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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
Can we count N ?!
Posted: Jun 8, 2013 9:55 AM
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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.

But they might retort with the question: what is the index of the set of
even numbers?

To which we'd have to get into non-standard integers (the ordinally
transfinite), while contending that there can be no such thing as
"unfinished sets", that the standard N and similar sets are incongruent
within a *set theory of infinite sets*, and that all infinite sets are
rather (extended-)countable, i.e. there is a bijection with the set N*, the
compactification of N, and that in omega we already have two directions of
counting, forwards from zero and backwards from w (actual infinity as a
doubly-potential infinity), etc. etc.

Anyway, the point of contention is that we do have (can produce) an answer
to the question of what is the index of the set of even numbers, it's the
standard that is inadequate to answer and even just ask that question.

Julio






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