By Ruth Barcan-Marcus, George J. W. Dorn, Paul Weingartner

BARCAN, R, / G, J, W, DORN / P, WEINGARTNER, EDS,: common sense, technique AND PHILOSOPHY OF technology VII, lawsuits OF THE 7th overseas CONGRESS, SALZBURG, AUSTRIA, AMSTERDAM, 1986, xiv 738 p, Encuadernacion unique, Nuevo,

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**Extra resources for Logic, Methodology and Philosophy of Science VII, Proceedings of the Seventh International Congress of Logic, Methodology and Philosophy of Science**

**Example text**

In view of the previous work we need here to prove the following results. 1 y models REA. e»; if definition by transfinite recursion on the type U of small types is assumed. 1 In CZF every set is a subset of a transitive set. ACZEL 46 ao ~ a. For the regularity of a it suffices to show that if {3 E V such that {3 E a then {3 ~ a and for each species F on V x V "Ix E (3 3y E a F(x, y) ::J 3{3' E a F'({3, (3') where F'({3, (3') is "Ix E {3 3y E (3' F(x, y) & 'Vy E {3' 3x E (3 F(x, y). So let ao E V be transitive.

For the regularity of a it suffices to show that if {3 E V such that {3 E a then {3 ~ a and for each species F on V x V "Ix E (3 3y E a F(x, y) ::J 3{3' E a F'({3, (3') where F'({3, (3') is "Ix E {3 3y E (3' F(x, y) & 'Vy E {3' 3x E (3 F(x, y). So let ao E V be transitive. io(x). Then A o E U and B« E Ao~ U so that if A = (Wx E Ao)Bo(x) then A E U. Now define h E A ~ V by transfinite recursion on A so that h(sup(a, I» = (sup u E Bo(a»h (f(u» for a EA o, fEBo(a)~A. Then a E V where a =sup(A,h). Let {3 E V.

In general WaEA B; can be characterised as the unique class W such that W= U aEA {a}xWBa. CONSTRUCTIVE SET THEORY 27 Fig. 1. 6. 6 of [2]. Define a class X to be n~I -closed if (i) wE X (ii) ILEA B. EA B. E X whenever A E X and B. E X for all a E A. (iii) I(b,c)EX for all b, c EA if A EX In the above the cartesian product ItEA B. is the set of functions f with domain A such that f(a) E B. for all a E A and the disjoint union L. EAB. is the set of pairs (a,b) such that aEA and bEBa • Also I(b,c)= {z E {0} b = c}.