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Franklin D. Tall

Publications and source records attributed to Franklin D. Tall.

At least 19 recordsLinked to original sources
arXiv2026

Complexity of deep computations via topology of function spaces

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.

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arXiv2024

On the undefinability of pathological Banach spaces

Motivated by Tsirelson's implicitly defined pathological Banach space, T. Gowers asked whether explicitly defined Banach spaces must include either $c_0$ or some $\ell^p$. J. Iovino and P. Casazza gave an affirmative answer for first-order continuous logic. We greatly extend their work to logics with much weaker requirements than compactness on their type spaces. Noteworthy is our extensive use of the topology of function spaces ($C_p$-theory) as developed by Arhangel'skii, and our use of double limit conditions studied by H. König and N. Kuhn.

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arXiv2023

An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of $σ$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.

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arXiv2022

A new topological generalization of descriptive set theory

We introduce a new topological generalization of the $σ$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^ωω$ by $^κκ$, for $κ$ regular uncountable, or replaced countable by $σ$-discrete. Instead we close the usual $σ$-projective sets under continuous images and perfect preimages together with countable unions. The natural set-theoretic axiom to apply is $σ$-projective determinacy, which follows from large cardinals. Our goal is to generalize the known results for $K$-analytic spaces (continuous images of perfect preimages of $^ωω$) to these more general settings. We have achieved some successes in the area of Selection Principles--the general theme is that nicely defined Menger spaces are Hurewicz or even $σ$-compact. The $K$-analytic results are true in ZFC; the more general results have consistency strength of only an inaccessible.

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arXiv2021

$C_p$-Theory for Model Theorists

We present applications of $C_p$-theory, the branch of general topology concerned with spaces of real-valued continuous functions, to model theory, mostly in the context of continuous logics. We include $C_p$-theoretic results and proofs in a self-contained way for model theorists who are not familiar with the techniques of this field. We further generalize some results of Casazza and Iovino, and of the authors, involving the definability of Banach spaces including isomorphic copies of $c_0$ or $\ell^p$, after a problem posed by Odell and Gowers.

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arXiv2020

Countable Tightness and the Grothendieck Property in $C_p$-Theory

The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangel'ski\uı by proving it undecidable whether countably tight spaces with Lindelöf finite powers are Grothendieck. We answer another of his questions by proving that $\mathrm{PFA}$ implies Lindelöf countably tight spaces are Grothendieck. We also prove that various other consequences of $\mathrm{MA}_{ω_1}$ and $\mathrm{PFA}$ considered by Arhangel'ski\uı, Okunev, and Reznichenko are not theorems of $\mathrm{ZFC}$.

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arXiv2020

The Strength of Menger's Conjecture

Menger conjectured that subsets of R with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Menger's conjecture for projective sets has consistency strength of only an inaccessible cardinal.

math.GN↗
arXiv2020

Model Theory for $C_p$-theorists

We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and $C_p$-theory which have applications to a classification problem involving Banach spaces not including $c_0$ or $l^p$, following recent results obtained by P. Casazza and J. Iovino for compact continuous logics. Using $C_p$-theoretic results involving Grothendieck spaces and double limit conditions, we extend their results to a broader family of logics, namely those with a first countable weakly Grothendieck space of types. We pose $C_p$-theoretic problems which have model-theoretic implications.

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arXiv2018

The Open Graph Axiom and Menger's Conjecture

Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Menger's conjecture restricted to this family of subsets of $\mathbb R$.

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arXiv2018

Completely Baire spaces, Menger spaces, and projective sets

W. Hurewicz proved that analytic Menger sets of reals are $σ$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $ω$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.

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arXiv2017

Omitting Types and the Baire Category Theorem

The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are known to be closely linked. We examine the precise relation between these two theorems. Working with a general notion of logic we show that the classical Omitting Types Theorem holds for a logic if a certain associated topological space has all closed subspaces Baire. We also consider stronger Baire category conditions, and hence stronger Omitting Types Theorems, including a game version. We use examples of spaces previously studied in set-theoretic topology to produce abstract logics showing that the game Omitting Types statement is consistently not equivalent to the classical one.

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arXiv2016

On the definability of Menger spaces which are not /sigma-compact

Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.

math.GN↗
arXiv2016

Hereditarily normal manifolds of dimension > 1 may all be metrizable

P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercompact cardinal, and, in addition, that the manifold is hereditarily collectionwise Hausdorff. We are able to omit these extra assumptions.

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arXiv2016

Definable versions of Menger's conjecture

Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces are /sigma-compact, but Menger continuous images of co-analytic spaces need not be. The general co-analytic case is still open, but many special cases are undecidable, in particular, Menger topological groups. We also prove that if there is a Michael space, then productively Lindelof Cech-complete spaces are /sigma-compact. We also give numerous characterizations of proper K-Lusin spaces. Our methods include the Axiom of Co-analytic Determinacy, non-metrizable descriptive set theory, and Arhangel'skii's work on generalized metric spaces.

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arXiv2016

PFA(S)[S] and countably compact spaces

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact, perfectly normal space is paracompact".

math.LO↗
arXiv2016

PFA(S)[S] for the masses

We present S. Todorcevic's method of forcing with a coherent Souslin tree over restricted iteration axioms as a black box usable by those who wish to avoid its complexities but still access its power.

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