David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Cahiers du Centre de Logique 17:99-118 (2010)
In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent theory of sets and an equivalent foundation for arithmetic was introduced by John Mayberry and developed by the current author in his doctoral thesis. In that thesis, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry<br>from the early 1980s.
|Keywords||set theory arithmetic|
|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
No citations found.
Similar books and articles
Alistair H. Lachlan & Robert I. Soare (1994). Models of Arithmetic and Upper Bounds for Arithmetic Sets. Journal of Symbolic Logic 59 (3):977-983.
Fredrik Engström (2004). Expansions, Omitting Types, and Standard Systems. Dissertation, Chalmers
Jan Krajíček (1995). Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press.
George Mills & Jeff Paris (1984). Regularity in Models of Arithmetic. Journal of Symbolic Logic 49 (1):272-280.
V. Kanovei (1995). Uniqueness, Collection, and External Collapse of Cardinals in Ist and Models of Peano Arithmetic. Journal of Symbolic Logic 60 (1):318-324.
Andreas Blass (1974). On Certain Types and Models for Arithmetic. Journal of Symbolic Logic 39 (1):151-162.
M. Krynicki & K. Zdanowski (2005). Theories of Arithmetics in Finite Models. Journal of Symbolic Logic 70 (1):1-28.
Charles Parsons (1987). Developing Arithmetic in Set Theory Without Infinity: Some Historical Remarks. History and Philosophy of Logic 8 (2):201-213.
Richard Pettigrew (2009). On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame Journal of Formal Logic 50 (2):141-152.
Added to index2009-08-11
Total downloads162 ( #23,324 of 1,911,681 )
Recent downloads (6 months)4 ( #180,081 of 1,911,681 )
How can I increase my downloads?