arXiv subjects

Paul Gorbow

Publications and source records attributed to Paul Gorbow.

5 recordsLinked to original sources

arXiv2026

Feasible constructivism

Dummett's argument for intuitionism is well known. There is a concern that the argument proves too much, specifically, that it supports the extreme and apparently incoherent position of strict finitism. The central question is how to explicate the notion that it is possible in practice to construct an arithmetical term or verify a statement. The strict finitist answer is plagued by the sorites paradox. We propose and develop feasibilism as a more plausible view, where computational feasibility, as captured by the class of polynomial-time problems, yields a robust and expedient explication of "possible in practice". In this approach, the complexity is bounded by a polynomial function of the input size, rather than bounded by a constant (as in strict finitism), thus resolving the sorites issues. We show that a system of strictly bounded arithmetic, introduced by Sam Buss, precisely formalizes the feasibilist view so as to satisfy Dummett's requirements.

math.LO↗

arXiv2026

Intentic Semantics for Potentialist Truthmaking

This draft introduces the technical machinery of a semantic framework for potentialist truthmaking based on our innovation of intentic states, which are structured partial models accounting for our distinction between non-hypothetical and hypothetical reasoning. The framework is developed for first-order logic in a purely relational language and is compatible with both classical and intuitionistic settings. Truthmaking is defined via a recursive construction over intentic states, yielding a semantic consequence relation that is shown to be sound and complete with respect to standard natural deduction. The resulting structure supports two natural extension relations, corresponding to truthmaking growth and hypothetical refinement, which are shown to satisfy the axioms governing Linnebo's bi-modal potentialist semantics. Moreover, we investigate the computational properties of the non-hypothetical fragment of natural deduction. Motivated by proof-theoretic and semantic considerations, we formulate a conjecture that non-hypothetical logic is decidable over Peano Arithmetic in a purely relational axiomatization, and more ambitiously over any fixed Peano Arithmetic theorem taken as an additional axiom. A schematic proof-search procedure is drafted to support this conjecture, identifying structural sources of finiteness. While preliminary, this analysis suggests a strong subformula discipline for normal non-hypothetical proofs and provides a proof-theoretic foundation for future work.

cs.LO↗

arXiv2025

Development Processes

Throughout mathematics there are constructions where an object is obtained as a limit of an infinite sequence. Typically, the objects in the sequence improve as the sequence progresses, and the ideal is reached at the limit. I introduce a view that understands this as a development process by which a dynamic mathematical object develops teleologically. In particular, this paper elaborates and clarifies the intuition that such constructions operate on a single dynamic object that maintains its identity throughout the process, and that each step consists in a transformation of this dynamic object, rather than in a genesis of an entirely new static object. This view is supported by a general philosophical discussion, and by a formal modal first-order framework of development processes. In order to exhibit the ubiquity of such processes in mathematics, and showcase the advantages of this view, the framework is applied to wide range of examples: The set of real numbers, forcing extensions of models of set theory, non-standard numbers of arithmetic, the reflection theorem schema of set theory, and the revision semantics of truth. Thus, the view proposed promises to yield a unified dynamic ontology for infinitary mathematics.

math.LO↗

arXiv2023

A genuinely untyped solution to the knower paradoxes

Kaplan and Montague have showed that certain intuitive axioms for a first-order theory of knowledge, formalized as a predicate, are jointly inconsistent. Their arguments rely on self-referential formulas. I offer a consistent first-order theory solving these knower paradoxes, with the following main features: - It solves the knower paradoxes by providing a faithful formalization of the principle of veracity (that knowledge requires truth), using both a knowledge and a truth predicate. - It is genuinely untyped. I.e. it is untyped not only in the sense that it uses a single knowledge predicate applying to all sentences in the language (including sentences in which this predicate occurs), but in the sense that its axioms quantify over all sentences in the language, thus supporting comprehensive reasoning with untyped knowledge ascriptions. - Common knowledge predicates can be defined in the system using self-reference. This fact, together with the genuinely untyped nature of the system and a technique based on Löb's theorem, enables it to support comprehensive reasoning with untyped common knowledge ascriptions (without having any axiom directly addressing common knowledge).

math.LO↗

arXiv2016

Feferman's Forays into the Foundations of Category Theory

This paper is primarily concerned with assessing a set-theoretical system, $S^*$, for the foundations of category theory suggested by Solomon Feferman. $S^*$ is an extension of NFU, and may be seen as an attempt to accommodate unrestricted categories such as the category of all groups (without any small/large restrictions), while still obtaining the benefits of ZFC on part of the domain. A substantial part of the paper is devoted to establishing an improved upper bound on the consistency strength of $S^*$. The assessment of $S^*$ as a foundation of category theory is framed by the following general desiderata (R) and (S). (R) asks for the unrestricted existence of the category of all groups, the category of all categories, the category of all functors between two categories, etc., along with natural implementability of ordinary mathematics and category theory. (S) asks for a certain relative distinction between large and small sets, and the requirement that they both enjoy the full benefits of the $\mathrm{ZFC}$ axioms. $S^*$ satisfies (R) simply because it is an extension of NFU. By means of a recursive construction utilizing the notion of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this construction yields a lower bound on the consistency strength of $S^*$. We also exhibit a basic positive result for category theory internal to NFU that provides motivation for studying NFU-based foundations of category theory.

math.LO↗