Re: Largest Set in ZFC?

On Mar 8, 8:45 am, "Peter Webb" <webbfam...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:

The definition is:

{S} = {S S}

and, of course {S S} we get from pairing..

Thanks. I was curious how that worked.
If S is a set then pairing says {S,S} is a set
and extensionality says {S,S} = {S}.

*** I had never even noticed the issue, interesting.

It seems AoI is deliberately vague about what
sets are members of X. Why is that?

I don't know the motivation of the actual people who wrote it, but the
reason I personally would give is this: We don't NEED to be any more
specific, since separation does the rest of the job for us to get to
the specific set w that we want.

It is not obvious to me that separation is powerful enough.
What if X is uncountable? We can only specify a countable
number of subsets of X. What if the subset we "want" can't
be specified with a finite formula?
And exactly which set do we want?

Saying "we don't need to be more specific" still sounds
like we are being deliberately vague. I think the reason
for vagueness is so we can "assume" the AoI set
contains limit ordinals (other than 0).

*** I know much less about this topic than you do,

I'm sure a lot of people disagree with you on this.

but it seems there is a
more fundamental reason for the vagueness of the definition of X. If anybody
could produce a specific such X, then they would have a model of ZF (a
collection of sets that satisfy ZF). This would prove the consistency of ZF.
AFAIK, there is no theoretical reason why such a set X could not be
described, but none ever has, and its seems pretty obvious that none ever
will. Hence the requirement to vague it up a bit.

We can derive omega from any set that satisfies AoI.
AoI says a set exists. It doesn't say more than one
such set exists. Since omega is defined as the "smallest"
set satisfying AoI, I see no reason to assume any
other set satisfies AoI.

Your argument suggests omega is a model of ZF.
Can't we prove omega exists in ZF?

I don't see how ZF can have a set with greater
cardinality than omega (without using Powerset).


Russell
- 2 many 2 count
.

Relevant Pages

  • Re: Largest Set in ZFC?
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    (sci.logic)
  • Re: Reals without infinity
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    (sci.math)
  • Re: Panu Raatikainens review of two of Chaitins books.
    ... "true for no reason"? ... > CALLED OMEGA, AND IT IS RANDOM, THEREFORE THE WHOLE OF MATHEMATICS IS ... That's much worse than anything Chaitin has written. ... All of mathematical knowledge can be expressed by Omega? ...
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  • Re: Why 2.00000001?
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  • Re: Why 2.00000001?
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    (talk.origins)