Scott-Karp analysis without sentences

Andreas Brunner; Charles Morgan; Darllan Concieção Pinto

Home/Archive/arXiv

arXiv · 2602.07646

Scott-Karp analysis without sentences

Andreas Brunner·Charles Morgan·Darllan Concieção Pinto

View original source ↗Read source document ↗Export metadata ↓

Abstract

Scott and Karp gave an analysis which provides a level-by-level equivalence between global similarity between two structures and local commonality in terms of sharing particular invariants. Scott and Karp's local invariants were certain infinitary formulae. We give a more abstract version of the local side of Scott-Karp analysis which avoids the use of infinitary languages. We show the resulting hierarchies are still provide desired equivalences in the classical setting. Moreover, the abstract nature of our analysis, as we show, makes it suitable to provide local level-by-level equivalents to Hjorth's much more general version of global similarity in the context of topological group actions on topological groups. We, furthermore, provide, analogously to the classical work of Ehrenfeucht and Fra\"iss\'e, novel game theoretic equivalents for Hjorth's global similarity relations and some natural variants.

Explore related subjects

math.LO

Keep this discovery

BibTeX RIS

Andreas Brunner, Charles Morgan, Darllan Concieção Pinto. 2026-02-07. Scott-Karp analysis without sentences. https://arxiv.org/abs/2602.07646

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

KEEP EXPLORING

Related papers

arXiv2026

There is no maximal $K$-degree

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$.

math.LO↗

arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗

arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

math.LO↗