[FOM] Three logical questions around ZF

Aatu Koskensilta aatu.koskensilta at xortec.fi
Fri Jun 15 05:50:09 EDT 2007

Timothy Y. Chow wrote:
> What about the existence of Grothendieck universes? Is the existence of
> an inaccessible enough?

With an inaccessible kappa we get the existence of three Grothendieck
universes: the empty set, the set of hereditarily finite sets and
V_kappa. Usually a more generous supply is assumed, i.e. that for every
set A there is an inaccessible kappa such that A is in V_kappa. For this
the existence of a single inaccessible does not suffice, obviously.

> Maybe that doesn't qualify as an "interesting set theoretic hypothesis."

Probably not in the sense intended in the original question. If we don't
want to deal with an inaccessible, we can note that any extension of ZFC
with an iterated truth predicate (and we can be generous, allowing
iteration along class sized well-orderings that become definable in the
process of iterating the truth predicate) is naturally interpretable in
Morse-Kelley set theory. I know of no "interesting set theoretic
hypothesis" that was unsolvable in ZFC but not in MK. Stuff like the
existence of a transitive, well-founded model of ZFC don't really count.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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