David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Clauses (1) and (2) guarantee the inclusion of all 'intuitive' natural numbers, and (3) guarantees the exclusion of all other objects. Thus, in particular, no nonstandard numbers, which would follow after the intuitive ones are admitted (nonstandard numbers are found in nonstandard models of Peano arithmetic, in which the standard natural numbers are followed by one or more 'copies' of integers running from minus infinity to infinity)1. What is problematic about this delimitation? I suspect that its hypothetical proponent would see its weakest point in the unexplained concept of successor. However, we logicians know better (or at least some of us are convinced that we do): it is clause (3) which harbours the neuralgic spot, by dint of resisting any reasonable logical formalization to the point of appearing utterly void! There is, of course, no problem with regimenting (1) - we only need an individual constant 0 and a unary predicate constant N (the one whose meaning we are interested in) and we postulate..
|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
No citations found.
Similar books and articles
Steven C. Leth (1988). Sequences in Countable Nonstandard Models of the Natural Numbers. Studia Logica 47 (3):243 - 263.
Vladimir Kanovei (1996). On External Scott Algebras in Nonstandard Models of Peano Arithmetic. Journal of Symbolic Logic 61 (2):586-607.
G. Aldo Antonelli (2010). Numerical Abstraction Via the Frege Quantifier. Notre Dame Journal of Formal Logic 51 (2):161-179.
Eric Steinhart (2002). Why Numbers Are Sets. Synthese 133 (3):343 - 361.
G. Landini (2011). Logicism and the Problem of Infinity: The Number of Numbers. Philosophia Mathematica 19 (2):167-212.
C. Ward Henson, Matt Kaufmann & H. Jerome Keisler (1984). The Strength of Nonstandard Methods in Arithmetic. Journal of Symbolic Logic 49 (4):1039-1058.
Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory. [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
Friedrich Waismann (1951/2003). Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. Dover Publications.
Erich H. Reck (2005). Frege on Numbers: Beyond the Platonist Picture. The Harvard Review of Philosophy 13 (2):25-40.
Shizuo Kamo (1981). Nonstandard Natural Number Systems and Nonstandard Models. Journal of Symbolic Logic 46 (2):365-376.
Added to index2010-12-22
Total downloads12 ( #148,182 of 1,689,927 )
Recent downloads (6 months)1 ( #183,782 of 1,689,927 )
How can I increase my downloads?