so that u F Is p - q There exists a finite eca such that sym (g) ::> H and 1f(a) = a otherwise. We now have

Transcription

1 131 Let q be the group of all permutations 1f of A such that 1f(S[i}) = S(i} for each i E: I', i.e. if 1f(a. ) = a. then i = j. Let J be the filter on q an Jm generated by [He : e e is finite }, where H [1f E: q: (YaE: e)[1fa= a l ]. e Let u be the class of all HS elements of lr. It follows from the definition of q that that sym (h) = q. If p q then S c S p - q remains to show that if p 1:. q then u F ISpl ISql. sym (Sp) = q for each p I and so that u F Is p I Is q I; hence it Let p 1:. q, let i E: P - q and assume that there is a one-to-one mapping g E: u of S P into S q Let n,n' be such that a. i e, a lo.n, e. l.n which interchanges a. and lon and so 1f(g) = g. Since where j =I i; therefore, There exists a finite eca such that sym (g) ::> H - e Let 1f be the permutation of A and 1f(a) = a otherwise. We now have Clearly 1f E: He' so that g(a. ) = a., an Jm g(a. ) = a. = 1f(a.) 1f(g(a.» = (1fg)(Ira.) g(a lo. nl) an Jm Jm an an contrary to the assumption that g is one-to-one. Bibliography [1] L. Bukovsky', The continuum problem and powers of alephs, Commentationes Math. Univ. Caro1inae, 6 (1965), [2] P. J. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. 50 (1963), and 51 (1964), [3] P. J. Cohen, Set theory and the continuum hypothesis, New York [4] W. B. Easton, Powers of regular cardinals, AnnalS of Math. Logic 1 (197 0), [5] R. Engelking and M. Karlowicz, Some theorems of set theory, Fund. Math. 57 (1965), [6] A. Fraenkel, Der Begriff "definit" und die Unabhiingigkeit des Auswahlaxioms, S.-B. d. Preuss. Akad. d. W. 1922, [7] A. Fraenkel, tiber die Axiome der Teilmengenbi1dung, Verh. des into Math.- Kongress, ZUrich 1932,

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6 c absolute cardinals 62 absolute formula 21 absolute operation 21 almost universal class 24 antichain 92 maximal a. 95 Aronszajn tree 91 atom 122 automorphism 74, 85 axiom of choice (AC) 15 axiom of constructibility 35 axiom of regularity 19 axioms of ZF 2-6 dense 49 open dense 66 dense below b 102 dependent choices (DC) 79 direct limit 105 elementary embedding 45 e. equivalence 44 embedding 7, 74 Embedding Theorem 125 extension of a tree 91 extensional class 21 e. relation 26 Boolean algebra 48 complete B.a. (cba) 48 homogeneous cba 76 Boolean values 53, 54 Boolean-valued model 53 branch 91 filter 50 F-complete f. 59 normal f. 115, 123 forcing 61 f. conditions 52 Fraenkel-Mostowski models 123 chain 15 countable chain condition (c.c.c.) 65 It-chain condition (x: - c i closed under an operation 23 closure 33 codes of Borel sets 80 collapsing algebra 71 compatible 48, 49 completion 14 constant function 45 constructible sets 33. ) 65 relative constructibility 39 continuum hypothesis 14 generalized c.h. (GCH) 19 Dedekind cut 14,47 Dedekind finite (D-finite) 13 generated 74, 86 generic 52 F-generic 99 g. extension 58 canonical g. ultrafilter 58 Godel operations 29 ground model 51 Hartog's number 10 homomorphism 7, 74 complete h. 74 natural h. 50 ideal 51 prime L. 5l inaccessible 36 weakly i. 12

7 d 137 induction 8 -induction 20 interpretation 58 isomorphism 7, 74 Isomorphism Theorem 25, 27 kernel 123 Lebesgue measurable (LM) 78 length of a tree 91 level of a tree 91 Levy algebra 72 Martin's axiom (MA) 99 maximum principle 15 model 20 transitive m. 21 m. of ZF 24 name 58 null set 81 order in a tree 91 order-preserving (o.p.) 7 ordinal definable (OD) 42 hereditarily o i, (HOD) 43 partially ordering {p,o. ) 6 separative p,o. 48 partition 56, 73 perfect set 110 Solovay real 83 canonical S.r. 85 sub algebra sum of p.o. sets 67 Suslin's problem 15, 90 Sus lin tree (ST) 92 normal S.t. (NST) 93 symmetric 115, 123 hereditarily s. (HS) symmetric extension 116 transfinite recursion 9 transitive 7 tree 91 t. closure (TC) 20 normal binary t. 94 ultrafilter (u.f.) 44, 51 countably complete u.f. 46 trivial u.f. 46 ultrapower well founded (w.f.) 26 well ordering (w.o.) 6 canonical w ;o. of W X W well ordering principle 15 reflection principle 28 RO (p) 50