David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
History and Philosophy of Logic 22 (3):135-161 (2001)
Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Matthias Schirn (2010). Consistency, Models, and Soundness. Axiomathes 20 (2-3):153-207.
Similar books and articles
Paolo Mancosu (ed.) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press.
Richard Zach (1999). Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic. Bulletin of Symbolic Logic 5 (3):331-366.
Matthias Schirn & Karl-Georg Niebergall (2003). What Finitism Could Not Be (Lo Que El Finitismo No Podría Ser). Critica 35 (103):43 - 68.
Thomas Hofweber (2000). Proof-Theoretic Reduction as a Philosopher's Tool. Erkenntnis 53 (1-2):127-146.
Richard Zach, Hilbert's Program. Stanford Encyclopedia of Philosophy.
William Demopoulos (1994). Frege, Hilbert, and the Conceptual Structure of Model Theory. History and Philosophy of Logic 15 (2):211-225.
Wilfried Sieg (1999). Hilbert's Programs: 1917-1922. Bulletin of Symbolic Logic 5 (1):1-44.
Wilfried Sieg & Mark Ravaglia, David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark.
Graham Priest (1997). On a Paradox of Hilbert and Bernays. Journal of Philosophical Logic 26 (1):45-56.
W. W. Tait (2010). Gödel on Intuition and on Hilbert's Finitism. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
Added to index2010-08-10
Total downloads22 ( #81,337 of 1,099,707 )
Recent downloads (6 months)1 ( #301,057 of 1,099,707 )
How can I increase my downloads?