Generating Custom Set Theories with Non-set Structured Objects

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Abstract

Set theory has long been viewed as a foundation of mathematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising mathematics in set theory. We wish to have set theory’s features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs.

Original languageEnglish
Title of host publicationIntelligent Computer Mathematics. CICM 2021
EditorsFairouz Kamareddine, Claudio Sacerdoti Coen
PublisherSpringer
Chapter19
Pages228-244
Number of pages17
ISBN (Electronic)9783030810979
ISBN (Print)9783030810962
DOIs
Publication statusPublished - 2021

Publication series

NameLecture Notes in Computer Science
Volume12833
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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