Re: Axiom of infinity and the set of all hereditary finite sets.

On Sep 30, 5:17 am, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On Sat, 29 Sep 2007 13:16:43 -0700, Zaljo...@xxxxxxxxx wrote:
On Sep 29, 10:57 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
Zaljo...@xxxxxxxxx writes:
Can you give me an example of a set x such that the set
{x,Ux,UUx,UUUx,.....} is infinite.
[...]

For example, let x be the set that one
might informally define by

x = {{1}, {{2}}, {{{3}}}, ...}.

For each natural number n, n is an element of U^n x but
not an element of U^k x for any k < n.

(for example, 3 is in UUUx but not in x, Ux, or UUx. Hence
UUUx is not equal to any of x, Ux or UUx. Etc.)

Let's see:

U1x= {1,{2},{{3}},{{{4}}},...........}
U2x= {2,{3},{{4}},.....}
U3x= {3,{4},{{5}},.....}

So the i-th Union of x always starts with member i.
so the i-th Union of x is never empty. Yes, I agree
this would be an example of a set x such that
{x,Ux,UUx,UUUx,....} is infinite.

Thank you.

Zuhair

The reason is because if the later set is not finite, then
this mean that for some x there is an infinite descending membership
of x

Oh? How do you prove that?

and this clearly violates Regularity.

Zuhair

--
Aatu Koskensilta (aatu.koskensi...@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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David C. Ullrich


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