FOM: Iterative conception of set: comment on Silver

Charles Silver csilver at
Thu Jan 22 07:00:15 EST 1998

On Thu, 22 Jan 1998, Moshe' Machover wrote:

> Silver mentions attempts by Boolos and Shoenfield (to which must be added
> Kreisel and others--all foreshadowed by Zermelo's 1930 paper) to justify
> the axioms of ZF by the iterative conception of sets.
> This justifiaction is not very convincing for the replacement axiom. At
> least, I for one have never been convinced by it. If I am not mistaken,
> Boolos also admits as much in a paper published 1989(?).

	Boolos's conception is much weaker than Shoenfield's.  Shoenfield
gives a kind of "proof" of replacement, but it relies on his 'principle of
cofinality', which in turn relies on being able to "visualize" the
completion of various stages of set construction.  I suspect that what you
are not convinced of is "visualization".  At least, that's the part that
bothers me.

Charlie Silver
Smith College

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