Baire property and axiom of choice
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BAIRE PROPERTY AND AXIOM OF CHOICE* BY
HAIM JUDAH Department of Mathematics and Computer Science Abraham Fraenkel Center for Mathematical Logic Bar-Ilan University, 5~900 Ramat-Gan, Israel
AND SAHARON
SHELAH**
Institute of Mathematics The Hebrew University of Jerusalem, 91904 s
Israel
ABSTRACT
We show that without using inaccessible cardinals it is possible to get models of "ZF + all sets of reals have the Baire property + DC(wl )" and "ZFC + all projective sets have the Baire property + the union of less than w2 many meager sets is meager", answering two well-known open questions of Woodin and Judah, respectively.
1.
Introduction
In 1979 Shelah proved t h a t in order to obtain a model in which every set of reals has the Baire property, a large cardinal assumption is not necessary, thus finding a deep a s y m m e t r y in the s t u d y of measure and category on the real line. Shelah * The authors would like to thank the Israel Academy of Sciences BSF for partial support. ** The second author would like to thank the Landau Center for Mathematical Analysis, supported by the Minerva Foundation (Germany). Received November 7, 1991 and in revised form January 6, 1993 435
436
H. JUDAH AND S. SHELAH
started from L and by a method called a m a l g a m a t i o n
Isr. J. Math.
he built a forcing notion
P satisfying (i) H O D ( L P) ~"every set of reals has the Baire property", (ii) L P and L have the same cofinalities; moreover P ~ccc, (iii) L P ~ CH. From (ii) it is possible to conclude that in H O D ( L P) there are uncountable wellordered sets of reals (namely, the constructible reals!). From this evidence it was natural to ask the following question: Woodin:
Can we get a model where every projective set of reals
has the Baire property and DC(0,1) holds? Recall here that DC(0,1 ) is the following sentence: if T~ is a relation such that
(VX)(3Y)(Ti(X,Y))
then there is a
sequence (Z~: a < 0,1) such that (w
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