January 10
Albert Visser,
Utrecht University
Interpreters as a species of Consistoids
Harvey Friedman shows that, over Peano Arithmetic as base theory, the consistency statement for a finitely axiomatised theory can be characterised as the weakest statement that, in combination with the base, interprets the given theory. Thus, we have a coordinate-free characterisation of these consistency statements modulo base-theory-provable equivalence.
Let us call a base theory that, in analogy to Peano Arithmetic, has such weakest extensions: Friedman-reflexive. We call such a weakest statement the interpreter of the finite theory. Interpreters are not always consistency statements, but they are still 'consistoids'.
Which theories are Friedman-reflexive and what more can we say about their consistoids and the associated provability-like notion? We will sketch some preliminary insights. (E.g., all complete theories are Friedman-reflexive.)
We discuss Friedman-reflexive sequential base theories. We introduce an example of an attractive very weak base theory that shares many properties with Peano Arithmetic, to wit Peano Corto. We have a look at what Friedman-reflexive sequential theories look like in general. It turns out that they may look somewhat different from Peano Arithmetic and its little brother Peano Corto.
Given an interpretation $K$ of a Friedman-reflexive base $U$ in a finitely axiomatised theory $A$, we can define an analogue of provability logic: the interpreter logic of $A$ over $U$, relative to $K$. All interpreter logics satisfy K4, aka the Löb Conditions. Two theories are irreconcilable iff they do not have finite extensions that are mutually interpretable. If $A$ and $U$ are irreconcilable, then their interpreter logic relative to $K$ contains at least Löb’s Logic. If one of $A$ or $U$ is sequential, then $A$ and $U$ are irreconcilable.