From Frege to Gödel

TL;DR: A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory was expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations.

TL;DR: This work presents a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice and shows how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-Statements.

TL;DR: It is shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title, and these results are applied to simplify methods used in reasoning in and about modal and intermediate logics.

TL;DR: Hjortland and Thomassen as mentioned in this paper developed a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of prooftheory and proposed a new theory of how inference rules determine the semantic content of logical constants.