# name
A classification of the set-theoretic total recursive functions of KPl

# metadata.dc.description[].value
Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 2023-2024. Tutor: Joost J. Joosten and Juan Pablo Aguilera

# metadata.dc.description.abstract[].value
The set theory KPl, which stands for Kripke-Platek-limit, roughly stipulates that
there are unboundedly many admissible sets. Admissible sets are models of the
Kripke-Platek set-theory KP which is a very weak fragment of ZFC. In [1], J. Cook
and M. Rathjen classify the provably total set functions in KP using a proof system
based on an ordinal notation system for the Bachmann-Howard ordinal relativized
to a fixed set. In this paper, we adapt this result to the KPl set theory. We consider
set functions which are provably total in KPl and Σ-definable by the same formula
in any admissible set. We prove that, if f is such a function then, for any set x
in the universe, the value f(x) always belongs to an initial segment of L(x), the
constructible hierarchy relativized to the transitive closure of x, at a level below
the relativized Takeuti-Feferman-Buchholz ordinal (the TFB ordinal is the prooftheoretic
ordinal of KPl).
To prove this result, we first construct an ordinal notation system based on [2] for
KPl relativized to a fixed set that we will use in order to build a logic dependent on
this fixed set where we will embed KPl. Thanks to this relativized system, we will
be able to bound the value of the function at this fixed set.

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

# metadata.dc.title[].value
A classification of the set-theoretic total recursive functions of KPl