Synthese 143 (3):223-253 (2005)

F. A. Muller
Erasmus University Rotterdam
.  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 skeptics 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 settheoreticians 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)
Reprint years 2005
DOI 10.1007/s11229-005-0800-0
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Translate to english
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 63,393
Through your library

References found in this work BETA

Philosophical investigations.Ludwig Wittgenstein & G. E. M. Anscombe - 1953 - Revue Philosophique de la France Et de l'Etranger 161:124-124.
Meaning.Paul Horwich - 1998 - Oxford University Press.
Models and Reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.
Model Theory.Wilfrid Hodges - 2008 - Stanford Encyclopedia of Philosophy.

View all 23 references / Add more references

Citations of this work BETA

The Implicit Definition of the Set-Concept.F. A. Muller - 2004 - Synthese 138 (3):417 - 451.

Add more citations

Similar books and articles

The Mathematics of Skolem's Paradox.Timothy Bays - 2006 - In Dale Jacquette (ed.), Philosophy of Logic. North Holland. pp. 615--648.
Adding Skolem Functions to Simple Theories.Herwig Nübling - 2004 - Archive for Mathematical Logic 43 (3):359-370.
Some Remarks on Finite Löwenheim‐Skolem Theorems.Martin Grohe - 1996 - Mathematical Logic Quarterly 42 (1):569-571.
On Gödel's Awareness of Skolem's Helsinki Lecture.Mark van Atten - 2005 - History and Philosophy of Logic 26 (4):321-326.
There is No Recursive Link Between the K-Size of a Model and its Cardinality.R. Barker - 2002 - Annals of Pure and Applied Logic 118 (3):235-247.
Reflections on Skolem's Paradox.Timothy Bays - 2000 - Dissertation, University of California, Los Angeles


Added to PP index

Total views
184 ( #57,145 of 2,449,019 )

Recent downloads (6 months)
4 ( #177,092 of 2,449,019 )

How can I increase my downloads?


My notes