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Found in Translation: at the limits of the Hudetz program

Toby Meadows

TL;DR

The paper tackles what it means for two theories to be interdefinable when many scientific theories resist articulation in first-order logic. It generalizes interpretation to a broad framework, the $ V^{*}$-framework, using $ZFCU$ and a universe of sets over structures with atoms to define definability in the language of mathematics. It demonstrates that many familiar equivalences (e.g., $Top$ vs $Nei$, $Bool$ vs $Stone$) persist under this framework, while revealing nontrivial inequivalences (e.g., $Set1$ vs $Set2$) and the coordinate effect where adding structure (like coordinates) can convert inequivalences into equivalences. The framework also exposes limitations with rigid structures and the confounding effects of $V=HOD$, suggesting directions for refinement, such as anchored or quasi-structured approaches and strength/weakness trade-offs in definability. Overall, the $ V^{*}$-framework offers a coherent translation-centered account of interdefinability that aligns with mathematical practice while clarifying where further development is needed for more exotic theories.

Abstract

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.

Found in Translation: at the limits of the Hudetz program

TL;DR

The paper tackles what it means for two theories to be interdefinable when many scientific theories resist articulation in first-order logic. It generalizes interpretation to a broad framework, the -framework, using and a universe of sets over structures with atoms to define definability in the language of mathematics. It demonstrates that many familiar equivalences (e.g., vs , vs ) persist under this framework, while revealing nontrivial inequivalences (e.g., vs ) and the coordinate effect where adding structure (like coordinates) can convert inequivalences into equivalences. The framework also exposes limitations with rigid structures and the confounding effects of , suggesting directions for refinement, such as anchored or quasi-structured approaches and strength/weakness trade-offs in definability. Overall, the -framework offers a coherent translation-centered account of interdefinability that aligns with mathematical practice while clarifying where further development is needed for more exotic theories.

Abstract

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.

Paper Structure

This paper contains 21 sections, 44 theorems, 41 equations.

Table of Contents

  1. Things that don't work well
  2. Straw equivalence
  3. Sticks equivalence
  4. Triviality
  5. When $V=HOD$
  6. Counterintuitive Cantor-Bernstein
  7. $V$-logic and the $\mathcal{V}^{*}$ framework
  8. Background Theory and Perspective
  9. Structured Sets and Theories
  10. Interpretations and Isomorphisms
  11. Definitional equivalence and Bi-interpretability
  12. Applications
  13. Equivalence
  14. Inequivalence
  15. A more concrete example of bi-interpretation failure and the coordinate effect
  16. ...and 6 more sections

Key Result

Proposition 2

If $T$ and $S$ are theories (as definable classes) that each have at least one definable structure within them, then $T$ and $S$ are straw equivalent.

Theorems & Definitions (97)

  • Definition 1
  • Proposition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Definition 8
  • Theorem 9
  • ...and 87 more