Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2212.11859

Librationism & its classical and extraclassical set theories

Frode A. Bjørdal
View original source ↗Read source document ↗Export metadata ↓

Abstract

Librationist set theory \pounds ${}$ is developed. It descends from semantics for truth, initiated by Kripke, and others. # extends \pounds, of Librationist closures of the paradoxes in Logic and Logical Philosophy 21(4), 323-361, 2012. Focus is on the paradoxes in theories of sets. A central result is that extension #$\mathscr{HR}(\mathbf{G})$, of #, accounts for $\mathbf{NBG}$ set theory, with global AC and Tarski's Axiom. # succeeds with defining an impredicative manifestation set $\mathbf{W}$, $\mathit{die \ Welt}$, so that #$\mathscr{H}(\mathbf{W})$ accounts for Quine's $\mathbf{NF}$. The points of view developed support the view that the truth-paradoxes and the set-paradoxes often have common origins, so that the librationist resolutions of set theoretic paradoxes are in some cases at the same time resolutions of corresponding truth theoretic paradoxes. Librationist set theories have the consequence that there are absolutely no uncountable sets, and so # retains the ideology of the author's 2012 article, that the set theoretic universe is countable. But the set within which e.g. $\mathbf{NBG}$ is interpreted "believes" that there are sets which are uncountable. That situation is analyzed almost as by Skolem, though it is not suggested that the notion of $\mathit{set}$ is imprecise: for the bijection from the set of finite von Neumann ordinals to the full universe is not a member of a classical set theory. The Diagonal Lemma is derivable; but the truth predicate of the system only fulfills slightly reformed Hilbert-Bernays-L\"ob derivability conditions, so the L\"ob rule for truth is not valid.

Explore related subjects

math.LO

Keep this discovery

BibTeXRIS

Frode A. Bjørdal. 2022-12-22. Librationism & its classical and extraclassical set theories. https://arxiv.org/abs/2212.11859

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↗