[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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