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Spring 1999 An Open Formalism against Incompleteness
Francesc Tomàs
Notre Dame J. Formal Logic 40(2): 207-226 (Spring 1999). DOI: 10.1305/ndjfl/1038949537


An open formalism for arithmetic is presented based on first-order logic supplemented by a very strictly controlled constructive form of the omega-rule. This formalism (which contains Peano Arithmetic) is proved (nonconstructively, of course) to be complete. Besides this main formalism, two other complete open formalisms are presented, in which the only inference rule is modus ponens. Any closure of any theorem of the main formalism is a theorem of each of these other two. This fact is proved constructively for the stronger of them and nonconstructively for the weaker one. There is, though, an interesting counterpart: the consistency of the weaker formalism can be proved finitarily.


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Francesc Tomàs. "An Open Formalism against Incompleteness." Notre Dame J. Formal Logic 40 (2) 207 - 226, Spring 1999.


Published: Spring 1999
First available in Project Euclid: 3 December 2002

zbMATH: 0972.03058
MathSciNet: MR1816889
Digital Object Identifier: 10.1305/ndjfl/1038949537

Primary: 03F30
Secondary: 03B80, 03F50

Rights: Copyright © 1999 University of Notre Dame


Vol.40 • No. 2 • Spring 1999
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