Aspects of Constructibility by K. J. Devlin

By K. J. Devlin

Keith Devlin - common nationwide Public Radio commentator and member of the Stanford collage employees - writes concerning the genetic development of mathematical pondering and the main head-scratching math difficulties of the day. And he one way or the other manages to make it enjoyable for the lay reader.

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Aspects of Constructibility

Keith Devlin - commonplace nationwide Public Radio commentator and member of the Stanford college employees - writes concerning the genetic development of mathematical pondering and the main head-scratching math difficulties of the day. And he someway manages to make it enjoyable for the lay reader.

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Then L Hence Con(ZF) + Con(ZFC + V = L). I ~ M and L f o r M. = ( L ) M. But as M ~ ZF, -38- Corollary 20 La For all infinite ~ a On, is @ Proof: L~ is evidently closed under the function . r. function. r. definition And, by lemma 14, this Moreover, it is easily seen of this function does not lead out of L . We can thus repeat the (trivial) proof of corollary 5(ii) for L to show that ~ e ) e~ is u. Since a(= On is a ~0 predicate, the result follows at once. I The following lemma is due, in the form stated, Corollary 21 (Condensation Suppose lim(~).

There is nothing special We conclude w i t h the following is U-absolute and about La in the above. set M will have the same property. is that [1-satisfaction functions, What we have really under very general conditions. important result. Let X ~ L , and let M be the set of all elements Lemma 27 Let e ~ ~ be a limit ordinal. of L Proof: which are L -definable By lemma 24, L Corollary from parameters has a complete in X. Then M ~ L . set of definable, canonical skolem functions. 28 Let ~ ~ ~ be a limit ordinal.

Is prolific. By the above lemmas and remark, we may assume For each ~ < ~I' T is countable, and for ~ > O, infinite. So, for each non-zero e < ml, let < that of the rationals. be a linear ordering of T Let X be the set of maximal branches of T. linear ordering on X by b < d +-+ ( ~ ) [ ( ~ x (b(e) < of order-type Define a c TI~)(x c h +-+ x e d) & d(~))], where b(e) denotes the unique element of r N b, and similarly for d(e)o It is easily seen that is a dense linear ordering without end-points.