By K. J. Devlin

Keith Devlin - average nationwide Public Radio commentator and member of the Stanford collage employees - writes in regards to the genetic development of mathematical pondering 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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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 . r. Proof: Define ~ by ~(~, u) = {~ e Fml~I~u~}. r. Define g by g(u) = {~ c PFmlulFr(~) = @ & ~u~}. r. (by our remark just prior to the lemma). __

But ~(v, u) = h v(g(u), u). r. QED. r. Proof: Define h by h(u, ~) = {x e Ul~u~(V0/X)}. r. h"({u} x Fmlu). QED. We may now study the That completes our analysis of 2 v and its semantics. constructible hierarchy in some detail. r. Proof: Since L = DefV(~), the result follows from lemma 13. [ Corollary 15 The predicate "y = L x ,, ZF is A 1 . e. "x ~ L") is El • ~ x e L ]. I Corollary 17 Let M be a transitive model of ZF, a ~ M ~On. Proof: By c o r o l l a r y 15, t h e p r e d i c a t e a e M-+ ( L ) M e M.

Hence ~(a) = {~(~)I~ This means that a c ran(q) = Le E LK+. ~(K) ~ L +. K % such that a e L%. Thus, by corollary Hence 28, cardinals ILKI = K. --I~) Let M be the smallest IMI = <. By the condensation Hence L _= LK+. ~ a nM} = {~(~)I~ ~(<) _c LK+. <, Since < ~ M QED. is s a} = {¢I~ s a} = a. -42- We conclude this chapter by remarking, L = H K for all infinite cardinals K. in the sense of L, then L world). K Hence, is a ZF--model for later use, that if V = L, then if K is an uncountable regular cardinal (both in the sense of L and in the real This implies that (regardless of whether or not V = L) L whenever K iS an uncountable regular cardinal of L.