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Proving Theorems from Reflection

Research output: Chapter in Book/Report/Conference proceedingChapter in a book

Original languageEnglish
Title of host publicationReflections on the Foundations of Mathematics
Subtitle of host publicationUnivalent Foundations, Set Theory and General Thoughts
EditorsS. Centron, D. Sarikaya, D. Kant
Publisher or commissioning bodySpringer, Cham
ISBN (Print)9783030156541
DateAccepted/In press - 23 Jul 2018

Publication series

NameSynthese Library in Philosophy
PublisherSpringer

Abstract

We review some fundamental questions concerning the real line of mathematical analysis, which, like the Continuum Hypothesis, are also independent of the axioms of set theory, but are of a less ‘problematic’ nature, as they can be solved by adopting the right axiomatic framework. We contend that any foundations for mathematics should be able to simply formulate such questions as well as to raise at least the theoretical hope for their resolution.

The usual procedure in set theory (as a foundation) is to add so-called strong axioms of in nity to the standard axioms of Zermelo-Fraenkel, but then the question of their justi cation becomes to some people vexing. We show how the adoption of a view of the universe of sets with classes, together with certain kinds of Global Reflection Principles resolves some of these issues.

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