Skolem functions of arithmetical sentences

Arithmetical Proof and Open Sentences

If the concept of proof (including arithmetic proof) is syntactically restricted to closed sentences (or their Gödel numbers), then the standard accounts of Gödel’s Incompleteness Theorems (and Löb’s Theorem) are blocked. In these standard accounts (Gödel’s own paper and the exposition in Boolos’ Computability and Logic are treated as exemplars), it is assumed that certain formulas (notably so ...

Skolem Functions for DQBF

We consider the problem of computing Skolem functions for satisfied dependency quantified Boolean formulas (DQBFs). We show how Skolem functions can be obtained from an elimination-based DQBF solver and how to take preprocessing steps into account. The size of the Skolem functions is optimized by don’t-care minimization using Craig interpolants and rewriting techniques. Experiments with our DQB...

Tests as Skolem Functions

Arithmetical Functions I: Multiplicative Functions

Truth be told, this definition is a bit embarrassing. It would mean that taking any function from calculus whose domain contains [1,+∞) and restricting it to positive integer values, we get an arithmetical function. For instance, e −3x cos2 x+(17 log(x+1)) is an arithmetical function according to this definition, although it is, at best, dubious whether this function holds any significance in n...

Skolem Functions and Elementary Extensions

In [4] Payne constructed such a pair of uncountable models for an uncountable language. Knight [1] announced that the language could be countable and presented her construction in [3] together with other examples one of which is a " nonSkolemizable elementary triple " of countable structures for a countable language. Payne's, Knight's and the present construction all seem to use rather differen...