adiazroj57@alumnes.ub.edu
Universitat de Barcelona
Departament de Filosofia
C/Montalegre, 6-8, 4th floor
Office: 4013
08001, Barcelona

Mathematical Logic and Foundations

In mathematical logic, I am presently interested in descriptive set theory. Descriptive set theory is the study of definable subsets of ℝ, and thereby of Polish spaces and their respective hierarchies of Borel sets. I am particularly interested in the applications of descriptive set theory in classifying the complexity of spacetime manifolds. I am working on something of this kind for Lorentzian manifolds which are also Einstein manifolds (Λ-vacuum solutions). I wish to explore the extent to which this can be done without the full Axiom of Choice.

Philosophical Logic

In philosophical logic, I am interested in formal theories of truth. I am particularly interested in the formalisation of the T-predicate in first-order theories of arithmetic and describing their class of fixed point models. My doctoral thesis is dedicated to the construction of fixed-point models for the T-predicate without the Axiom of Choice. It is well known that Dependent Choice (DC_ω) is sufficient for defining ω-sequences by transfinite recursion. I conjecture, and am working towards proving, that DC_ω is sufficient to regiment fixed-point models of the T-predicate and obtain all the results typically desired in this area. Thus our results will be consistent with the Solovay Model L(ℝ)^V[G] in which every set of reals is Lebesgue-measurable. I believe this might change the way we think about truth and solutions to certain semantic paradoxes.

Moreover, generalised descriptive set theory is the study dedicated to the generalisation of regularity properties for subsets of ℝ to subsets of κ^κ, for some cardinal κ. Therewith, Higher Solovay Models of the form L(V_κ+1)^M can be obtained in which Generalised Dependent Choice (DC_κ) holds [see Straffelini, C. & Thei, S. https://arxiv.org/pdf/2507.19129]. Moreover, (DC_κ) is sufficient for defining κ-sequences by transfinite recursion. Thus, so long as the choice of κ is not arbitrary, this does not appeal to the full Axiom of Choice. I therefore additionally aim to explore 'generalised' fixed-point models of the T-predicate using (DC_κ) to obtain results about the T-predicate in the higher transfinite hierarchy, still in a manner consistent with Higher Solovay Models, and thus still without appeal to the Axiom of Choice.

My project is funded by a PREDOCS-UB grant, and is supervised by José Martínez Fernández.

Aside from research, I like classical music, literature, history, non-analytic philosophy, art, sports, etc.

• Law, Bachelor of Laws - Queen Mary, University of London (2018 - 2022)
• Philosophy, Master of Arts - King's College London (2022 - 2023)
• Mathematical Logic, Master of Science - University of Barcelona (present)
• Analytic Philosophy (CCiL), Ph. D - University of Barcelona (present)