Librationism & its classical and extraclassical set theories

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.