Multiple infinities - one more look

multiple infinities - lets check them "really"

I was reading about cardinalities and the diagonal argument.

For me, here is how things are. For integer sequences such as 1,2,3,..
and 2,3,4,... there is a possibility that one can always insert a new
number at the end. Instead of single numbers, if we consider ordered
pairs or triplets, we can still insert new members only at the end
because there is a way of enumerating them as a single sequence by
means of a diagonal traversal. This traversal is made up of mini
traversals with alternating direction in which we traverse a finite
(?) number of steps in each direction (diagonal argument for
rationals). Programmatically, it is possible to run a for loop to
generate this sequence as long as we don't run out of memory and time.
Ordered N-tuples require a for loop nesting level of N.

Instead of ordered pairs and triplets, if we consider "ordered
infinite" sets, we can still produce a sequence using the diagonal
traversal but each mini traversal takes infinite number of steps right
from the first mini traversal. It is possible to write a for loop
here, but the nesting level N hits infinity.

Actually even in the case of ordered pairs (rationals), a single mini
traversal takes infinite number of steps when the numbers in the pairs
hit infinity.

What's big deal here? Does hitting infinity in the first mini
traversal automatically qualify it as cardinality of the spatial
continuum?

If something is countable, why should be called infinite? I don't
think it is possible to count infinite number of things even by
imagination or logic. Doesn't counting fail because there are no
adjacent numbers at infinity just like we don't have adjacent real
numbers? If there are adjacent numbers at infinity, then adding one to
infinity should produce a new number. So I believe infinite is always
uncountable by definition.

Purnam adah purnam idam
purnat purnam udacyate
purnasya purnam adaya
purnam evavasisyate - Isha Upanishad

(http://en.wikipedia.org/wiki/Infinite)

- venkat
.

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