[FOM] The defence of well-founded set theory

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Wed Oct 5 06:36:29 EDT 2005

    Roger Bishop Jones asked  for references relevant to a defense of 
the conception  os sets as  items in a well-founded cumulative 
hierarchy.  I am writing a separate, lengthy (sorry!) historical 
bibliography on the topic,  but one recent book defends a version of 
the  view he might find congenial: Marcus Giaquinto's "The Search for 
Certainty: a philosophical account of the foundations of mathematics" 
(Oxford University Press, 2002: ISBN 0-19-875244-X.  Pp. xii+286.)

    I was annoyed by some of Giaquinto's discussion: his treatment of 
Russell, in particular, seemed to me both unsympathetic and in places 
misleading.  On the whole, though, the book is clearly written (in a 
brisk style more academic writers should emulate).  I think it would 
make a good text for an upper-division undergraduate or beginning 
graduate unit on the philosophy of mathematics, and good collateral 
reading for a set theory course.

    From  Jones's perspective.  Jones in a couple of replies has said 
he is not a "platonist" and doesn't want  to think about the question 
of what sets REALLY exist (as a matter of metaphysical fact), but 
only questions of what  sets exist
	"in the context of some existential presuppositions (i.e.
	only in the context of an understanding about the domain
	of discourse). ...  I do believe that the sentences of set
	theory have a definite objective truth value once the
	semantics of the language of set theory has been made
	sufficiently definite (for example, but not necessarily,
	by identifying the domain of discourse with the sets described
	in "the iterative conception of set")."
(((***I personally*** have doubts about trying to distinguish in this 
way between "existence on a conception" and  existence in some 
metaphysically more absolute sense, but))) Giaquinto gives an 
interesting exposition of the iterative conception viewed in what I 
think may be a Jonesian way.  He argues that both the iterative 
conception and his "non-absolutist" view of it can be found in 
Zermelo's later work.
Allen Hazen
Philosophy Department
University of Melbourne

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