David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Philosophical Logic 28 (2):165-174 (1999)
Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very weak fragment of finite-set theory, so weak that finite-set induction is not assumed. Many basic features of finiteness fail to hold in these weak fragments, conspicuously the principle that finite sets are in one-one correspondence with a proper initial segments of a (any) natural number structure. In the last part of the paper, we propose two prima facie evident principles for finite sets that, when added to these fragments, entail this principle.
|Keywords||foundations of arithmetic predicativism finiteness natural numbers induction|
|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
Herman Dishkant (1986). About Finite Predicate Logic. Studia Logica 45 (4):405 - 414.
Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.
Andreas Blass (1995). An Induction Principle and Pigeonhole Principles for K-Finite Sets. Journal of Symbolic Logic 60 (4):1186-1193.
Klaus Sutner (1990). The Ordertype of Β-R.E. Sets. Journal of Symbolic Logic 55 (2):573-576.
Eric Steinhart (2002). Why Numbers Are Sets. Synthese 133 (3):343 - 361.
Shaughan Lavine (1995). Finite Mathematics. Synthese 103 (3):389 - 420.
Øystein Linnebo (2004). Predicative Fragments of Frege Arithmetic. Bulletin of Symbolic Logic 10 (2):153-174.
Maricarmen Martinez (2001). Some Closure Properties of Finite Definitions. Studia Logica 68 (1):43-68.
Paul Strauss (1991). Arithmetical Set Theory. Studia Logica 50 (2):343 - 350.
Solomon Feferman & Geoffrey Hellman (1995). Predicative Foundations of Arithmetic. Journal of Philosophical Logic 24 (1):1 - 17.
Added to index2009-01-28
Total downloads16 ( #164,110 of 1,725,579 )
Recent downloads (6 months)4 ( #167,283 of 1,725,579 )
How can I increase my downloads?