David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
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
No references found.
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 downloads17 ( #110,773 of 1,410,009 )
Recent downloads (6 months)1 ( #177,059 of 1,410,009 )
How can I increase my downloads?