Axiomatic Set Theory à la Dijkstra and Scholten

  • Ernesto Acosta
  • Bernarda Aldana
  • Jaime Bohórquez
  • Camilo Rocha
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 735)

Abstract

The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents \(\mathsf {Set}\), a first-order logic axiomatization for set theory using the approach of Dijkstra and Scholten. What is novel about the approach presented in this paper is that symbolic manipulation of formulas is an effective tool for teaching an axiomatic set theory course to sophomore-year undergraduate students in mathematics. This paper contains many examples on how argumentative proofs can be easily expressed in \(\mathsf {Set}\) and points out how the rigorous approach of \(\mathsf {Set}\) can enrich the learning experience of students. The results presented in this paper are part of a larger effort to formally study and mechanize topics in mathematics and computer science with the algebraic approach of Dijkstra and Scholten.

Keywords

Axiomatic set theory Dijkstra-Scholten logic Derivation Formal system Zermelo-Fraenkel (ZF) Symbolic manipulation Undergraduate-level course 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ernesto Acosta
    • 1
  • Bernarda Aldana
    • 1
  • Jaime Bohórquez
    • 1
  • Camilo Rocha
    • 2
  1. 1.Escuela Colombiana de Ingeniería Julio GaravitoBogotáColombia
  2. 2.Pontificia Universidad JaverianaCaliColombia

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