[FOM] Non-absoluteness and Replacement

joeshipman@aol.com joeshipman at aol.com
Tue May 22 13:43:37 EDT 2007

Randall Holmes has argued that the intuitive picture of the cumulative 
hierarchy V, along with the intuitive principle that any "Universe" we 
have our hands on can be extended, do not in fact suffice to justify 
the Axiom of Replacement.

Specifically, if we have a definable proper class function F with 
domain a set A, it is not possible to argue that F[A] must be a set on 
the grounds that you can just extend the universe so that the old class 
of ordinals becomes a set ordinal, because if F is defined by a formula 
which contains quantifiers over the whole universe V, the meaning of 
the quantifiers changes when the universe is extended past the old 
V_omega, and the class F[A] may have a different extension after the 
extension. (The pun was unintentional, but I don't see a better way to 
say this...).

Holmes points out that you can at least motivate Sigma_2 Replacement, 
along these lines, because Sigma_2 formulas are absolute in an 
appropriate sense.

Can anyone suggest an example of a (previously known) theorem in ZFC 
whose proof requires more than Sigma_2 Replacement?

I don't think it will be easy to find such an example.

It is reminiscent of the situation in a recent post of mine, where I 
argued against large cardinal axioms which are defined with reference 
to arbitrarily large sets, as too vague, and suggested they be replaced 
with "internal" axioms (so instead of assuming "there exists a 
supercompact cardinal", assume "there exists kappa such that V_kappa |= 
'there exists a supercompact cardinal' ").  I don't think we lose any 
consistency strength in this way, and we get to avoid formulas higher 
than Sigma_2.

-- JS

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