Research

Plural Level Theory

My aim is to produce a cumulative hierarchy of sets that conceives stages as plurals rather than sets. To do so, the project is grounded on the approach to plural logic championed by Øystein Linnebo, Salvatore Florio and Sam Roberts and on the Level Theory recently developed by Tim Button.  So far I produced two plural level theories:

• PLT: for full plural logic (published, see above);

• CPLT: for critical plural logic (manuscript);

Contact me if you'd like to discuss it!

The (Pre-)History of the Iterative Conception of Set

I'm writing three papers on the pre-history of the Iterative Conception of Set and of the Axiom of Foundation, hopefully to become a book. 

• Part 1: the pre-history of the notion of layer, that is, from Russellian types to set-theoretic ranks and the cumulative hierarchy;

• Part 2: the pre-history of the Axiom of Foundation, the hallmarks of the iterative conception;

• Part 3: a comprehensive history and analysis of axiomatizations of the conceptions: stage and level theories and much more;

Contact me if you want to check them!

Axiomatizing the Weak Iterative Conception of Set

With Neil Barton we are axiomatizing the weak iterative conception of set in Gödel's constructibilist sense (i.e., a level theory for L) and in a countabilist sense. The model for the axiomatization is again Tim Button's Level Theory. 

As part of this project we are also drawing an explicit connection between Gödel's Constructible Universe and Russell's Ramified Type Theory. The link is similar to the one between the cumulative hierarchy V and Simple Type Theory, extensively analyzed by the literature on the iterative conception (see my Pre-History Part 1).

Is the Concept of Set Semantically Indeterminate?

With Pablo Dopico we are writing a paper arguing that set is a semantically indeterminate concept. The kind of indeterminacy we are after is labelled, after Dummett, "indefinite precisifiability". Our argument involves an axiomatic understanding of the extension of the set-concept and touches on large cardinals and the inner model program. A version of the paper is currently under review, contact us if you'd like to check it!