The goal of this talk will be to present the philosophical consequences of a new Synthetic Model of Charles Sanders Peirce's Continuum. This model was built by the second author. As far as we know this is the first mathematical model that captures the main properties of Peirce's Continuum[1] namely: 1) Inextensibility; 2) Reflexivity; 3) Supermultitudiness and weldedness; 4) Modality and Potentiality; 5) Genericity.

Via this model, we first intend to show the consistency of these properties and how they are actually realized on it.

Quite unexpectedly, the said Model is constructed in the context of ZFC (Zermelo-Fraenkel axioms for set theory), which is the most widespread framework for mathematics. In fact, the said Model uses ZFC to obtain, through an appropriate interpretation of the construction, a Synthetic Model of the Continuum despite the analytical character of ZFC itself.

In short, the Model is built with set theoretic tools, but using MereoTopological concepts and interpretations, together with a key inversion idea of the set-theoretical membership relation.

From this, we will finally point to the fact that there are far reaching implications for the philosophy of mathematics and the foundations of Geometry.