December 3
Mateusz Łełyk,
University of Warsaw
Varieties of truth definitions
In the talk we address the following problem: how many essentially different truth definitions (for the language of arithmetic) are there? Formally, a truth definition for us is just a sentence $\phi$ in some language $L$, which extends the elementary arithmetic (a.k.a. $I\Delta_0 + \exp$) and such that for some $L$-formula $\Theta(x)$, $$\phi\vdash \psi\equiv\Theta(\ulcorner\psi\urcorner),$$ for every sentence $\psi$ in the language of arithmetic. In other words $\phi$ is a sentence which can define a truth predicate for arithmetic (via a formula $\Theta(x)$). We investigate the structure of the definability relation between so defined truth definitions. To be more precise: we say that a truth definition $\phi$ (in a language $L$) defines a truth definition $\phi'$ (in a language $L'$) if and only if there are $L$-formulae $A_1,\ldots,A_n$ such that $\phi\vdash \phi'[A_1/R_1,\ldots,A_n/R_n]$, where $R_i$'s are all the non-arithmetical predicates from the language $L'$ and $\phi'[A_1/R_1,\ldots,A_n/R_n]$ denotes the result of translating $\phi'$ by substituting $A_i$ for each occurrence of $R_i$. We note that this translation does not relativize the quantifiers in $\phi'$ and keeps the arithmetical symbols unchanged. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we (slightly) generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not $\Sigma_2$-definable in the standard model of arithmetic.
This is joint work with Piotr Gruza which was published in here.