[FOM] Replacement
Allen Hazen
allenph at unimelb.edu.au
Sun Aug 19 04:14:54 EDT 2007
I haven't responded to Thomas Forster's request because I don't know
the thought processes of "real" replacement deniers. (I have moods
in which I am sceptical about set theory, but in those moods I don't
stop with denying replacement!) But there is something curious about
the arguments for (or lines of thought "motivating") replacement that
puzzles me. Maybe what I want is a historical account, maybe a
conceptual explanation.
In the context of the other axioms-- so ZC (and of classical logic)--
Replacement is equivalent to collection,
Collection: For any set x, if for every y in x there is a z such
that ...yz...
then there is a set w such that for every y in x
there is a z in w such that ...yz...
Now, this is SIMPLER (you don't have a clause in the antecedent about
every y having a unique z such that ...yz...). If you think that the
idea of the cumulative hierarchy motivates the axioms, Collection
seems just as well motivated by it as Replacement. If you like
thinking about set theory as a description of the cumulative
hierarchy, then, it would seem you would probably prefer Collection
to Replacement. Yet most introductory set theory texts give you
Replacement. Why?
Is it JUST history, that Replacement was formulated first, before the
cumulative hierarchy was as much in the foreground of discussion as
it is now? Or is the fact that Replacement has a different
motivation (in terms of "Limitiation of Size": cf. Boolos's
"Iteration again" (which is also in "Logic, Logic and Logic"))
important here?
Comments: I'd be interested in Randy Holmes's arguments about the
limitations of the usual cumulativist arguments for Replacement: can
I add my voice to Thomas's in asking him to publish? Second, in the
course of discussion, the existence of Cartesian products was
mentioned as an important application of Replacement, but then
pointed out that it could be proven just from Powerset and
Separation: there is also a very pretty proof-- it's on page 12 of
Barwise's "Admissible Sets and Structures"-- that avoids the appeal
to Powerset, using instead two applications of a weak form (Delta-0)
of Collection (and something very elementary like Pairset).
Allen Hazen
Philosophy Department
University of Melbourne
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