arXiv subjects

Antonio Montalbán

Publications and source records attributed to Antonio Montalbán.

17 recordsLinked to original sources

arXiv2026

The barrier Ramsey theorem

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of $α$-largeness, where $α$ is a countable ordinal equipped with a system of fundamental sequences. To extend this approach the more appropriate notion is barrier largeness. Since the complexity of barriers can be measured by countable ordinals, we define Ramsey ordinals and, using appropriate iterations of the Veblen functions, we are able to compute them.

math.LO↗

arXiv2022

The $ω$-Vaught's Conjecture

We introduce the $ω$-Vaught's conjecture, a strengthening of the infinitary Vaught's conjecture. We believe that if one were to prove the infinitary Vaught's conjecture in a structural way without using techniques from higher recursion theory, then the proof would probably be a proof of the $ω$-Vaught's conjecture. We show the existence of an equivalent condition to the $ω$-Vaught's conjecture and use this tool to show that all infinitary sentences whose models are linear orders satisfy the $ω$-Vaught's conjecture.

math.LO↗

arXiv2022

The structural complexity of models of arithmetic

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $ω$ and that non-standard models of true arithmetic must have Scott rank greater than $ω$. Other than that there are no restrictions. By giving a reduction via $Δ^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $ω$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $α>ω$ is realized as the Scott rank of a model of $T$.

math.LO↗

arXiv2022

Herd immunity under individual variation and reinfection

We study a SEIR model considered by Gomes et al. \cite{Gomes2020} and Aguas et al. \cite{Aguas2020} where different individuals are assumed to have different levels of susceptibility or exposure to infection. Under this heterogeneity assumption, epidemic growth is effectively suppressed when the percentage of population having acquired immunity surpasses a critical level - the herd immunity threshold - that is lower than in homogeneous populations. We find explicit formulas to calculate herd immunity thresholds and stable configuration, and explore extensions of the model.

q-bio.PE↗

arXiv2021

A Feiner Look at the Intermediate Degrees

We say that a set $S$ is $Δ^0_{(n)}(X)$ if membership of $n$ in $S$ is a $Δ^0_{n}(X)$ question, uniformly in $n$. A set $X$ is low for $Δ$-Feiner if every set $S$ that is $Δ^0_{(n)}(X)$ is also $Δ^0_{(n)}(\emptyset)$. It is easy to see that every low$_n$ set is low for $Δ$-Feiner, but we show that the converse is not true by constructing an intermediate c.e. set that is low for $Δ$-Feiner. We also study variations on this notion, such as the sets that are $Δ^0_{(bn+a)}(X)$, $Σ^0_{(bn+a)}(X)$, or $Π^0_{(bn+a)}(X)$, and the sets that are low, intermediate, and high for these classes. In doing so, we obtain a result on the computability of Boolean algebras, namely that there is a Boolean algebra of intermediate c.e. degree with no computable copy.

math.LO↗

arXiv2020

The determined property of Baire in reverse math

We define the notion of a determined Borel code in reverse math, and consider the principle $DPB$, which states that every determined Borel set has the property of Baire. We show that this principle is strictly weaker than $ATR$. Any $ω$-model of $DPB$ must be closed under hyperarithmetic reduction, but $DPB$ is not a theory of hyperarithmetic analysis. We show that whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $DPB$, then for every $Z \in M$, there is a $G \in M$ such that $G$ is $Δ^1_1$-generic relative to $Z$.

math.LO↗

arXiv2019

On the inevitability of the consistency operator

We examine recursive monotonic functions on the Lindenbaum algebra of $\mathsf{EA}$. We prove that no such function sends every consistent $φ$ to a sentence with deductive strength strictly between $φ$ and $(φ\wedge\mathsf{Con}(φ))$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function $f$, if there is an iterate of $\mathsf{Con}$ that bounds $f$ everywhere, then $f$ must be somewhere equal to an iterate of $\mathsf{Con}$.

math.LO↗

arXiv2017

On the structure of the Wadge degrees of BQO-valued Borel functions

In this article, we give a full description of the Wadge degrees of Borel functions from $ω^ω$ to a better quasi ordering $\mathcal{Q}$. More precisely, for any countable ordinal $ξ$, we show that the Wadge degrees of $\mathbfΔ^0_{1+ξ}$-measurable functions $ω^ω\to\mathcal{Q}$ can be represented by countable joins of the $ξ$-th transfinite nests of $\mathcal{Q}$-labeled well-founded trees.

math.LO↗

arXiv2016

The uniform Martin's conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them "on a cone." We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^ω$ and Wadge degrees of functions $ω^ω\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

math.LO↗

arXiv2016

Borel Functors and Infinitary Interpretations

We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: Every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show the complexities are maintained in the sense that if the functor is $\mathbfΔ^0_α$, then the interpretation that induces it is $Δ^{\mathtt{in}}_α$ up to $\mathbfΔ^0_α$ equivalence.

math.LO↗

arXiv2015

Independence in computable algebra

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.

math.LO↗

arXiv2015

Computable functors and effective interpretability

Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.

math.LO↗

arXiv2013

Classes of structures with no intermediate isomorphism problems

We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle on a cone.

math.LO↗

arXiv2013

Analytic equivalence relations satisfying hyperarithmetic-is-recursive

We prove, in ZF+$\bfΣ^1_2$-determinacy, that for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_E$ we have that a real $X$ computes a member of the equivalence class if and only if $\om_1^X\geq\om_1^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$.

math.LO↗

arXiv2013

Priority arguments via true strages

We describe a variation of Ash's $η$-system, and give a new proof of Ash's metatheorem. As an application, we prove a generalization of Ash and Knight's theorem on pairs of structures.

math.LO↗

arXiv2012

Computing Maximal Chains

In 1967 Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk's original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets.

math.LO↗

arXiv2010

The Veblen functions for computability theorists

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where alpha is a fixed computable ordinal and phi the two-placed Veblen function. For the former statement, we show that omega iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA_0^+ over RCA_0. To prove the latter statement we need to use omega^alpha iterations of the Turing jump, and we show that the statement is equivalent to Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.

math.LO↗