arXiv subjects

Toby Meadows

Publications and source records attributed to Toby Meadows.

5 recordsLinked to original sources

arXiv2026

Loose Inductive Set Theories

We prove that KP, with Full Set Induction, is neither solid nor tight. More generally, every consistent theory obtained from KPC by adding a quantifier bounded set of sentences has a model with a parameter-free definable class surjection from its natural numbers onto its universe. Using such enumerations to define generic extensions by relative interpretations, we show that theories with the stable definable ground property are not tight. Ground definability over MOST and continuum coding over ZFC^{Π_3} yield applications to weak inductive set theories with large cardinals and forcing axioms. Finally, solidity for quantifier bounded extensions of MOST with Full Set Induction implies a Ground Axiom formulated using a fixed finite level of genericity; under the same solidity hypothesis, the corresponding extension of full ZFC proves the ordinary Ground Axiom.

math.LO↗

arXiv2026

Foundations with Imagination

We show that countable set theory, $ZFC^{-}+\forall x\ |x|\leqω$, is unable to eliminate imaginaries. In other words, this theory cannot provide representatives for arbitrary definable equivalence relations. We also see that $ZFC^{-}$ and ZFC^{-}+\existsκ(Inacc(κ)\wedge\forall x\ |x|\leqκ)$ also fail to eliminate imaginaries.

math.LO↗

arXiv2025

Found in Translation: at the limits of the Hudetz program

This paper aims to provide an analysis of what it means when we say that a pair of theories, very generously construed, are equivalent in the sense that they are interdefinable. With regard to theories articulated in first order logic, we already have a natural and well-understood device for addressing this problem: the theory of relative interpretability as based on translation. However, many important theories in the sciences and mathematics (and, in particular, physics) are precisely formulated but are not naturally articulated in first order logic or any obvious language at all. In this paper, we plan to generalize the ordinary theory of interpretation to accommodate such theories by offering an account where definability does not mean definability relative to a particular structure, but rather definability without such reservations: definable in the language of mathematics.

math.LO↗

arXiv2025

Internal Categoricity and the Generic Multiverse

John Steel's theory, MV, of the generic multiverse provides a foundation for mathematics that aims to neutralize the effects of incompleteness brought on by forcing arguments. Jouko Väänänen's development of internal categoricity arguments provides opportunities to argue that the subject matter of some theory is, in some sense, determined. This paper investigates whether MV is internally categorical.

math.LO↗

arXiv2025

Teasing apart definitional equivalence

In a recent paper, Enayat and Le lyk [2024] show that second order arithmetic and countable set theory are not definitionally equivalent. It is well known that these theories are biinterpretable. Thus, we have a pair of natural theories that llustrate a meaningful difference between definitional equivalence and bi-interpretability. This is particularly interesting given that Visser and Friedman [2014] have shown that a wide class of natural foundational theories in mathematics are such that if they are bi-interpretable, then they are also definitionally equivalent. The proof offered by Enayat and Le lyk makes use of an inaccessible cardinal. In this short note, we show that the failure of bi-interpretability can be established in Peano Arithmetic merely supposing that one of our target theories are consistent.

math.LO↗