David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Synthese 143 (3):223 - 253 (2005)
Remarkably, despite the tremendous success of axiomatic set-theory in mathematics, logic and meta-mathematics, e.g., model-theory, two philosophical worries about axiomatic set-theory as the adequate catch of the set-concept keep haunting it. Having dealt with one worry in a previous paper in this journal, we now fulfil a promise made there, namely to deal with the second worry. The second worry is the Skolem Paradox and its ensuing 'Skolemite skepticism'. We present a comparatively novel and simple analysis of the argument of the Skolemite skeptic, which will reveal a general assumption concerning the meaning of the set-concept (we call it 'Connexion M'). We argue that the Skolemite skeptic's argument is a petitio principii and that consequently we find ourselves in a dialectical situation of stalemate. Few (if any) working set-theoreticians feel a tension -- let alone see a paradox -- between, on the one hand, what the Löwenheim-Skolem theorems and related results seem to be telling us about the set-concept, and, on the other hand, their uncompromising and successful use of the set-concept and their continuing enthusiasm about it, in other words: their lack of skepticism about the set-concept. Further, most (if not all) working set-theoreticians have a relaxed attitude towards the ubiquitous undecidability phenomenon in set-theory, rather than a worrying one. We argue these are genuine philosophical problems about the practice of set-theory. We propound solutions, which crucially involve a renunciation of Connexion M. This breaks the dialectical situation of stalemate against the Skolemite skeptic
|Keywords||Philosophy Philosophy Epistemology Logic Metaphysics Philosophy of Language|
|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
Paul Horwich (1998). Meaning. Oxford University Press.
Wilfrid Hodges, Model Theory. Stanford Encyclopedia of Philosophy.
Gottlob Frege, Gottfried Gabriel, Brian Mcguinness & Hans Kaal (1982). Philosophical and Mathematical Correspondence. Revue Philosophique de la France Et de l'Etranger 172 (1):64-64.
Hilary Putnam (1980). Models and Reality. Journal of Symbolic Logic 45 (3):464-482.
Citations of this work BETA
No citations found.
Similar books and articles
Jairo José Da Silva (2002). The Axioms of Set Theory. Axiomathes 13 (2):107-126.
Johannes Heidema (1990). An Axiom Schema of Comprehension of Zermelo–Fraenkel–Skolem Set Theory. History and Philosophy of Logic 11 (1):59-65.
F. A. Muller (2004). The Implicit Definition of the Set-Concept. Synthese 138 (3):417 - 451.
Timothy Bays (2000). Reflections on Skolem's Paradox. Dissertation, University of California, Los Angeles
Gregory H. Moore (1980). Beyond First-Order Logic: The Historical Interplay Between Mathematical Logic and Axiomatic Set Theory. History and Philosophy of Logic 1 (1-2):95-137.
A. Paseau (2003). The Open-Endedness of the Set Concept and the Semantics of Set Theory. Synthese 135 (3):379 - 399.
Ignagio Jane (2001). Reflections on Skolem's Relativity of Set-Theoretical Concepts. Philosophia Mathematica 9 (2):129-153.
F. A. Muller (2005). Deflating Skolem. Synthese 143 (3):223--53.
Added to index2009-01-28
Total downloads35 ( #101,199 of 1,781,482 )
Recent downloads (6 months)1 ( #295,005 of 1,781,482 )
How can I increase my downloads?