9 research outputs found
Functional interpretation and inductive definitions
Extending G\"odel's \emph{Dialectica} interpretation, we provide a functional
interpretation of classical theories of positive arithmetic inductive
definitions, reducing them to theories of finite-type functionals defined using
transfinite recursion on well-founded trees.Comment: minor corrections and change
A herbrandized functional interpretation of classical first-order logic
We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio
Metastability in the Furstenberg-Zimmer tower
According to the Furstenberg-Zimmer structure theorem, every
measure-preserving system has a maximal distal factor, and is weak mixing
relative to that factor. Furstenberg and Katznelson used this structural
analysis of measure-preserving systems to provide a perspicuous proof of
Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, the
transfinite construction of the maximal distal factor of a separable
measure-preserving system can extend arbitrarily far into the countable
ordinals. Here we show that the Furstenberg-Katznelson proof does not require
the full strength of the maximal distal factor, in the sense that the proof
only depends on a combinatorial weakening of its properties. We show that this
combinatorially weaker property obtains fairly low in the transfinite
construction, namely, by the th level
Herbrandized modified realizability
Realizability notions in mathematical logic have a long history, which can be tracedback to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations ofintuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticatednotions such as Kreisel’s modified realizability and various modern approaches. Inthis context, our work aligns with the lineage of realizability strategies that emphasizethe accumulation, rather than the propagation of precise witnesses. In this paper, weintroduce a new notion of realizability, namely herbrandized modified realizability.This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, whichenables the gathering of witnesses into nonempty finite sets. We also show that theprevious analysis can be extended from logic to (Heyting) arithmetic.The authors are grateful to Fernando Ferreira for interesting discussions on the topic.
They extend their gratitude to the anonymous referee for providing valuable suggestions, which inspired the
addition of Sect. 4.3 to the manuscript. Both authors acknowledge the support of Fundação para a Ciência e
a Tecnologia under the Projects: UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020 and are also
grateful to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (Universidade de
Lisboa). The first author is also grateful to LASIGE - Computer Science and Engineering Research Centre
Herbrandized modified realizability (Universidade de Lisboa). The second author also benefitted from Fundação para a Ciência e a Tecnologia doctoral Grant 2022.12585.BD.info:eu-repo/semantics/publishedVersio
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