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Reduced Set Theory

Matthias Kunik

Abstract

We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation and with only finite or countably infinite ordinals. Still a considerable part of mathematics can be formalized within this system.

Reduced Set Theory

Abstract

We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation and with only finite or countably infinite ordinals. Still a considerable part of mathematics can be formalized within this system.
Paper Structure (5 sections, 19 theorems, 46 equations)

This paper contains 5 sections, 19 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Some notations and constructions with pure sets
  3. Formal mathematical systems
  4. The system RST of reduced set theory
  5. Models for RST

Key Result

Theorem 2.2

The following statements are equivalent for any set $T$. For every set $A$ the so called transitive closure $\mathcal{TC}(A)$ of $A$ is a transitive set such that $A \subseteq \mathcal{TC}(A)$, the smallest transitive set $T$ with $A \subseteq T$, i.e. $\mathcal{TC}(A) = \cap\,(\,\{ T \,:\,A \subseteq T \textrm{ and } T \textrm{ is a transitive set}\,\}\

Theorems & Definitions (44)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Definition 3.1
  • ...and 34 more