Re: CH yet again.


Bill Taylor wrote:
|Still, your injunction to become more familar with L or 0# is too
|much for me, alas. Regarding L, I have been trying to grok it
|for many years now; I can follow the definition OK, but as soon as
|I try to "understand" it, I find myself at sea - I just cannot
|connect it to anything familiar in set theory, or see how it
|"feels" in itself or what consequences it might possibly have.
|It seems to be some fault in my psychological make-up, sadly.
|
|And as for 0#, I cannot even get to the point where someone
|could present me with a definition! I seem to have reached
|the limits of my understanding of abstract objects with these.

Well, if your complaint is merely that you can't feel them in
your bones, maybe you should increase your tolerance for
things that leave you cold, as it were.

It just strikes me that the appearances of L and 0# on usenet
tend always to be in relation to the same little set of facts about
them. And in fact I don't know that much more about them than
has appeared here, but just enough to "get" certain things, like
V=L implying the axiom of choice. It implies the axiom of choice
because it implies that a specific formula defines a one-to-one
correspondence between ordinals and all sets. So if V=L you
can always choose the element of a set that associates with
the smallest ordinal.

There's the fact that L is a model of ZFC. Then there's the fact that
V=L contradicts certain large cardinal axioms, whose proof I haven't
seen except in the loosest thumbnail sketch. 0# is a common
example of a set of natural numbers often believed to exist but
(if it exists) not in L. I don't know the definition of 0# either.
There's
some kind of regularity thought to occur in L... some types of
questions about n-tuples of ordinals whose answer is always
the same... and 0# encodes it, in the sense that each natural
number is in 0# iff some corresponding question about L has an
answer of "yes".

We're here spashing around in the "kiddie" end of the pool, where
the water is under a meter deep, and you state your preference for
avoiding this kind of daunting challenge. Haven't you had students
who were like this with things like limits of series? Don't you find
a way to gently prod them into getting over that initial threshold
of intimidation and getting on with it?

I agree that the Woodin stuff appears elaborate enough that one
might decide reasonably enough that it's not worth the trouble to
digest it just to extract an example of what you're asking for from
it. I don't know... maybe it would be a good hobby for awhile.

Keith Ramsay

.

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