Re: Is one-to-one mapping valid for comparing infinite-sized sets?

On Sep 16, 3:00 am, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Mon, 15 Sep 2008 23:31:43 +0200, Herman Jurjus wrote:
Dave Seaman wrote:
On Mon, 15 Sep 2008 19:27:57 +0200, Herman Jurjus wrote:
Michael Stemper wrote:
In article <4e603bf6-4b66-4597-bad1-a52d8bbb8...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, venkat.6...@xxxxxxxxx writes:

two sets of infinite size. Not because of the practical difficulties
in comparing or the time it takes for the comparing process to
complete. I think, atleast it should be proved logically that such
mapping is valid between two infinite sets.
Please define what you mean by the term "valid".

My attempt to rephrase venkat's question:
"Is there a convincing argument (an informal one will do) why
cardinality is an accurate formalization of the intuitive notion of
'size', for infinite sets?"

Changing the name of every element of a set does not change the size of the
set.
So if they have the same cardinality, they have the same size.
That's one direction. Now the other.

If two sets have the same size, then the elements of one are merely a
renaming of the elements of the other.

That's correct. But my question is about confirming the size (or
cordinality) itself, and about using 1-1 mapping as the technique for
this. If it is a finite set, whose size is an integer, 1-1 mapping
works perfectly and we have the size and cardinality compared
correctly. We can prove this both logically and experimentally.

How do we extend this to infinite sets? The problem here is - while n
and n+1 refer to different integers, infinite and infinite+1 do not.
When did the size of the set suddenly change from integer to non-
integer? May be 1-1 mapping requires the sizes under comparison to be
strictly integers. How can one say it can work with non-integer sizes?

Hope my question is clear now.

- venkat



Neither of those arguments is rigorous, but what can you expect when
"size" is undefined?

--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>

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