630 Citations
Kurt Gödel and the Foundations of Mathematics: Gödel's Functional Interpretation and Its Use in Current Mathematics
- Philosophy
- 2011
This paper discusses applied aspects of Gödel’s functional (‘Dialectica’) interpretation which originally was designed for foundational purposes. The reorientation of proof theory towards…
P I C C P M
- Mathematics
- 2007
Hilbert’s original aim, to show the consistency of the use of such ideal elements by finitistic means, which would su ffice to eliminate these ideal elements from proofs of purely universal (‘real’)…
CONTOURS OF THE HISTORY OF MATHEMATICS: DIVERGING MATHEMATICAL SCHOOLS OF THOUGHT
- PhilosophyPONTE International Scientific Researchs Journal
- 2019
Although many people appreciate mathematics as a discipline exhibiting sound reasoning with a universal appeal, the mere fact that this academic discipline has a history shows that there must have…
NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM
- PhilosophyThe Bulletin of Symbolic Logic
- 2019
A historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction are presented.
Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth
- PhilosophyStud Logica
- 2018
It is shown that to any natural system in one cluster, there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth.
Approaches to analysis with infinitesimals following Robinson, Nelson, and others
- Economics
- 2017
This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement…
Formal Universes
- ArtLogic, Rewriting, and Concurrency
- 2015
This essay addresses the concerns of the foundations of mathematics of the early 20th century which led to the creation of formally axiomatized universes with computational models of mental experiments with the infinite in set-theory and symbol-manipulation systems.
The concept of axiom in Hilbert’s thought
- Philosophy
- 2011
David Hilbert is considered the champion of formalism and the mathematician who turned the axiomatic method into what we know nowadays. Hilbert compares in 19001 the axiomatic method with the genetic…
On the computational content of the Bolzano‐Weierstraß Principle
- Mathematics, ChemistryMath. Log. Q.
- 2010
An explicit solution of the Godel functional interpretation as well as the monotone functional interpretation of BW for the product space Πi ∈ℕ[–ki, ki] (with the standard product metric) results in optimal program and bound extraction theorems for proofs based on fixed instances of BW.