Re: An exercise in set theory: How high do we get by autonomously iterating the powerset operation?
- From: Rupert <rupertmccallum@xxxxxxxxx>
- Date: 22 May 2007 20:56:44 -0700
On May 22, 10:26 pm, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
On 2007-05-22, in sci.logic, Rupert wrote:
On Apr 19, 1:01 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
This line of thought suggests the following mathematical question. What is the
least ordinal alpha with the property that alpha > omega and if there is a
well-ordering of type beta in V_alpha, beta < alpha? (Hint: consider the
fixed point of a suitable ordinal function.)
It's the first fixed point of the beth function, isn't it?
Yep.
In an attempt to have discussion about logic in sci.logic on a not utterly
inane level, let's try another exercise. As everyone knows NBG (with global
choice) is a conservative extension of ZFC. This is not difficult to prove,
but in fact I don't know of any place one could actually read a proof. So
the exercise is to prove this.
Here's what is to be done. Start with a countable model of ZFC and add to it
as classes all definable collections. This gets us a model of NBG without
global choice. The only subtlety in showing this is in showing that classes
defined with conditions with free class variables are also included. Alas,
this doesn't (necessarily) get us a model in which global choice holds, since
there is no guarantee that there is a choice function for the universe among
the classes we added. So, to remedy this apply a bit of forcing, using the
(proper class) of choice functions, ordered by inclusion -- or reverse
inclusion, depending on whether you like the "less-is-more" convention or not
-- as the forcing notion. To finish the proof, show that the set portion of
the model was not disturbed by the forcing, effectively by noting that the
intersection of any set and the new global choice function is already included
in the original model (it's just one of the set-sized choice functions).
This exercise is a rather nice introduction to forcing.
(I'll give some credit also to anyone who points me to a proof in the
literature.)
--
Aatu Koskensilta (aatu.koskensi...@xxxxxxxxx)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
There's a widely-used textbook which has a proof - but I won't reveal
it just now, so as not to spoil the exercise for those who want to try
it.
.
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