Skolem and the Skeptic

@article{Benacerraf1985SkolemAT,
  title={Skolem and the Skeptic},
  author={P. Benacerraf and C. Wright},
  journal={Aristotelian Society Supplementary Volume},
  year={1985},
  volume={59},
  pages={85-138}
}
structure that all progressions have in common in virtue of being progressions. It is not a science concerned with particular objects-the numbers. The search for which independently identifiable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.1 The argument which precedes these conclusions is well known. Suppose that someone has received satisfactory definitions of'l' (or '0'), 'number', 'successor', '+', and 'X' on the basis of which the laws of arithmetic… Expand
- The Mathematics of Skolem's Paradox
Publisher Summary This chapter presents a discussion on Skolem's paradox. In its simplest form, Skolem's Paradox involves a (seeming) conflict between two theorems of modern logic: Cantor's theoremExpand
Chapter 10 Absoluteness and the Skolem Paradox
When seen in the “correct” light, the contradictions of set theory are by no means disastrous, but instructive and fruitful. For instance, the antinomies of Russell and Burali-Forti live on in theExpand
Reflections on Skolem ’ s Paradox
The Löwenheim-Skolem theorems say that if a first-order theory has infinite models, then it has models which are only countably infinite. Cantor's theorem says that some sets are uncountable.Expand
Models and recursivity
It is commonly held that the natural numbers sequence 0, 1, 2, . . . possesses a unique structure. Yet by a well known model theoretic argument, there exist non-standard models of the formal theoryExpand
The Mathematics of Skolem ’ s Paradox
In 1922, Thoralf Skolem published a paper entitled “Some Remarks on Axiomatized Set Theory.” The paper presents a new proof of a model-theoretic result originally due to Leopold Löwenheim and thenExpand
Deflating skolem
TLDR
It is argued that the Skolemite skeptic’s argument is a petitio principii and that consequently the authors find ourselves in a dialectical situation of stalemate, and propound solutions, which crucially involve a renunciation of Connexion M. Expand
Absoluteness and the Skolem Paradox
When seen in the “correct” light, the contradictions of set theory are by no means disastrous, but instructive and fruitful. For instance, the antinomies of Russell and Burali-Forti live on in theExpand
A Defence of Predicativism as a Philosophy of Mathematics
ADefence of Predicativism as a Philosophy of Mathematics T. J. Storer A speci cation of a mathematical object is impredicative if it essentially involves quanti cation over a domain which includesExpand
Categoricity by convention
TLDR
This paper provides a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and shows that the solution generalizes, giving us full second- order logic and thereby securing the categoricity or quasi-categoricality of second-order mathematical theories. Expand
The significance of non-standard models
TLDR
This paper shall argue for the following two points: (1) the philosophical problems that Shapiro takes unintended interpretations to pose first-order formulations are not solved by his second- order formulations, and (2) Shapiro has not shown that unintended interpretations demonstrate that first- order theories are inadequate. Expand
...
1
2
3
4
...

References

SHOWING 1-10 OF 22 REFERENCES
Models and Reality
  • H. Putnam
  • Philosophy, Computer Science
  • J. Symb. Log.
  • 1980
TLDR
It is argued that the resolution of the antinomy—the only resolution that I myself can see as making sense— has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language. Expand
Skolem's Criticisms of Set Theory
In 1915 Leopold Ldwenheim formulated and gave a flawed proof of a result which for my purposes I express as follows: If a first order (FO) formula is satisfiable it is satisfiable in a denumerableExpand
Philosophy of mathematics: Russell's mathematical logic
Mathematical logic, which is nothing else but a precise and complete formulation of formal logic, has two quite different aspects. On the one hand, it is a section of Mathematics treating of classes,Expand
Philosophy of mathematics: What numbers could not be
THE attention of the mathematician focuses primarily upon mathematical structure, and his intellectual delight arises (in part) from seeing that a given theory exhibits such and such a structure,Expand
The Hanf Number of Second Order Logic
TLDR
This work proves, among other things, that the number mentioned above cannot be shown to exist without using some IIl,() instance of the axiom of re- placement, and shows that the nonconstructive proof is basically the only one possible for L2, second order logic. Expand
Peano's Axioms and Models of Arithmetic
Publisher Summary The chapter discusses Peano's axioms and models of arithmetic and presents, how models of a similar kind can be set up in a perfectly constructive way when some very restrictedExpand
What the numbers could not be
General cargo must be stowed differently when being transported by ships, or by railway waggons and trucks, respectively. To reduce the time a ship is tied up in port, a transhipping station isExpand
1908a] 'Investigations in the Foundations of Set Theory
  • in van Heijenoort
  • 1967
Frege's Conception of Numbers as Objects
Where the articles do not appear in English, quotat own translations. With the exception of [1922] and [1970], a date f letter in Skolem's works is as they appear in the Bibliography in S Barwise
  • The Hanf Number of Second Order Logic' 3
  • 1970
...
1
2
3
...