Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept of function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
The Representational Foundations of Computation.Michael Rescorla - 2015 - Philosophia Mathematica 23 (3):338-366.
More on Putnam's Models: A Reply to Belloti. [REVIEW]Timothy Bays - 2007 - Erkenntnis 67 (1):119--35.
Similar books and articles
The Standard Model and Beyond: Interrelation Between Theory and Reality.Vladimir Slobodenyuk - unknown
Towards a Theory of Models In Physical Science.John Forge - 1982 - Philosophy Research Archives 8:321-338.
Theories of Truth Without Standard Models and Yablo's Sequences.Eduardo Alejandro Barrio - 2010 - Studia Logica 96 (3):375-391.
Sequences in Countable Nonstandard Models of the Natural Numbers.Steven C. Leth - 1988 - Studia Logica 47 (3):243 - 263.
Standard Quantification Theory in the Analysis of English.Stephen Donaho - 2002 - Journal of Philosophical Logic 31 (6):499-526.
The Model of Set Theory Generated by Countably Many Generic Reals.Andreas Blass - 1981 - Journal of Symbolic Logic 46 (4):732-752.
On the Standard Part of Nonstandard Models of Set Theory.Menachem Magidor, Saharon Shelah & Jonathan Stavi - 1983 - Journal of Symbolic Logic 48 (1):33-38.
Theories of Arithmetics in Finite Models.M. Krynicki & K. Zdanowski - 2005 - Journal of Symbolic Logic 70 (1):1-28.
Extending Standard Models of ZFC to Models of Nonstandard Set Theories.Vladimir Kanovei & Michael Reeken - 2000 - Studia Logica 64 (1):37-59.
Added to index2009-01-28
Total downloads62 ( #83,431 of 2,158,397 )
Recent downloads (6 months)3 ( #133,350 of 2,158,397 )
How can I increase my downloads?