[FOM] The meaning of `set'

Muller F.A. F.A.Muller at phys.uu.nl
Mon Feb 20 06:46:57 EST 2006

 Dear all,

 All postings about the similarities between
 finite and infinite sets, the colloquial use
 of the word `set', about the acceptability
 of the axioms of ZFC or extensions of it,
 and more, orbit around one deep philosophical
 problem: the problem of meaning, of what
 meaning `is', of how expressions obtain

 Since Wittgenstein it has become a triviality
 to say that our use of expressions is 
 constitutive for their meaning. But only
 very recently has Paul Horwich made an attempt
 to propound a theory of meaning based on this
 triviality. The word `theory' is meant here
 in the sense in which philosophers use it,
 so mathematicians are warned there are
 differences with how they use the word
 `theory'. Nonetheless the mentioned theory
 of meaning can be formulated as having
 five postulates; on the basis of those
 postulates then several outstanding problems
 in the philosophy of language can be

 Important for readers of this list is
 that this theory has been applied to 
 mathematics, in particular to the word
 `set' and to set-theory in general.
 For those who are interested, here are
 the references:

 `The Implicit Definition of the Set-Concept'
      Synthese 138 (2004) 417-451
 `Deflating Skolem'
      Synthese 143 (2005) 223-253

 I submit that without a clear view of how
 expressions obtain meaning in language,
 the discussions concerning the topics mentioned
 in the opening paragraph of this posting
 are destined NOT to go to the heart of the

 Many things said by for example Harvey Friedman 
 about the set-concept fit the view expressed
 above and expounded in the two references seamlessly.


 --> F.A. Muller

 Inst History & Foundations of Science & Math.
 Utrecht University
 f.a.muller at phys.uu.nl

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