[FOM] The defence of well-founded set theory
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.
University of Melbourne
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