Cambridge Summer School in Mathematical Logic: Held in - download pdf or read online

By D. van Dalen (auth.), A. R. D. Mathias, H. Rogers (eds.)

ISBN-10: 354005569X

ISBN-13: 9783540055693

ISBN-10: 3540368841

ISBN-13: 9783540368847

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Extra info for Cambridge Summer School in Mathematical Logic: Held in Cambridge/England, August 1–21, 1971

Example text

YI @ n Vx(ylx • ( B(ith = ~x), which + is contradictory. B. ~ e ~ Vx(~x B = Bx). 44 As a corollary we have the 45 somewhat surprising From we LS3 can result derive e = B v ~ ~ the V3 B. (weak) c o n t i n u i t y properties: WC-N: V~ ~ x A(~,x) ~ V~ 3y 3x VB(~y : ~y ~ A(~,x)) WC-F: V~ 3 a A(~,a) ~ Ve ~ y 3a VB(Sy = [y ~ A(B,a)) and where a is Consider lawlike the following V~ 3 x A ( ~ , x ) ((0)0)0. tions It stronger Some ~ about variable. 3~ turns the selection V~ A(~,~(~)) out nature continuity properties principle: that , where ~ is of the continuous.

The class of K*. 5. Lemma. (i) l < ' x E K* (ii) let for then e of type Vx(e (00)0 C K*) ~ e E e x be defined by e x = l ~ ' e ( ~ * ~), K* X (iii) One now K* easily is the least class satisfying (i) a n d (ii). 6. Theo r e m . K* C CONT. By c l a s s i c a l E CONT means we h a v e one can } ~ K*. 5. each axiom ~). ¢(n* there n with ¢ n is an x o such that Cxo ~ K*. ~ K* an x such that ¢ ^ ~ K* , so an a p p l i c a t i o n n*x of d e p e n d e n t choices provides us w i t h Similarly we find for of the a ~ s u c h that Vx(¢= x~ ~ K*).

The following the principle same grounds of intensional as LS3 for lawless A~ ~ ~ the general justified conditions for Troelstra's The principle For various justified notions ~ V~ 3 y lawless, The principle of continuous 3x ~ as being generated is also sequences. is what we called WC-N: Vn(~0~y : TonY ~ A(n,x)). sequences this principle has been of the principle of intensional continuity. Inspired by Myhill operator for the purpose conditionpart the sequence continuity of choice on the basis on sequences: must be obeyed.

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Cambridge Summer School in Mathematical Logic: Held in Cambridge/England, August 1–21, 1971 by D. van Dalen (auth.), A. R. D. Mathias, H. Rogers (eds.)


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