Related papers: A Constructive Interpretation of Ramsey's Theorem …

We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated…

It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…

There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a…

Goedel's functional "Dialectica" interpretation can be used to extract functional programs from non-constructive proofs in arithmetic by employing two sorts of higher-order witnessing terms: positive realisers and negative counterexamples.…

G\"odel's Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent…

G\"odel's Dialectica interpretation is a fundamental tool for the extraction of computational content from proofs, and plays a central role in today's proof mining program. In the past decades, it has also been studied from the perspective…

By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for…

Ramsey Theorem [6] for pairs is intuitionistically but not classically provable: it is equivalent to a subclassical principle [2]. In this note we show that Ramsey may be restated in an intuitionistically provable form, which is informative…

G\"odel's Dialectica interpretation was conceived as a tool to obtain the consistency of Peano arithmetic via a proof of consistency of Heyting arithmetic in the 40s. In recent years, several proof-theoretic transformations, based on…

In this expository note we present simple proofs of the lower bound of Ramsey numbers (Erd\"os theorem), and of the estimation of discrepancy. Neither statements nor proofs require any knowledge beyond high-school curriculum (except a minor…

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0…

G\"odel's Dialectica has been introduced and developed in the tradition of the so-called functional interpretations. Only recently has it been related with the a priori unrelated notion of differentiation, by taking a program-theoretic…

We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature.

During the last twenty years or so a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with…

In this paper we give a new proof of the Ne\v{s}et\v{r}il-R\"odl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate…

A theorem, usually attributed to Barr, yields that (A) geometric implications deduced in classical L_{\infty\omega} logic from geometric theories also have intuitionistic proofs. Barr's theorem is of a topos-theoretic nature and its proof…

We introduce the notion of a G\"odel fibration, which is a fibration categorically embodying both the logical principle of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the…

This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…

We use G\"odel's Dialectica interpretation to analyse Nash-Williams' elegant but non-constructive "minimal bad sequence" proof of Higman's Lemma. The result is a concise constructive proof of the lemma (for arbitrary decidable…

A major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemer\'edi's theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs…