# Infinities

John K Clark (johnkc@well.com)
Sun, 16 Nov 1997 23:05:06 -0800 (PST)

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On Sun, 16 Nov 1997 Keith Elis <hagbard@ix.netcom.com> Wrote:

>big numbers (positive) have a meaningful upper limit that we can use
>to define its value. Infinity does not.

Infinity may not have a limit but it does have a precise meaning. Remove one
element from set A and make a new set B, if you can find a one to one
correspondence between A and B then set A has an infinite number of elements.
In no place does this say that all numbers that poses this property must be
equal, and in fact we know that they are not.

>Does it really make sense to say that all infinities are not equal?

Yes because it also has a precise definition, if 2 infinities can be put in
a one to one correspondence with each other then they are equal, if they can
not then one is larger than the other. It's really not a matter of dispute,
some infinities can be put in such a correspondence and some can not.

>There doesn't seem to be a worthwhile way to deal with this unless
>every infinity is just plain old infinity, unbounded, and -- to try
>to keep it within our semantic framework -- equal.

Things would certainly be simpler, and more boring, if we could do that, but
we can't because it's not true.

On Sun, 16 Nov 1997 Dan Fabulich <daniel.fabulich@yale.edu> Wrote:

>Suppose there are the same number of decimals in [0, 1] and [0, 2].

There are. The number of decimals from [0,1] is equal to the number from
[0,2] or [0,99999].

>We should be able to subtract the number of decimals in [0, 1] from
>the number of decimals in [0, 2]. If all infinities are equal, then
>these represent the same number, and the answer should be zero.

If A is an infinite number then A -A = A not zero, A+A = A too, and A*A =A,
but 2^A is not equal to A, it's a higher infinity.

On Sun, 16 Nov 1997 wolfkin@ldl.net Wrote:

>>Me:
>>You've made a one to one correspondence between all the points in
>>the short line and all the points in the long line, so they must
>>have an equal number of points.

>It seems obvious that because we admitted that the longer line *was*
>longer, after you have used all of the (infinity of) points on the
>shorter line, there will be points on the longer line in between the
>lines you drew across.

Yes, any fool can see that the longer line must have more points in it than
the shorter line, but if you can find a one to one correspondence between all
the points in the short line and all the points in the long line, and in my
last post I showed how to do it, then the fool must be wrong. The reason
trans-finite mathematics is so important is that it teaches us that something
can be blindingly obvious and also completely untrue.

>>Me:
>>we've used all the integers but there are still points remaining,
>>so there must be more points on a line than integers.

>But doesn't this apply to the odd number example above?

No. I showed how to match up every integer with an odd integer, you can't
show me any integer that doesn't have a odd mate, so the number of all
integers and the number of odd integers must be the same. In my other example
even after using all the integers I could still find a point on the line that
did not have an integer mate, so there must be more points than integers.

>If we use what appears to be this same logic, since we have used all
>of the integers, by pairing them up with only half of the *same*
>integers, it follows that the infinity of the integers is equal to
>half of itself.

True, if A is the number of integers then A/2 = A

>Therefore the whole argument must be wrong.

Why?

John K Clark johnkc@well.com

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