arXiv subjects

J. P. Aguilera

Publications and source records attributed to J. P. Aguilera.

4 recordsLinked to original sources

arXiv2022

Functorial Fast-Growing Hierarchies

Fast-growing hierarchies are sequences of functions obtained through various processes similar to the ones that yield multiplication from addition, exponentiation from multiplication, etc. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. We show that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by ordinal collapsing functions, thus establishing a connection between two fundamental concepts in Proof Theory. Using this fact, we obtain a restatement of the subsystem $Π^1_1$-CA$_0$ of analysis as a higher-type wellordering principle.

math.LO↗

arXiv2019

Long Borel Games

It is shown that Borel games of length $ω^2$ are determined if, and only if, for every countable ordinal $α$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $α$-many iterated powersets above a limit of Woodin cardinals.

math.LO↗

arXiv2019

$F_σ$ Games and Reflection in $L(\mathbb{R})$

It is shown that determinacy of $F_σ$ games of length $ω^2$ is equivalent to the existence of a transitive model of KP + AD which contains the reals and reflects $Π_1$ facts about the next admissible set.

math.LO↗

arXiv2019

The Order of Reflection

Extending Aanderaa's classical result that $π^1_1<σ^1_1$, we determine the order between any two patterns of iterated $Σ^1_1$- and $Π^1_1$-reflection. We show that this \emph{linear reflection order} is a prewellordering of length $ω^ω$. This requires considering the relationship between linear and some \emph{non-linear} reflection patterns, such as $σ^1_1\wedgeπ^1_1$, the pattern of simultaneous $Σ^1_1$- and $Π^1_1$-reflection.

math.LO↗