[FOM] The liar and the semantics of set theory (expansion)
Rupert McCallum
rupertmccallum at yahoo.com
Thu Oct 3 01:09:30 EDT 2002
--- Rupert McCallum <rupertmccallum at yahoo.com> wrote:
> In Kelly-Morse set theory plus "the universe is totally
> indescribable",
> we can prove there's a V_kappa that reflects all the second-order
> properties of the universe (without parameters), I believe. I think
> you
> can probably prove that even in KM plus "the universe is Mahlo", but
> I'm not quite sure how and I am a little uncertain about this
> conjecture.
>
No, I'm just being thick. If the universe is Mahlo, any cardinal
infallible for parameter-free second-order formulas is Mahlo, so you
can't prove such a cardinal from KM plus "the universe is Mahlo". An
cardinal infallible for parameter-free second-order formulas is
extremely Mahlo (it is a cardinal kappa which is kappa-hyper-Mahlo, for
example). Now I'm wondering whether it's pi-1,1-indescribable.
So (1) we have "extremely Mahlo" as a lower bound for the consistency
strength of a cardinal infallible for parameter-free second-order
formulas, and "totally indescribable" as an upper bound for its
consistency strength, which I would certainly hope could be improved,
and (2) "extremely Mahlo over the class of totally indescribable
cardinals" as a lower bound for the consistency strength of a cardinal
infallible for second-order formulas with parameters, and currently no
upper bound for the consistency strength (which hopefully is somewhere
short of "0=1").
Still a bit of work to be done.
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