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

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, $0^{\#}$.