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Topic: Can we count N ?!
Replies: 5   Last Post: Jun 10, 2013 12:06 PM

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LudovicoVan

Posts: 3,201
From: London
Registered: 2/8/08
Re: Can we count N ?!
Posted: Jun 9, 2013 5:28 AM
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"Julio Di Egidio" <julio@diegidio.name> wrote in message
news:kovcsm$g74$1@dont-email.me...

> It seems easy enough to count the subsets of N, i.e. P(N), by traversing
> the complete binary tree breath-first and a mapping between nodes and
> subsets of N. Something like (details omitted):
>
> ""
> "*"
> "0*", "1*"
> "00*", "01*", "10*", "11*"
> "000*", "001*", "010*", "011*", "100*", "101*", "110*", "111*"
> ...
>
> ""
> "1"
> "01", "11"
> "001", "011", "101", "111"
> "0001", "0011", "0101", "0111", "1001", "1011", "1101", "1111"
> ...
>
> {}
> {0}
> {1}, {0,1}
> {2}, {1,2}, {0,2}, {0,1,2}
> {3}, {2,3}, {1,3}, {1,2,3}, {0,3}, {0,2,3}, {0,1,3}, {0,1,2,3}
> ...
>
> Now, the immediate objection to any such attempts is that: infinite
> subsets are not captured.
>
> To which we reply, informally (as we cannot be more formal than the
> objection is!), that, despite we cannot but write down few entries of any
> infinite sequence, the sequence itself cannot be more "incomplete" than
> N={0,1,2,3,...} itself is, i.e. that infinite subsets are surely captured
> as long as N itself is fully captured. Namely, the objection is
> illogical, as an attempt at its formalisation would possibly show.


Illogical within the potentially infinite setting:

We can capture N = {0,1,2,3...} (indeed, what would such notation otherwise
mean?) by a most simple sequence of finite sets:

{0}
{0,1}
{0,1,2}
{0,1,2,3}
...

By "capturing" we mean that every initial segment of N is produced.

By the same token, the set of even numbers is captured by the sequence of
subsets of N shown initially, because the following is a subsequence of it
(i.e. these entries get produced):

{0}
{0,2}
{0,2,4}
{0,2,4,6}
...

By the same token, the sequence of subsets of N shown initially captures
*all* subsets of N, finite and (potentially) infinite.

Bottom line, within the potentially infinite, P(N) is countable.

Julio






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