Re: probabilistic definition of an ultrafilter?

On Sat, 07 Mar 2009 14:05:44 -0800, Ben Crowell
<crowell09@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

Can anyone tell me whether this idea for defining a free ultrafilter
is right, wrong, stupid, trivially equivalent to some other method...?
Obviously it can't be an explicit *construction* of a free ultrafilter
within ZFC, because no such construction exists. So either it's wrong
or it just doesn't qualify as an explicit construction, for the correct
technical definition of "construction."

Let U be the set of subsets of the integers defined so that X is a
member of U iff X is, with unit probability, a superset of any finite
set of integers defined by a randomly chosen first-order logical
formula.

Well, to start, I have no idea what that means.

There's no "natural" or "standard" definition of
"randomly chosen first-order formula". If we're talking
about actual mathematical probability theory then before
we can say anything about "randomly chosen" we need
to specify, or somehow postulate the existence of,
a "distribution".

Since there are only countably many formulas this
means we have to start with a number P(phi) >= 0,
one for each formula phi, such that the sum of the
P(phi) = 1. Hence P(phi) > 0 for at least one phi.

Now there are two possibilities:

Case 1 There exists phi with P(phi) = 0.

This is a very un-natural notion of "randomly
chosen formula" - some formulas are chosen
with positive probability but some are
chosen with probability zero?

Whether or not anything makes sense in Case 1,
it certainly can't be what you meant by "randomly
chosen first-order formula" - this would be like
someone talking about a "random positive integer"
and then it turns out that P(1) = 1/3 but P(2) = 0;
the choice is not "random" in the rough intuitive
sense of the word.

Case 2: P(phi) > 0 for all phi.

In this case your condition says that X is a superset
of every finite set of integers. Again, not what you meant.

We need:
(1) U is closed under supersets.
(2) U is closed under intersections.
(3) For any set of integers, U contains either that set or its
complement.

1 clearly holds. I would conjecture that 3 holds, since it seems
unlikely that any X has a superset-probability 0<P<1. I would
also conjecture that 2 holds, based on naive multiplication of
probabilities. Even if 2 and 3 do hold for a particular first-order
language, and a particular definition of "randomly chosen," I suspect
it would be pretty much impossible to prove them.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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