FOM: Comment on Parson/Davis
wtait at ix.netcom.com
Sat Mar 14 14:33:11 EST 1998
In reply to Moshe' Machover (3/14)
>I agree. My guess is that Zermelo (like Cantor) thought in terms of
>limitation of size.
Of course Cantor's 1883 construction of the number classes involved more
than Replacement: Essentially, the n+1st number class N is defined in
terms of the nth number class M by
*If X is a subclass of N of power less than or equal to M, then its Sup
is in N*
where by Sup I mean greater than each element in X. This implicitly
involves limitation of size, but obviously more, since N is of power
greater than M.
>The strongest argument against the view that in 1908 he had in mind the
>cumulative hierarchy is the absence of any postulate ensuring
>well-foundedness--Parsons alludes to this in his mention of
>Mirimanoff--which is surely the hallmark of the cumulative hierarchy.
Mirimanoff defined the notion of a well-foiunded set; but it was v
Neumann who first introduced the axiom of regularity in print. Michael
Hallett (in his introduction to the translation of Zermelo's 1930 paper)
refers to Bernays 1941 ``A system of set theory Part II'' in JSL for
evidence that Zermelo ``was in possession of what has become known as the
`von Neumann conception' as early as 1915''. But I am not sure what that
extends to. Hallett is mostly speaking of the representation of ordinals
by sets and I haven't looked at the Bernays paper.
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