Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.13519

Varieties of truth definitions

Piotr Gruza·Mateusz Łełyk
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $α$ which extends a weak arithmetical theory (which we take to be EA) such that for some formula $Θ$ and any arithmetical sentence $φ$, $Θ(\ulcornerφ\urcorner)\equiv φ$ is provable in $α$. We say that a sentence $β$ is definable in a sentence $α$, if there exists an unrelativized translation from the language of $β$ to the language of $α$ which is identity on the arithmetical symbols and such that the translation of $β$ is provable in $α$. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not $Σ_2$-definable in the standard model of arithmetic. We conclude by remarking that no $Σ_2$-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Piotr Gruza, Mateusz Łełyk. 2023-11-22. Varieties of truth definitions. https://arxiv.org/abs/2311.13519

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Natural Term Logic

In this paper we develop a formal system called Natural Term Logic (NTL). NTL aims to represent key aspects of the logical and grammatical mechanisms of natural language as well as grammatical transformations which preserve core logical meaning. NTL can be seen as a refinement of the ideas of Quine's paper `Variables Explained Away' and the technical concepts introduced by Bealer and Zalta. NTL is more fine-grained than Bealer's first-order intensional logic (BL): there is a many-to-one correspondence $ν$ between NTL terms and closed BL terms as well as a canonical map $β$ which assigns to each closed BL term a corresponding NTL term. The map $ν$ can be seen as assigning a core logical content of the NTL term. We define a series of reductions on NTL terms which intuitivelyy speaking capture meaning-preserving syntactic transformations ( transformations which preserved the basic logical meaning of a term) and our main result is that each NTL term $T$ reduces to a unique normal term $N$. The reductions fall into the structural, predicative and pushing-in categories. Predicative reductions decompose NTL terms so that predication is only applied to a primitive term (such terms are called prenormal). A key ingredient in the proof is the fact that $βνN = N$ when $N$ is normal. This suggests that within NTL the normal form of a term expresses the core logical content of the term.

math.LO↗
arXiv2026

Hyper-hyperfiniteness and complexity

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite, then the complexity of hyperfinite countable Borel equivalence relationsis as high as possible, namely, $Σ^1_2$-complete. We also establish an implication between the question of the effectivity of hyperfiniteness and its complexity.

math.LO↗
arXiv2026

Coordinate recognition: General theory, Groups, and other surprises

A class of structures \emph{recognizes coordinates} if any reduced product of structures from said class witnesses a certain kind of rigidity phenomenon. We provide several equivalent characterizations of this property. This property has (at least) two remarkable consequences, one set-theoretic and one model-theoretic, for reduced products of structures of the said class. First, under appropriate set-theoretic assumptions every isomorphism between such reduced products associated with the Fréchet ideal lifts (modulo a finite change) to an isomorphism between products of the original structures. Second, with an additional mild assumption, it implies a strong quantifier elimination result. Of note, we show that a class recognizes coordinates if and only if an individual formula witnesses a certain syntactic property. We also consider many concrete classes of structures and determine whether or not they recognize coordinates. We place heavy emphasis on well-known classes of groups, such as permutation groups, acylindircally hyperbolic groups, quasisimple groups, free products, and graph products, but we also discuss other classes of structures.

math.LO↗