Juan P. Aguilera
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View article: Large cardinals beyond HOD
Large cardinals beyond HOD Open
Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the se…
View article: The metamathematics of separated determinacy
The metamathematics of separated determinacy Open
Determinacy axioms are mathematical principles which assert that various infinite games are determined. In this article, we prove three general meta-theorems on the logical strength of determinacy axioms. These allow us to reduce a metamat…
View article: Induction on Dilators and Bachmann-Howard Fixed Points
Induction on Dilators and Bachmann-Howard Fixed Points Open
One of the most important principles of J.-Y. Girard's $Π^1_2$-logic is induction on dilators. In particular, Girard used this principle to construct his famous functor $Λ$. He claimed that the totality of $Λ$ is equivalent to the set exis…
View article: Monotone versus non‐monotone projective operators
Monotone versus non‐monotone projective operators Open
For a class of operators , let denote the closure ordinal of ‐inductive definitions. We give upper bounds on the values of and under the assumption that all projective sets of reals are determined, significantly improving the known results…
View article: Large cardinals, structural reflection, and the HOD Conjecture
Large cardinals, structural reflection, and the HOD Conjecture Open
We introduce exacting cardinals and a strengthening of these, ultraexacting cardinals. These are natural large cardinals defined equivalently as weak forms of rank-Berkeley cardinals, strong forms of Jónsson cardinals, or in terms of princ…
View article: The Limits of Determinacy in Higher-Order Arithmetic
The Limits of Determinacy in Higher-Order Arithmetic Open
We prove level-by-level upper and lower bounds on the strength of determinacy for finite differences of sets in the hyperarithmetical hierarchy in terms of subsystems of finite-and transfinite-order arithmetic, extending the Montalbán-Shor…
View article: Fundamental Logic Is Decidable
Fundamental Logic Is Decidable Open
It is shown that Holliday’s propositional Fundamental Logic is decidable in polynomial time and that first-order Fundamental Logic is decidable in double-exponential time. The proof also yields a double-exponential–time decision procedure …
View article: The Logic of Correct Models
The Logic of Correct Models Open
For each $n\in\mathbb{N}$, let $[n]ϕ$ mean "the sentence $ϕ$ is true in all $Σ_{n+1}$-correct transitive sets." Assuming Gödel's axiom $V = L$, we prove the following graded variant of Solovay's completeness theorem: the set of formulas va…
View article: Reflection Properties of Ordinals in Generic Extensions
Reflection Properties of Ordinals in Generic Extensions Open
We study the question of when a given countable ordinal $α$ is $Σ^1_n$- or $Π^1_n$-reflecting in models which are neither $\mathsf{PD}$ models nor the constructible universe, focusing on generic extensions of $L$. We prove, amongst other t…
View article: Gödel-Dummett linear temporal logic
Gödel-Dummett linear temporal logic Open
We investigate a version of linear temporal logic whose propositional fragment is Gödel-Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: firs…
View article: Modern perspectives in Proof Theory
Modern perspectives in Proof Theory Open
Open AccessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Aguilera J. P., Pakhomov F. and Weiermann A. 2023Modern perspectives in Proof TheoryP…
View article: The Löwenheim-Skolem theorem for Gödel logic
The Löwenheim-Skolem theorem for Gödel logic Open
We prove the following Löwenheim-Skolem theorems for first-order Gödel logic: For the Gödel logic G[0,1], a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω(=sup{ℵ0,2ℵ0,…}). For an arbitr…
View article: The Π21$\Pi ^1_2$ consequences of a theory
The Π21$\Pi ^1_2$ consequences of a theory Open
We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond the ordinal numbers, resulting in an analog of ordinal analysis aimed at the study of theorems of complexity . This is done by replacing the use…
View article: Locally hyperarithmetical induction
Locally hyperarithmetical induction Open
We compute the closure ordinals of hyperarithmetical inductive definitions of sets of integers and of locally hyperarithmetical inductive definitions of sets of integers.
View article: A Gödel Calculus for Linear Temporal Logic
A Gödel Calculus for Linear Temporal Logic Open
We consider Gödel temporal logic ($\sf GTL$), a variant of linear temporal logic based on Gödel--Dummett propositional logic. In recent work, we have shown this logic to enjoy natural semantics both as a fuzzy logic and as a superintuition…
View article: Time and Gödel: Fuzzy temporal reasoning in PSPACE
Time and Gödel: Fuzzy temporal reasoning in PSPACE Open
We investigate a non-classical version of linear temporal logic whose propositional fragment is Gödel--Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural …
View article: Ackermann and Goodstein go functorial
Ackermann and Goodstein go functorial Open
We present variants of Goodstein's theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that t…
View article: The $Π^1_2$ Consequences of a Theory
The $Π^1_2$ Consequences of a Theory Open
We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity $Π^1_2$. This is done by replacing the …
View article: The $\Pi^1_2$ Consequences of a Theory
The $\Pi^1_2$ Consequences of a Theory Open
We develop the abstract framework for a proof-theoretic analysis of theories\nwith scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis\naimed at the study of theorems of complexity $\\Pi^1_2$. This is done by\nreplacin…
View article: Long Games and $\sigma$-Projective Sets.
Long Games and $\sigma$-Projective Sets. Open
We prove a number of results on the determinacy of $\\sigma$-projective sets\nof reals, i.e., those belonging to the smallest pointclass containing the open\nsets and closed under complements, countable unions, and projections. We first\np…
View article: Long Games and $σ$-Projective Sets
Long Games and $σ$-Projective Sets Open
We prove a number of results on the determinacy of $σ$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the …
View article: GAMES AND REFLECTION IN
GAMES AND REFLECTION IN Open
We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a tr…
View article: THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY
THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY Open
We determine the consistency strength of determinacy for projective games of length ω 2 . Our main theorem is that $\Pi _{n + 1}^1 $ -determinacy for games of length ω 2 implies the existence of a model of set theory with ω + n Woodin card…
View article: Determined admissible sets
Determined admissible sets Open
It is shown, from hypotheses in the region of Woodin cardinals, that there is a transitive model of Kripke–Platek set theory containing in which all games on are determined.
View article: Projective Games on the Reals
Projective Games on the Reals Open
Let $M^\sharp_n(\mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(\mathbb{R})$ the class-sized model obtained by iterating th…
View article: Long Borel Games
Long Borel Games Open
It is shown that Borel games of length $\omega^2$ are determined if, and only if, for every countable ordinal $\alpha$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $\alpha$-many iterated powerse…
View article: $F_\sigma$ Games and Reflection in $L(\mathbb{R})$
$F_\sigma$ Games and Reflection in $L(\mathbb{R})$ Open
It is shown that determinacy of $F_\\sigma$ games of length $\\omega^2$ is\nequivalent to the existence of a transitive model of KP + AD which contains the\nreals and reflects $\\Pi_1$ facts about the next admissible set.\n
View article: STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES
STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES Open
Given a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ , we define a topology $\tau _{ + \lambda } $ in such a way that τ + 0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the res…