arXiv subjects
Publications and source records attributed to Toby Meadows.
We show that countable set theory, $ZFC^{-}+\forall x\ |x|\leq\omega$, 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^{-}+\exists\kappa(Inacc(\kappa)\wedge\forall x\ |x|\leq\kappa)$ also fail to eliminate imaginaries.
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.
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 V\"a\"an\"anen'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.
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.