[FOM] Might there be no inaccessible cardinals?
Kenny Easwaran
easwaran at berkeley.edu
Fri Nov 2 18:58:46 EDT 2007
> "In ZFC we can neither prove nor disprove the existence of inaccessible
> cardinals."
>
> Can anyone (Mosterin perhaps?--I don't have his email address) enlighten
> me as to the meaning of "cannot" ("can neither") here?
I think it just means what it says - ZFC+no inaccessibles and
ZFC+inaccessibles are both consistent. Of course, there's no
non-question-begging mathematical proof for either of these claims.
> In the case of "prove" there is no question: "cannot" means it is
> impossible, since there are models of ZFC too small to include an
> inaccessible cardinal.
This is true if there are models of ZFC at all - that is, we know that
if ZFC is consistent, then so is ZFC+no inaccessibles.
As for showing that ZFC doesn't prove the non-existence of
inaccessibles, this is tantamount to showing the consistency of
ZFC+inaccessibles. Of course, there is no proof in ZFC that can show
this, and any proof of the consistency of ZFC+inaccessibles will most
likely involve some sort of axiom that begs the question.
You're right that there's some further points we can make in the case
of "prove" - if we grant the consistency of ZFC (which is in some
sense a precondition for this question being at all interesting), then
we can show that ZFC doesn't prove the existence of an inaccessible.
By Godel, we can't do anything just from the consistency of ZFC to
prove the consistency of inaccessibles.
But we might wonder why one would grant the consistency of ZFC. Most
likely this is because so many people have worked so productively with
ZFC and have tested it, and never found a contradiction. However, the
same seems true for ZFC+inaccessibles. As long as this system is
consistent, then the original claim is correct - in ZFC we can neither
prove nor disprove the existence of inaccessibles.
As for whether ZFC+inaccessibles is consistent - well, there won't be
any useful proof that it is (unless maybe someone proves its
consistency from some other interesting system whose place in the
consistency hierarchy is not currently known), but on the other hand,
there are plenty of good reasons to think that it is consistent. I
don't know anything about "fine structure" and "inner model theory",
but they seem to give people in the know a good deal of evidence that
this theory (and stronger ones) is consistent. Also, the most natural
"proofs" of the consistency of ZFC involve using an inaccessible, so
if we have reason to believe that ZFC has natural models, then we have
reason to believe inaccessibles are at least consistent.
Finally, I don't know of any set theorists that believe ZFC is
consistent, but think ZFC+inaccessibles isn't. Does anyone else know
if there are any?
Kenny Easwaran
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