Jeff Schatz

This paper has two goals. First, we reconstruct Wittgenstein’s views on what counts as a legitimate irrational – since, as he repeatedly points out, and in agreement with mathematicians such as Emile Borel, not just every infinite string of digits qualifies as one. Once his conception (‘full-blooded intensionalism’) is sketched out, and its specificity is highlighted by comparing it with two other cognate views (‘extensionalism’ and ‘quasi-intensionalism’), our second objective is to examine how his type of intensionalism impacts his attitude towards Cantor’s theorem. In this regard, the more general claim we argue for is that, despite appearances to the contrary, Wittgenstein was not a revisionist about set-theoretical practice.

In recent years, set theory has been uniquely interesting for philosophers due to the presence of non-proof-based forms of reasoning not often found in other areas of mathematics. While many areas of ‘core’ mathematics take their axioms as given, and focus on proving theorems from these axioms, in set theory the matter of which axioms should be accepted remains a live research question. Historically, much of this effort focused on identifying and justifying the now-standard theory ZFC, or its widely accepted extension ZFC+LCs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZFC+LCs$$\end{document}; today, the aim of axiom selection has shifted to finding new, defensible axioms capable of settling independent questions like the Continuum Hypothesis. Given that traditional proof methods cannot be any help in defending new axioms candidates—since axioms must be accepted prior to any proving—an important epistemological question is what methods are used to justify axiom choices in set theory, and whether those methods are rationally defensible.

This paper examines contemporary approaches to maximization in axiom selection debates in set theory, applying these methods to the current debate between V=Ult(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=Ult(L)$$\end{document} and forcingForcing axioms. The paper first surveys the motivations for maximization as a tool in axiom selection debates, focusing in particular on the formal approach of Maddy. It then examines an important current debate in set-theoretic practiceSet-theoretic practice between strong axiom candidates capable of settling the value of the continuum: the inner modelInner model axiom V=Ult(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=Ult(L)$$\end{document} and strong forcingForcing axioms. Finally, the paper will apply the formal notions of maximizeMaximize to these axiom candidates, showing that–under highly plausible conjectures–forcingForcing axioms strictly maximize over V=Ult(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=Ult(L)$$\end{document}. The paper will then briefly conclude with an examination of the implications of this result for axiom selection as a whole.

This article examines Kevin Scharp’s formal solution to the alethic paradoxes, ADT, which stands for ascending and descending truth. One of the main supposed benefits of ADT over its competitors is that it alone can validate the uses of truth concepts in theoretical contexts, such as truth-theoretic semantics. The appendixes contain a new consistency proof for ADT, and additionally show that it is conservative. As a result of its conservativity, the article argues that ADT faces a problem in accounting for certain mathematical uses of truth. Thus, Scharp’s theory needs to be amended in order to fulfill its aim of replicating all substantive uses of truth.