منابع مشابه
The Axiom of Choice
We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...
متن کاملThe Axiom of Choice
We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...
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1. INTRODUCTION. In this paper we introduce the reader to two remarkable results in the theory of sets. Both are more than fifty years old, but neither one appears to be well known among nonspecialists. Each one states that a certain proposition implies the Axiom of Choice. First we describe the results, then review definitions, then, finally, present the proofs, most of which are straightforwa...
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This theorem is well known to be equivalent to the axiom of choice (though there does not seem to be a proof of this fact in the literature) and it has been suggested as an alternative for this axiom. The purpose of this note (which is purely methodological) is to propose a simpler but equivalent formulation of (A) as a substitute for the Zermelo axiom. The simplicity lies in the fact that we m...
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We present two proofs, one proof-theoretic and one model-theoretic, showing that 0 adding the BE1-collection axioms to any bounded first-order theory R of arithmetic 0 yields an extension which is VC1-conservative over R. Preliminaries. A theory of arithmetic R contains the non-logic symbols 0, S, +, = , and 6 . R may contain further non-logical symbols; in particular, S2 is a theory of arithme...
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ژورنال
عنوان ژورنال: Notre Dame Journal of Formal Logic
سال: 1960
ISSN: 0029-4527
DOI: 10.1305/ndjfl/1093956555