (IF) first-order logic simply by exempting from the arguments of f(yl,yz, ...) those variables y, for which the existential quantifier

286 Jaakko Hintikka

in question/which now can be written in general (3 xfd zl, V z2, ...) is inde- pendent of (V y,). For IF logic, see Hintikka 1996. And in IF first-order logic the rule of functional instantiation is no longer dispensable, i.e. it does no longer follow from the rules of ordinary first-order logic.

The functions f that can be introduced by the rule of functional instantia- tion are familiar from the theory of first-order logic. They are there known as the Skolem functions of So. In the context of second-order logic, the force of a first-order sentence So is to assert that all the Skolem functions of So exist. (The converse implication obtains trivially: If all the Skolem functions of So exist, then So is true.) What happens in a step of functional instantiation is merely that a set of particular Jane Doe functions are introduced to exemplify the Skolem functions that are known to exist if So is true.

The rule of functional instantiation can be considered a strengthened form of the axiom of choice. For instance, for all inferential purposes, the step from (2) to ( 5 ) does the same job as the step from (2) to (4).

The way in which the rule of functional instantiation was motivated deserves a comment. This motivation had a Janus face. On the one hand (or should I say, face?), it can be motivated in a way that are usually appealed to in order to motivate the axiom of choice. This motivation was sketched above. And this motivation was not a mere rehash of the “intuitions” that are sup- posed to prompt us to accept the axiom. The motivation turned on what it means for the sentence of a first-order language to be true. It did not incorpo- rate a tacit or overt appeal to intuitions, but an unspoken appeal to the truth- conditions of quantified sentences.

The new rule offers arguably certain practical advantages in actual applied first-order reasoning. I will not consider them here, however. Instead, it is interesting to note the rule of functional instantiation cannot be objected to on the basis of constructivistic ideas. If anything, constructivistic principles there- fore serve to strengthen the line of thought that led to the rule. For a con- structivist, an existential quantifier (3 x) must be interpreted as requiring that there is a method of actually constructing a suitable x. But if so, we can cer- tainly move always in our reasoning from (3x)S[x] to S[b] for some b. And the only difference between this case and (2) is that in (2) the construction of y must of course depend on x, in other words, that there must be a construc- tive function f(x) to produce the “witness individual”. Hence we can surely reason in terms of there being such a function f as in (4), and likewise in the general case.

The fact that the axiom of choice can be considered as being a consequence of the rule of functional instantiation puts a very dubious light on the wide-

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spread contemporary view that the axiom of choice is “a simple, fundamental principle and must be postulated, not proved”. This mistake seems to go back all the way to Zermelo. (See Moore 1982, p. 123.)

To this implementation of the axiom of choice on the first-order level it might be objected that the use of function constants makes the new rule of inference really a higher-order rule. When this objection has been made to me, the only thing I have been able to say that I do not understand the sense of the terms “first-order” and “higher-order” any longer. For the only clear sense that has ever been given is in terms of the entities - individuals or higher-order enti- ties - over which we quantify in logic. But by that criterion the rule of func- tional instantiation is a first-order one. No higher-order quantifiers occur any- where. Hence, if we are to believe Quine (“to be is to be a value of a bound variable”) we do not have to worry about the existence of any higher-order enti- ties, either. Furthermore, as was pointed out, in ordinary first-order logic all ist consequences are first-order consequences, for it is there merely a derived rule.

It is nevertheless not a part of my agenda to argue that the rule of func- tional instantiation is a first-order rule, as long as it is acknowledged as a purely logical rule that implements the axiom of choice.

But now it might seem that I have to prove that higher-order logic is logic in a sense that makes it independent of set theory. For otherwise it might still be maintained that there is a hidden higher-order and aforriori set-theoretical import to the rule of functional instantiation.

There is a different way, however, of showing that the rule of functional instantiation cannot be considered as a set-theoretical principle. This argument can be sketched as follows: Assume that we are given some usual first-order axiomatization of set theory AX. We can extend mutaris murandis to that first- order theory AX our rule of functional instantiation formulated above. Part of the murandis is due to the fact that the new constant c that is introduced must now be set constant. Hence we have to express f(yI,y2, ...) in terms c. This is of course doable, even if explicit rules for doing so are not given here.

In AX we can use Gijdelization and formulate ist own syntax in it. Among other things, we can in AX formulate a predicate C[x] that says that Skolem functions of the sentence S with the Godel number x=g(S) exist. I will not prove the existence of C[x] here. Proving its existence obviously requires a complicated argument, even though this existence is so plausible as to be almost obvious.

However, I can do slightly better here than to complain about the narrowness of margins or to use the well-known proof method of hand-waving. I can refer to a related argument which is in fact sketched in Hintikka (1998), secs. 4-7.

288 Jaakko Hintikka

It is equally obvious that the diagonal lemma holds. (For this lemma in arithmetic, see e.g. Mendelson 1987, pp. 149-160.) Using this lemma, we can ascertain that there is a number n which is the Godel number of

(7) 4 3 n l

where n is the numeral that expresses n. Now if (7) is false, the Skolem func- tions of the sentence with the Godel number n must all exist. Hence this sen- tence is true. But that sentence is (7) itself, and hence cannot be false. Hence it must be true.

What that means is that there exists in AX a true sentence whose Skolem functions do not all exist. But the rule of functional instantiation asserts pre- cisely that they do exist. Hence axiomatic set theory turns out to be incom- patible with the rule of functional instantiation when this rule is transposed to set theory. But this rule was seen to be nothing but a generalized and strength- ened form of the axiom of choice. Whatever motivation can be given to the usual form of the axiom can be extended to justify the rule of functional instan- tiation.

In this sense the axiom of choice is incompatible with any usual first-order axiomatic set theory, even though this axiom is a natural component of axiomatic set theory. Not only do the usual axiomatizations of set theory fail to capture a natural set-theoretical assumption. The full force of this assump- tion cannot possibly be accommodated within first-order axiomatic set theory. The axiom of choice is not a set-theoretical principle.

In the light of hindsight it was a pity that the axiom of choice was first for- mulated by Zermelo in a set-theoretical context and that it lived its early log- ical life in the context of problems and controversies concerning set theory.

But why do the usual set-theoretical formulations fail to capture the full force of the assumption behind them? They do not even capture the full force of an axiom schema like (2) because the classes of sets y satisfying S[a,y] for some values need not be sets, nor does the class of such classes.

We can thus also see that the rule of functional instantiation is much stronger than the usual formulations of the axiom of choice in axiomatic set theory.

It might perhaps be thought that this result is merely due to the special way functions are dealt with in axiomatic set theory. This is not the right explana- tion, however. The incompatibility cuts much deeper.

What is going on can be seen from the original reasons given above to show why the Skolem functions of a true sentence S must exist. Their existence is precisely what it means for S to be true. These functions produce the “witness

Is the Axiom of Choice a Logical or Set-Teoretical Principle? 289

individuals” that show the truth of S. As soon as the existence of the Skolem functions of each true S can be expressed in a first-order axiomatic set theory AX, they cannot all exist for all true sentences, for then we could express the truth of S in AX itself. This is what Tarski’s (1958) famous theorem shows to be impossible. The incompatibility of stronger forms of the axiom of choice and first-order axiomatic set theory is thus a consequence of the impossibility of defining set-theoretical truth in set theory - in effect, of the impossibility of doing the model theory of set theory by set-theoretical means.

For the same reason the truth of sentences like (7) in all the purported mod- els of first-order axiomatic set theory is highly pathological. The sentence (7) “says” that ist Skolem functions do not all exist. But that existence is what it means for (7) to be true. Hence (7) says in a strikingly natural sense that it is itself false. What saves first-order axiomatic set theory from the ignominy of an outright inconsistency is therefore a pathological interpretation.

The situation would be in principle the same if a sentence of the form (2) were true in a model of set theory but that we also had to countenance the truth of the negation of (4). Notice that I am not talking about the fact that it can- not be proved in set theory that (2) entails (4). This would simply mean that the axiomatization of the set theory we are dealing with is incomplete - an unfortunate but inevitable predicament. What is going on is an interpretational conundrum. What could it possibly mean to say that (2) is true but (4) false? The truth of (4) is what the truth of (2) means. The only conclusion that can be drawn here is that the very interpretation of a first-order axiomatic set the- ory is awry. What are there called functions cannot possibly be interpreted as to be what we usually mean by’these terms.

A mathematician might describe this situation by saying that in any model of a first-order axiomatic set theory there are sentences which are true on the first-order reading of set-theoretical axioms even though they are false on the obvious combinatorial reading of set-theoretical formulas. It is for instance not due to the existence of too large sets. The paradox I have uncovered is com- binatorial rather than set-theoretical.

These considerations show that the failure of first-order axiomatic set the- ory to accommodate the principles on which the axiom of choice is based is a telling black mark against this entire approach to set theory.

290 Jaakko Hintikka

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