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Re: Matheology § 288
Posted:
Jun 20, 2013 1:22 PM
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On 20 Jun., 15:41, "Julio Di Egidio"
> So, there is no such thing as an "additional step" to take, we
> just consider the limit of the sequence...
No. You seem to have not yet understood. But I enjoy to explain it
again, and again, and again, ... because the story is incredible.
Every set of the sequence
1,
1, 2
1, 2, 3
...
is the union of all its predecessors and its last {n}.
There are infinitely many sets, so there are infinitely many unions.
None of these infinitely many unions yields |N, although every union
adds something. The set of all these sets exists like the set of all
finite approximations of 1/9:
0.1
0.11
0.111
...
Why should it not exist?
After you pause, exhausted, since you failed to get |N but got only
sets which a lacking aleph_0 natural numbers, you pray and do a last
union. This last union cannot supply anything new, because the union
of a set with its subsets cannot yield a larger set. (And we know,
none of the sets is |N. All sets are subsets of one set because of
inclusion monotony. Each of the sets is infinitely far away from being
|N.) But your prayer has been accepted. The last union, you would not
believe it, yields |N.
I love this story really so much!
Regards, WM
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