TY - JOUR

T1 - Applicable mathematics in a minimal computational theory of sets

AU - Avron, Arnon

AU - Cohen, Liron

N1 - Publisher Copyright:
© A. Avron and L. Cohen.

PY - 2018

Y1 - 2018

N2 - In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored. In this work we first improve that framework by enriching it with means for coherently extending by definitions its theories, without destroying its static nature or violating any of the principles on which it is based. Then we turn to investigate within the enriched framework the power of the minimal (predicatively acceptable) theory in it that proves the existence of infinite sets. We show that that theory is a computational theory, in the sense that every element of its minimal transitive model is denoted by some of its closed terms. (That model happens to be the second universe in Jensen’s hierarchy.) Then we show that already this minimal theory suffices for developing very large portions (if not all) of scientifically applicable mathematics. This requires treating the collection of real numbers as a proper class, that is: a unary predicate which can be introduced in the theory by the static extension method described in the first part of the paper.

AB - In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored. In this work we first improve that framework by enriching it with means for coherently extending by definitions its theories, without destroying its static nature or violating any of the principles on which it is based. Then we turn to investigate within the enriched framework the power of the minimal (predicatively acceptable) theory in it that proves the existence of infinite sets. We show that that theory is a computational theory, in the sense that every element of its minimal transitive model is denoted by some of its closed terms. (That model happens to be the second universe in Jensen’s hierarchy.) Then we show that already this minimal theory suffices for developing very large portions (if not all) of scientifically applicable mathematics. This requires treating the collection of real numbers as a proper class, that is: a unary predicate which can be introduced in the theory by the static extension method described in the first part of the paper.

UR - http://www.scopus.com/inward/record.url?scp=85060223121&partnerID=8YFLogxK

U2 - 10.23638/LMCS-14(4:1)2018

DO - 10.23638/LMCS-14(4:1)2018

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AN - SCOPUS:85060223121

SN - 1860-5974

VL - 14

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

IS - 4

M1 - 1

ER -