What is a mathematical theory?

Since Hilbert' s and Skolem's work in foundations of mathematics we got used to mathematizing the concept of a theory as a theory formalized in first order logic

Varieties of truth definitions

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $\alpha$ which extends a weak arithmetical theory (which we take to be EA) such that for some formula $\Theta$ and any arithmetical sentence $\varphi$, $\Theta(\ulcorner\varphi\urcorner)\equiv \varphi$ is provable in $\alpha$. We say that a sentence $\beta$ is definable in a sentence $\alpha$, if there exists an unrelativized translation from the language of $\beta$ to the language of $\alpha$ which is identity on the arithmetical symbols and such that the translation of $\beta$ is provable in $\alpha$. 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 generalize the result of Pakhomov and Visser showing that the set of (G\"odel codes of) definitions of truth is not $\Sigma_2$-definable in the standard model of arithmetic. We conclude by remarking that no $\Sigma_2$-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic

Interpretierte Theorien und Reduktionen

Theories in the philosphy of science are often described from a syntactical or semantical point of view. In this text both descriptions are generalised by interpreted theories. The corresponding interpreted reductions unify the usual attempts to describe intertheoretical reductions. Furthermore leads the chosen framework to interesting results in various versions of (anti-)reductionism. Approximative Reductions are identified as a special case, as well as truthlikeness