perfect subsets of definable sets of real numbers.by richard mansfield

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Perfect Subsets of Definable Sets of Real Numbers. by Richard Mansfield Review by: Yiannis N. Moschovakis The Journal of Symbolic Logic, Vol. 40, No. 3 (Sep., 1975), p. 462 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2272192 . Accessed: 16/06/2014 07:51 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org This content downloaded from 194.29.185.216 on Mon, 16 Jun 2014 07:51:48 AM All use subject to JSTOR Terms and Conditions

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Page 1: Perfect Subsets of Definable Sets of Real Numbers.by Richard Mansfield

Perfect Subsets of Definable Sets of Real Numbers. by Richard MansfieldReview by: Yiannis N. MoschovakisThe Journal of Symbolic Logic, Vol. 40, No. 3 (Sep., 1975), p. 462Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272192 .

Accessed: 16/06/2014 07:51

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 194.29.185.216 on Mon, 16 Jun 2014 07:51:48 AMAll use subject to JSTOR Terms and Conditions

Page 2: Perfect Subsets of Definable Sets of Real Numbers.by Richard Mansfield

462 REVIEWS

J. I. FRIEDMAN. Proper classes as members of extended sets. Mathematische Annalen, vol. 83 (1969), pp. 232-240.

In this paper the author introduces the formal theory STC, the set theory of proper classes. STC has the language of set theory augmented by the predicate M, where "Mx" is read "x is an extended set." The axioms of STC are NGBM (von Neumann-Bernays-Gbdel set theory with choice, in the language of STC), axioms asserting the closure of M under subsets, un- ordered pairs, power set, and UM(UMX = U(x r) M)), and the axioms of: Infinity (3x)(xe M, Tx ' M, x is infinite) (Tx = x u Ux u UUx ); Universes {x: Tx ' M U zjE V, for zE V; Maximality xE Kq- (x M,3 v x is unnata); where Kf = {xE V-M: Tx M u U<Y Ky}; M: = {xE M: Tx ' M Uy<,Ky}; x is unnat,6 <-> (3y < 1)TTMx nM My(TMx = x J

UMX u UMUMX .); x is natq <> (Vy < P)TMx n My -< My. It is shown that the M: , and the K: (the extended proper classes), form a double cumulative

hierarchy and that M = UqMq = {x E V:(Vp)x is natq}; V-M = UqKq = {x E V:](3)x is unnat4j.

The main result of this paper is that Tarski's axiom of inaccessibility, Al, (Vs)(3y)(s < y, y is strongly inaccessible), is a theorem of STC and that STC is interpretable in NBG augmented by Al. Only outline proofs are given.

Finally, the comparative strengths of fragments of STC are considered and it is shown that STC-{Maximality, Universes}, STC-{Maximality}, and STC are not equivalent.

J. B. PARIS

RICHARD MANSFIELD. Perfect subsets of definable sets of real numbers. Pacific journal of mathematics, vol. 35 (1970), pp. 451-457.

In this paper Mansfield develops a method which is very important in the applications of modern set-theoretic techniques to descriptive set theory. Fix an infinite set A, where usually A = K is an infinite cardinal number. A tree on w x A is a set T of finite sequences from w x A. The set of paths through Tis defined by [T] = {(a,f): ae 'Ec, fE '?A & (Vn)[(<a(0),f(0)>, * * *, <a(n - 1), f(n - 1)>) E T]} and the projection proj[T] of [T] onto O'c is naturally taken to be proj[T] = {aE 'co: (3f)(a, f) E [T]}. We take L(T) to be the smallest model of Zermelo- Fraenkel set theory which contains T and all ordinals.

MAIN THEOREM. If T is a tree on co x A and proj[T] has an element not in L(T), then proj[T] has a perfect subset (and hence has cardinality 2No). The proof is by a simple and elegant argument which uses Boolean-valued models. Mansfield gives two direct applications of his result. Corollary 1. Every Z' subset of 'co which has a non-constructible member has a perfect subset. Corollary 2. If there is a measurable cardinal, then every E' subset of 'ct which has a non-ordinal definable member has a perfect subset.

It is clear, however, that the main theorem is interesting in its own right and has wide applicability beyond these two specific corollaries. YIANNIS N. MOSCHOVAKIS

YOEMON SAMPEI. A proof of Mansfield's theorem by forcing method. Commentarii mathe- matici Universitatis Sancti Pauli, vol. 17 no. 2 (1969), pp. 99-103.

The author gives a proof by forcing for a result of Mansfield whose original proof used Boolean-valued models: Every Y2 subset of 'co which has a non-constructible member has a perfect subset. YIANNIs N. MOSCHOVAKIS

E. M. KLEINBERG The independence of Ramsey's theorem. The journal of symbolic logic, vol. 34 (1969), pp. 205-206.

It is proved that the negation of Ramsey's theorem is consistent with the axioms for Zermelo- Fraenkel set theory (without the axiom of choice). Here Ramsey's theorem is taken to be the assertion that for any infinite set A and any positive integer n, if the n-element subsets of A are partitioned into two sets, then there is an infinite subset of A all of whose n-element subsets lie in the same set. The proof is in essence the observation that the negation of Ramsey's theorem (with n = 2) is a consequence of the following proposition, which is consistent by work of Cohen: There is a Dedekind-finite set A which can be mapped onto co in such a way that the inverse image of each element of co is finite. JAMES E. BAUMGARTNER

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