Zach Weber

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Extensionality and restriction in naive set theory

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

What Is an Inconsistent Truth Table

ABSTRACT Do truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions.

Real Analysis in Paraconsistent Logic

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

Explanation And Solution In The Inclosure Argument

In a recent article, Emil Badici contends that the inclosure schema substantially fails as an analysis of the paradoxes of self-reference because it is question-begging. The main purpose of this note is to show that Badicis critique highlights a necessity condition for the success of dialectic about paradoxes. The inclosure argument respects this condition and remains solvent.

Notes on Inconsistent Set Theory

The purpose of this paper is to highlight and discuss two ideas that play in to the axiomatic development of a paraconsistent naive set theory. We will focus on aspects of the theory that can be read right off the axioms, concerning intensional identity and unrestricted set existence. Both relate to inconsistency. To begin I lay out a relevant background logic, placing a strong emphasis on the restrictions such a logic must have in order to support an inconsistent set theory. The sections that follow proceed on the understanding that, while highly inconsistent, a good deal of control is being exerted on the theory through the weakened logic. The two features of a fully naive theory, identity and self-reference, dovetail throughout.

On closure and truth in substructural theories of truth

Closure is the idea that what is true about a theory of truth should be true (and therefore expressible) in it. Commitment to closure under truth motivates non-classical logic; commitment to closure under validity leads to substructural logic (nontransitive or noncontractive). These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will consider whether a noncontractive approach faces a similar revenge problem with respect to closure under provability, and argue that if a noncontractive theory is to be genuinely closed, then it must be free of all contraction, even in the metatheory.

Figures, Formulae, and Functors

This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs—which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the so-called ‘maverick’ approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and (b) category theory itself addresses questions of representation and information preservation over mappings. We obtain a mathematical answer to a philosophical question: a good mathematical representation can be characterized as a category theoretic natural transformation. Assuming that this is not some reductio against the maverick approach to these issues, this in turn moots some of the disagreement in the philosophical debate and provides better questions with which to go on.

Review of 'Inconsistent Geometry', by Chris Mortensen

and causal facts. I’m pretty sceptical about what strikes me as a reification of the pragmatic side of explanation, and maybe Daly hasn’t devoted as much space to that scepticism as I might have, but that’s not something to pursue in the context of this review. Rather, it’s that there is a methodological question that this raises which is much neglected. How can you argue for one view rather than the other? You can’t use the principles of explanatory virtue, because that is precisely what is at issue. Perhaps you can use other non-deductive argument forms of the kind discussed. But how do we evaluate them? Should philosophical methodology be treated as a kind of package, where we should look for the most coherent whole? What if some methodologies are neither supported nor undermined by others? There is a general question of meta-methodology that really deserves consideration. In any case, this book is very welcome. There’s nothing quite like it and, for its scope and its clear and balanced approach, I think it’s something that every advanced undergraduate would benefit from reading, and many other philosophers as well. I’ll certainly be trying it out as the basis for a course.