Download Aspects of Constructibility by K. J. Devlin PDF

By K. J. Devlin

ISBN-10: 3540065229

ISBN-13: 9783540065227

Keith Devlin - general nationwide Public Radio commentator and member of the Stanford collage employees - writes in regards to the genetic development of mathematical considering and the main head-scratching math difficulties of the day. And he by some means manages to make it enjoyable for the lay reader.

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Partially order T by I , of T, would imply the failure of the Souslin property for X, since, if we let x~ be the lower end-point of I , each E, then the pairs determine an uncountable collection of pairwise disjoint intervals of Xo The existence of an uncountable antichain of T would likewise contradict the Souslin property for X, since I and J will be Tincomparable iff they are disjoint intervals of X, of course, Thus T must be an (~I' ml )-tree with no cofinal branch and no uncountable antichain.

We seek b s L such that [ ( ~ x e a ) ( ~ y e b)~(y, x)] L. Pick ~ with For each x ~ a, let f(x) be the least ordinal B >. ~ such that (-~y e LB)~L(Y, x). Let y exceed all f(x), x e a. that [ ( ~ x ~ a)(~y Thus ( ~ x s a)(~y ~ L ) L(y, x). It follows at once Y is (by lemma i) as sought. ~ Ly)~(y, x)] L, so b = L Y That completes the proof of Theorem 2. We have shown above that ZF ~ ~L for every axiom ~ of ZF. ZF ~ AC L. We show next that In fact, we do this in a very strong way, by exhibiting an ~ - f o r m u l a which defines (in L) a well-ordering of the entire class L.

In particular, Fml Hence, so is Fml~u = F~(PFMIu' u) (as a function of ~, u). r. r. predicate. The set of variables occuring free in ~, if Fml(~) Define Fr(~) @, otherwise. r. If ~ is a formula, x is a variable, and t is a constant, ~(x/t) denotes the result of replacing each free occurrence of x in ~ by t. Define Sub(~, x, t) = ~ ~(x/t), if Fml(~) & Vbl(x) & Const(t) [ ~, otherwise. r. Formalising our earlier notation, let ~u ~ mean that ~ is a statement of ~ u which is true in under the obvious interpretation of the members of Const u .

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