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Publication - Professor Philip Welch

    Proving Theorems from Reflection

    Citation

    Welch, PD, 2018, ‘Proving Theorems from Reflection’. in: S Centron, D Sarikaya, D Kant (eds) Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer, Cham

    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.

    Full details in the University publications repository