By K. J. Devlin

Keith Devlin - commonplace 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 by some means manages to make it enjoyable for the lay reader.

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**Example text**

Lermna 4 R ~ V n is ~0ZF iff it is rud. Proof: (÷). By lemma 1 (3), (7) & (8), and an easy induction on the ~ F definition -33- o~ R. (÷) By ind. on a rud definition of ~R' using lerm~as 3 and 2, we see that ~ [ZF, whence so is R. r. , then R is [ZF. Proof: (i) follows from lemma 4. r. ~ ZF Since the predicate On(x) is [0 , and hence (by lemma 4), rud, lemma I (I) leads us to call a function f: On n + V rud (resp. ) iff it is the restriction to On n of a rud (resp. ) function f: V n + V. And in view of lemma I (i), we shall often "define a rud (resp.

C is closed and cofinal B < ~, ~ e C B, so f(~) # B. in K. Hence Let C = {~ c ~ I ~ E ~ B < ~ CB}. Let ~ s C f(e) ~ ~. q E. Then, for each But this contradicts the regressive nature of f. QED For the present purpose, following By S ~ combinatorial stationary principle 0 we mean the assertion of Jensen. that there is a sequence ~ for each e, and, whenever X is stationary in ~io sets are important by virtue of the ~ w I is given, ~~ such that the set {~ s ~ I X q ~ = S } -49- Note that 0 immediately implies 2m , since every subset of ~ must figure = ~I (~i many times, in fact) We shall prove We require in the 0 - s e q u e n c e • that 0 holds two simple in L. ~~

We write ~M for ~M(~) and ~n(M) for ~M(M). Similarly for H and A. ÷ If $(v) is a formula of the M-language, II$II~M denotes the relation {*I~ E M & l=M$(~)} Let ~ be a class of structures of the form M = . uniformly ~M for M E ~ A relation R -~Vm is iff there is a ~n(~) formula ~ such that M ~ ~ -> R n M m = II~I~. The interpretations of the constants, ~, occurring in a formula of an M-language are usually called parameters. from". We use the abbreviation wpf to mean "w1~h parameters Hence if we say $ is a formula ~ X, we mean that every constant in ~ is of the form ~ for some x c X. *