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

On Sep 16, 11:02 am, fishfry <BLOCKSPAMfish...@xxxxxxxxxxxxxxxx>
wrote:
In article
<945c7441-c892-41fe-96ab-f0d500d70...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,



 venkat.6...@xxxxxxxxx wrote:
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.

This is actually a very good question. It was Galileo way back in 1638
who pointed out that the counting numbers 1, 2, 3, 4, ... can be put
into 1-1 correspondence with the square numbers 1, 4, 9, 16, ...

This is known as Galileo's paradox ...http://en.wikipedia.org/wiki/Galileo's_paradox

Galileo concluded that the idea of comparing sets using 1-1
corrrespondence only worked for finite sets, and not for infinite sets.
So your intuition is actually very good. One of the best mathematicians
in the world essentially came to the same conclusion you did, 370 years
ago.

A couple of hundred years later, Cantor took this idea further. He
noticed that some infinite sets can't be put into 1-1 correpondence with
other infinite sets. For example even though there is a 1-1
correspondence between the counting numbers and the rational numbers --
which is at first glance surprising -- there is NOT a 1-1 correspondence
between the counting numbers and the real numbers. That is also very
surprising!

So now we just regard 1-1 correspondence as ONE WAY of talking about the
relative size of sets. It doesn't mean that the two sets are literally
the same size -- whatever that might mean for infinite sets. It just
means that we USE this 1-1 correspondence trick whenever it's handy; and
we don't use it when it's not handy.

So sometimes we care more about the fact that N (natural numbers) is a
proper subset of Q, the rationals; and other times we care more about
the fact that there is a 1-1 correspondence between N and Q.

We are not actually saying that N and Q are the same in the precise way
that two finite sets have the same number of elements -- because we
realize that infinite sets are funny, they can be in 1-1 correspondence
with a proper subset of themselves. Rather, we just USE 1-1
correspondence whenever it helps us reason about things we're interested
in. And, the concept turns out to be extremely useful. So mathematicians
use it. But they do understand that reasoning about infinite sets is
different than reasoning about finite sets.

Does that help?

Yes. It helps a lot. Thanks.

We called something as paradox when it seems go against our intuition
but still can't be disproved using the existing notions. The 1-1
mapping as a tool to compare size of non-integer sized sets seems to
me as a flaw, accepted without question. May be it is just extended
from finite sets to infinite sets by the very intuition thinking
nothing much changes at larger scale of numbers. Who knows, numbers
might not retian their basic properties at infinite distance of
observation, and using 1-1 mapping for infinite sets might just
produce more paradoxes, not because we are not capable of disproving
them, but because we may want these paradoxes to surprise ourselves,
to keep finding more such things, and keep brushing questions as
intuition related. At some point, I feel, philosophy and physical
purpose should constrain mathematics.

- venkat
.

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