A transcendental view on the continuum: Woodin's conditional platonism

TL;DR

This paper explores the philosophical and mathematical modeling of the continuum's intuitive, non-constructible nature within set theory, focusing on Woodin's approach and its relation to classical philosophical views.

Contribution

It proposes a framework for understanding the continuum's transcendence using Woodin's set-theoretic methods, bridging philosophical intuition and formal set theory.

Findings

Modeling of the continuum's transcendence within ZFC

Application of Woodin's large cardinal axioms and $oldsymbol{ ext{Ω}}$-logic

Insights into the philosophical implications of set-theoretic continuum

Abstract

One of the main difficulty concerning the nature of the continuum is to do justice, inside the set theoretical Cantorian framework, to the classical conception (from Aristotle to Thom, via Kant, Peirce, Brentano, Husserl and Weyl) according to which the continuum is a non-compositional, cohesive, primitive, and intuitive datum. This paper investigates such possibilities, from G\"{o}del to Woodin, of modelling inside a ZFC-universe the transcendence of the intuitive continuum w.r.t. its symbolic determination. Keywords: constructive universe, continuum, G\"{o}del, forcing, Kant, large cardinals, $\Omega$-logic, projective hierarchy, $V = L$, Woodin, $0^{\#}$.