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No minimal transitive model of Z−

✍ Scribed by W. Marek; M. Srebrny

Publisher
John Wiley and Sons
Year
1975
Tongue
English
Weight
279 KB
Volume
21
Category
Article
ISSN
0044-3050

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✦ Synopsis

NO MINIMAL TRANSITIVE MODEL OF Z-

by W. MABEK and M. SREBRNY in Warszawa (Poland) 1 1 1 this paper we compare the strength of higher order arithmetics, fragments of ZEHMELO-FRAENKEL set theory and of KEUEY-MORSE theory of classes. We prove t h a t 2is H conservative extension of second order arithmetic (without! choice) and that thew is no minimal transitive model of Z-.

1. Preliminaries

8 , denotes second-order arithmetic as formulated in SHOENFIELD [6 1. A, denotes full scwnd-order arithmetic, i.e. A, with the following schema of choice

w h e ~* t ~ l.('') = {na : J(n, m ) E Y ) and J ( . , .) is a fixed pairing function for natural numbers. %-is ZERMELO set theory without the power sct axiom. ZFis ZERMELO-FRAENKEL set, thcwy without the power set axiom. C denotes the following scheme of substitutionclioic*c~ : ( x ) ~ ( E z ) @(x, z ) + ( E l ) (Fund/) & (lorn(/) = y ( x ) ~ @ ( x , 1 ~) ) .

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Transitive Permutation Groups of Minimal
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✍ Avinoam Mann; Cheryl E. Praeger 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 133 KB

For a permutation group G on a set S, the mo¨ement of G is defined as the maximum cardinality of subsets T of S for which there exists an element x g G x Ž such that T is disjoint from its translate T that is, when such subsets have . bounded cardinality . It was shown by the second author that, if