Methodology

Logic, Methodology and Philosophy of Science VII, by Ruth Barcan-Marcus, George J. W. Dorn, Paul Weingartner

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}.

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