AG Horsten Calendar Overview

Speaker:
Ismael Ordóñez Miguéns (University of Santiago de Compostela)

Set-theoretic potentialism is the philosophical position that arises from the combination of two different approaches towards quantification: an expansionist approach and a modal approach. According to the former, given a plurality of sets where the first-order quantifiers are interpreted, there is another set which is not in the plurality. Thus, the domain is “expanded” by adding the “new” set. In turn, the second  approach uses modal operators to describe this expansionist phenomenon. As a result, potentialism achieves a balance between two conflicting theses: (i) no plurality of sets contains every set, but (ii) we can achieve absolute quantification ―quantification over every set― by attaching modal operators to quantifiers.

Potentialism is very promising, as its mathematical strength reveals: the modal theory obtained to describe the expansionist process partially interprets Zermelo-Fraenkel set theory. However, the proposal has some limitations. Its bottom-up approach has difficulties in justifying the strongest axioms of this standard theory: the axioms of Infinity and Replacement. One way to overcome this limitation is to develop modal abstraction principles. In this talk, I will explain (i) how modal abstraction principles can be formalized, (ii) how they recover the axiom of Infinity, and (iii) how they relate to Cantor’s Principle. Finally, I will discuss the main drawbacks of the proposal: the collapse of absolute quantification and the potentialist translation.

Speaker:
Ismael Ordóñez Miguéns (University of Santiago de Compostela)

Set-theoretic potentialism is the philosophical position that arises from the combination of two different approaches towards quantification: an expansionist approach and a modal approach. According to the former, given a plurality of sets where the first-order quantifiers are interpreted, there is another set which is not in the plurality. Thus, the domain is “expanded” by adding the “new” set. In turn, the second  approach uses modal operators to describe this expansionist phenomenon. As a result, potentialism achieves a balance between two conflicting theses: (i) no plurality of sets contains every set, but (ii) we can achieve absolute quantification ―quantification over every set― by attaching modal operators to quantifiers.

Potentialism is very promising, as its mathematical strength reveals: the modal theory obtained to describe the expansionist process partially interprets Zermelo-Fraenkel set theory. However, the proposal has some limitations. Its bottom-up approach has difficulties in justifying the strongest axioms of this standard theory: the axioms of Infinity and Replacement. One way to overcome this limitation is to develop modal abstraction principles. In this talk, I will explain (i) how modal abstraction principles can be formalized, (ii) how they recover the axiom of Infinity, and (iii) how they relate to Cantor’s Principle. Finally, I will discuss the main drawbacks of the proposal: the collapse of absolute quantification and the potentialist translation.