The wonder of colors and the principle of Ariadne

Walter Carnielli

Carlos di Prisco

pp. 309-317

in: Marcos Silva (ed), How colours matter to philosophy, Berlin, Springer, 2017

Abstract

This paper surveys some results on combinatorial aspects of infinite Ramsey-type problems inspired by finite properties, and intends to explain the relevance of an alternative set-theoretical principle formulated in the language of colors, the so-called Principle of Ariadne. This principle, intended to be a rival of the Axiom of Choice, can be consistently added to the usual axiom stock of ZF set theory under certain conditions. Such a new axiom, which preserves all the finite contents of mathematics but deviates from the standard in the infinite contents, may help us to understand the finite-infinite divide in mathematics, making clear that there is more than one way to generalize from finite principles of order (or choice) to the infinite. In other words, several infinite principles are possible starting from the same finitary content.