Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.23003

MSO logic of the real order with the set quantifiers ranging over the Borel sets

Mirna Džamonja
View original source ↗Read source document ↗Export metadata ↓

Abstract

A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic quantifier is allowed to range over all subsets of the reals, the resulting MSO theory is undecidable. He conjectured that when we allow the monadic quantifier to range over the Borel subsets of the reals, the resulting MSO theory is decidable. We confirm this conjecture. Namely, the MSO theory of the real order where the set quantifier is allowed to range over the Borel sets, is decidable. If we only ask for the decidability in the language where each level of the Borel hierarchy is allowed a quantifier to denote sets of that level in the hierarchy, then we obtain a weaker MSO theory, which is not inly decidable but also interpretable in S2S.

Explore related subjects

math.LO

Keep this discovery

BibTeXRIS

Mirna Džamonja. 2025-12-28. MSO logic of the real order with the set quantifiers ranging over the Borel sets. https://arxiv.org/abs/2512.23003

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↗