Comments on the Foundations of Set Theory. by Paul J. CohenReview by: Donald A. MartinThe Journal of Symbolic Logic, Vol. 40, No. 3 (Sep., 1975), pp. 459-460Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272188 .

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REVIEWS 459

RICHARD MONTAGUE. Set theory and higher-order logic. Formal systems and recursive functions, Proceedings of the Eighth Logic Colloquium, Oxford, July 1963, edited by J. N. Crossley and M. A. E. Dummett, Studies in logic and the foundations of mathematics, North- Holland Publishing Company, Amsterdam 1965, pp. 131-148.

Professor Montague first considers second-order theories corresponding to Peano arithmetic, the theory of real closed fields, and Zermelo-Fraenkel set theory, and then goes on to discuss cumulative type theory of possibly transfinite order. He points out that for the three theories considered, the standard models are the only models of a corresponding second-order theory. Given an arbitrary second-order theory he defines its first-order analogue to be its first-order consequences. Under this definition the theory of real closed fields is the first-order analogue of its corresponding second-order theory. The other two theories are not.

In considering higher-order logic, there is a problem which arises from the metatheory. Montague uses ZF set theory with individuals but does not assume there are arbitrarily large sets of individuals. By naively defining the cumulative hierarchy over the set A, one allows the possibility that memberships can occur among the members of A. If there were a set B of individuals in one-to-one correspondence with A, this problem could be avoided by using the hierarchy over B and transferring all relations to B by means of the correspondence. Since it is unnatural always to assume the existence of such a B, Montague adopts another route. He calls upon an abstract second-order characterization of the cumulative hierarchy discovered by Scott and Montague, and defines a higher-order sentence 'p to be true in a model with universe A if and only if there exists an abstract type structure with individuals A and rank at least as large as the largest type occurring in p which makes 'p true under the now obvious interpretation. The point is that the abstract characterization allows the interpretation of membership by a binary relation which ignores the spurious memberships.

RICHARD MANSFIELD

PAUL J. COHEN. Comments on the foundations of set theory. Axiomatic set theory, Proceed- ings of symposia in pure mathematics, vol. 13 part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 9-15.

The author discusses some of his views on basic problems in set theory, without making an

attempt to state in detail his positions and the arguments for them and "without an attempt to

convert the listener on the spot" (p. 9). A large part of the paper is concerned with the issue of Realism versus Formalism in set

theory. Cohen mentions some of the main defects in each of these views, and then declares

that he has chosen the Formalist position. Having done so he attempts to deal with the "great

weakness in the Formalist position... the fact that it must explain how the purely formal

axioms which constitute set theory can be successful" (p. 13). He holds "that the axioms are an extrapolation from the language of more finitistic mathematics" (p. 13). This is perhaps more an explanation of the source of the axioms than of their success. Cohen explains why the

axioms are produced (extrapolation) and why we continue to use them (because they are successful) but seemingly not why they are successful.

Cohen, as a Formalist, holds that "we do set theory because we feel we have an informal consistency proof for it" (p. 14). He goes on to sketch a plan for a consistency proof of set theory apparently analogous to Gentzen's proof for number theory. It is not clear to the

reviewer whether our "informal consistency proof" is to be interpreted as an intuition that

Cohen's sketch can be completed. Despite his Formalism, Cohen does not feel that we should dismiss the problem of finding

new axioms but instead holds that we should "develop our mystical feeling for which axioms

should be accepted" (p. 15). In particular, he thinks that the negation of the continuum hypothesis might eventually be accepted as an axiom. This seems rather unlikely, since Cohen's fundamental work and later results obtained by his methods show that the negation of the

continuum hypothesis has little deductive power, and the power of a candidate for axiomhood is at least as important as its plausibility.

Cohen concludes this interesting paper by remarking that there are few "operational' differences between his view and Realism, and by pointing out the quasi-empirical nature of

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460 REVIEWS

mathematics as he sees it. We have no assurance that our set-theoretic axioms will not be found wanting and be rejected, but this lack of certainty should not keep us from venturing onto the "unsafe ground of set theory" (p. 15). DONALD A. MARTIN

WILLIAM B. EASTON. Powers of regular cardinals. Annals of mathematical logic, vol. 1 no. 2 (1970), pp. 139-178.

J. R. SHOENFIELD. Unramified forcing. Axiomatic set theory, Proceedings of symposia in pure mathematics, vol. 13 part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 357-381.

The earlier treatments of forcing proceeded by generating recursively a ramified hierarchy of constructible sets. Shoenfield introduces a version of forcing which does not require the ramified hierarchy. As a result, one can use shorter proofs, with many fewer details.

Although Shoenfield motivates his treatment, so that one can understand the relationship between it and the earlier treatments of forcing, the resulting formulation is not at all intuitive. As a result, it does not seem practical to give a comprehensible summary of it in the small compass of this review. We will illustrate the flavor. Let M be a countable model of ZFC (Zermelo-Fraenkel plus the axiom of choice). A set C of M is called a "notion of forcing" if it is partially ordered by some relation, <, of M and has a largest element. A definition identifies certain "generic" subsets of C. There is an existence theorem to the effect that for each notion of forcing, C, and for each p e C, there is a generic subset, G, of C which contains p. For each such C and G, a definition is given of M[G], which is a countable model of ZFC which includes M and contains G, and is the smallest such. Conceivably M[G] is one of the models that could be generated by ramified forcing, but Shoenfield does not shed much light on this. It turns out that M and M[G] have the same ordinals and hence the same constructible sets. Cardinals of M[G] are cardinals of M; for the converse, some chain conditions need to be invoked.

One section apiece is devoted in turn to independence proofs for the axiom of constructibility, the axiom of choice, and the continuum hypothesis. In each case, a suitable C and G are defined, so that the statement whose independence is to be proved can be shown to be false in M[G] (for the axiom of choice, a subset of M[G] is used). In connection with the continuum hypothesis, Shoenfield derives not only the original result of Cohen but also a later, stronger result of Solovay. Finally, Shoenfield turns to a still stronger result, that given in the Easton paper, which is also being reviewed here. Easton proves: Let M be a countable model of ZFC in which the GCH holds, and let G be a function in M such that for cardinals (i) a < P implies G'a < G'/3; (ii) XG', is not cofinal with any cardinal less than or equal to X,, . Then there is an extension N of M, whose cardinals are precisely those of M, in which 21- G,, for regular cardinals a.

The status of singular cardinals is still an open question. Easton's proof is a generalization of the proofs by Cohen and Solovay of their more limited

results. Indeed, Easton acknowledges considerable personal assistance by Solovay. What keeps Easton's proof from being a straightforward generalization is that in his case the conditions constitute a class in M, rather than a set. The difficult points in the earlier proofs by Cohen and Solovay become more intractable, and difficulties arise even in such matters as the treat- ment of equality. We might mention that Easton takes conditions to be sets of quadruples <iya-q> for which y < Ma and -q < S In this way, the function G determines the model which is generated.

Shoenfield's proof of Easton's result borrows heavily from Easton's proof. Sets like Easton's conditions appear, but without the i, so that they are sets of triples. Shoenfield's C, the "notion of forcing," is now a class, rather than a set, which causes difficulties. Analogously to the Easton proof, one difficulty is with inequality.

For Shoenfield, as well as for Easton, G must be a function of M. Neither author raises the question of other possibilities for G. Of course, this question would involve interpretational complications. Easton alludes to some similar problems, and cites a discussion by Hafnal. It seems worthwhile to look at a specific case.

Let a logic L have a non-denumerable number of constants and axioms. Specifically, let each

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