A model of second-order arithmetic satisfying AC but not DC

Author:

Friedman Sy-David1,Gitman Victoria2,Kanovei Vladimir3

Affiliation:

1. Kurt Gödel Research Center, University of Vienna, Währinger Strasse 25 A-1090 Vienna, Austria

2. The City University of New York, CUNY Graduate Center, Mathematics Program, 365 Fifth Avenue, New York, NY 10016, USA

3. Institute for the Information Transmission Problems (IITP), 19, Build.1, Bolshoy Karetny per., Moscow 127051, Russia

Abstract

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 (1979)].

Publisher

World Scientific Pub Co Pte Lt

Subject

Logic

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