FOM: Iterative conception of set: comment on Silver wtait at
Wed Jan 21 22:55:38 EST 1998

Moshe' is right about Boolos. B's conception of the iterative conception 
(and Shoenfield's too, which incidently precedes Boolos's) presuppposes a 
given `external' hierarchy of stages--a system of ordinals or at least a 
well-founded partial ordering along which the iterations of power set are 
defined. In Boolos's conception, even the axiom of infinity is justified 
only because he assumes that there is an infinite ordinal (or 
whatever)---and so his scepticism about replacement is a bit of a 
mystery. But there is an autonomous conception of this iterative 
hierarchy, suggested by Zermelo's 1930 paper (which Moshe' mentions) and 
developed in lectures and papers by Godel, starting in 1933 (*1933 in his 
Collected Papers, Vol III), according to which the ordinals are obtained 
by closure under ordinal operations previously postulated. On this 
conception, not only infinity, but unordered pairs, replacement, 
inaccessible cardinals and much more are obtained. How much more is an 
open question. This idea leads to a `constructive' conception of set 
theory---constructing the numbers from below--- which contrasts with a 
conception according to which axioms are justified by their global 
consequences. (E.g. see Steel 12/19/97.)

Bill Tait


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