FOM: Further comment on the iterative conception

Moshe' Machover moshe.machover at kcl.ac.uk
Thu Jan 22 06:27:11 EST 1998

Further to my previous posting, I would like to add that the failure of the
iterative conception of set *per se* to justify Replacement is no light
matter.

Replacement is of course naturally justified on the grounds of Limitation
of Size. Cantor himself had this idea: 'Two equivalent [ie equipollent]
multiplicities are either both "sets" or are both inconsistent [ie proper
classes]' (Letter to Dedekind, in van Heijenoort, p. 114).

But the Power-set axiom is not so naturally justified on grounds of
limitation of size.

This is why the idea of a transfer (or reflection) principle from V_\omega
upwards to V is one of the most alluring foundational goals: it would
clearly justify *both* Replacement and Power-set, as it combines the two
conceptions into one. The idea is that in the cumulative hierarchy

(*) hereditarily finite : ordinary infinite :: ordinary infinite : Absolute.

Quite a few people (including myself) have had this idea, as it is most
natural. But no-one so far has been able to restrict the [false] principle
(*) in a natural way so as to make it correct. No-one has had the required
combination of technical skill and foundational insight.

This is why Harvey Friedman's project in this direction is, as far as I am
concerened, the most exciting promise in f.o.m. for a very long time.

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