# name
Higher Structural Reflection and Very Large Cardinals

# metadata.dc.description[].value
Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 2023-2024. Tutor: Joan Bagaria Pigrau

# metadata.dc.description.abstract[].value
One line of research in set theory aims at deriving large cardinal axioms from strengthened
forms of reflection principles. This research is often motivated by the foundational goal
of justifying the large cardinal axioms. The most comprehensive attempt in this direction is
the program of structural reflection (SR), initiated by Joan Bagaria, whose ultimate goal is
to formulate all large cardinal axioms as instances of a single, general structural reflection
principle that is conceptually compelling.
The basic version of SR already gives the hierarchy of large cardinals from supercompact
cardinals, through C(n)-extendible cardinals, up to Vopěnka’s Principle. A stronger version
of SR, the exact structural reflection principle (ESR), is studied by Bagaria and Philipp
Lücke, which gives almost huge cardinals, and beyond. However, ESR differs in form from
the basic version of SR, rather than being direct generalization of the same principle.
In this thesis we formulate the level by level version and the capturing version of SR
(CSR). CSR is a direct generalization of the basic version of SR. We introduce and study
the m-supercompact cardinals, the C(n)-m-fold extendible cardinals, and the capturing
version of VP, and show that the pattern of correspondence between large cardinals and the
basic version of SR also extends to the higher realm. We also apply our results to answer
several open questions concerning ESR. Finally, we note that CSR, when generalized to its
ω-version, leads to inconsistency.

# metadata.dc.source[].value
Màster Oficial - Pure and Applied Logic / Lògica Pura i aplicada

# metadata.dc.title[].value
Higher Structural Reflection and Very Large Cardinals