Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2104.07405

On logical parameterizations and functional representability in local set theories

Enrique Ruiz Hernández·Pedro Solórzano
View original source ↗Read source document ↗Export metadata ↓

Abstract

There is a well-known inclusion $ι_\mathscr{E}$ of a topos $\mathscr{E}$ in the linguistic topos $\mathscr{T}(Σ)$ of its internal language $Σ$ that proves both toposes to be equivalent. There is also a canonical translation $η_S$ for any local set theory $S$ into the local set theory $Σ$ of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from $\mathscr{T}(S)$ to $\mathscr{T}(Σ)$. Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that $ι_{\mathscr{T}(S)}$ parameterizes $\mathscr{T}(η_S)$ in such a way that syntactic $S$-functions are represented by themselves in $Σ$.

Explore related subjects

math.LOmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Enrique Ruiz Hernández, Pedro Solórzano. 2024-02-14. On logical parameterizations and functional representability in local set theories. https://arxiv.org/abs/2104.07405

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↗