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Constructive Set Theories

Jalali Keshavarz, Raheleh | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 45440 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematics Sciences
  6. Advisor(s): Ardeshir, Mohammad
  7. Abstract:
  8. One of the most important variants of classical logic is intuitionistic logic, which has arisen from the contemporary studies in the foundation of mathematics. Suppose we want to investigate the foundation of mathematics from a constructive point of view. For this purpose, we can choose two different ways. The first one is to modify the classical set theory, ZFC, in a way that is valid from a constructive point of view and is enough to formulate constructive mathematics, as well. The second approach, which is somehow different from the first one, is a foundational and dependent codification of the same foundation in a way that completely reflects constructive views.In this work we will investigate the first approach, which will result constructive set theories such as CZF and IZF. (If we choose the second approach, Martin Löf’s type heory will arise.) Constructive set theories are fundamental approaches to constructive mathematics.The language of these theories is the same language as the classical set theory,which is strengthened with a set of axioms and constructive logic as the base logic. Furthermore, since these theories do not explicitly use Brouwerian “types”,the obtained mathematics is so much like the classical mathematics based on the Zermelo-Frankel axioms and hence they are more user friendly than the theories which are obtained from the second approach. For this purpose, John Myhill in 1973 presented a set theory based on intuitionistic logic nd used the most famous foundation, ZFC.In this work we will concentrate on set theories based on constructive logic and we will nvestigate the studies on this subject.
  9. Keywords:
  10. Intuitionistic Logic ; Martin-Lof Theory ; Constructive Sets Theory ; Classic Sets Theory ; Intuitionistic Sets Theory

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