# Re: Infinities

Keith Elis (hagbard@ix.netcom.com)
Sun, 16 Nov 1997 11:59:46 -0500

John K Clark wrote:

> -----BEGIN PGP SIGNED MESSAGE-----
>
> On Fri, 14 Nov 1997 Harvey Newstrom <harv@gate.net> Wrote:
>
> >It seems obvious to me from a physical standpoint, that the 2 x
> >infinity inches volume is twice as large as the 1 x infinity inches
> >volume. It would take two of the former added together to equal the
> >latter. If you superimposed the 1 x infinity inches volume inside
> >the 2 x infinity inches volume, you would have 1 x infinity inches
> >volume left over.
>
> Infinite numbers do not obey the same laws of arithmetic that finite numbers
> do, however, you do count them, that is determine how big they are, in
> exactly the same way, by putting them in a one relationship with something
> else. I know that if I can put each of the fingers on my right hand in a one
> to one correspondence with some apples and have no apples or fingers left
> over, then there must be 5 apples. In the same way I can put the odd integers
> in a one to relationship with all the integers, both the odd AND the even,
> so there must be an equal number of both.
>
> 1 -1
> 3- 2
> 5- 3
> 7- 4
> 9- 5
> .
> .
>
> Or you can prove that all lines are composed of the same number of points
> regardless of length. Draw 2 parallel lines, a short one and a long one below
> it, pick a point midway along the short line but above it.
>
>
> /\
> / \
> /________\
> / \
> /________________\
>
>
> Draw a line from that point to any place on the short line, then continue it
> until you hit the long line. 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.
>
> But not all infinities are equal. Let's try to put the integers in a one to
> one correspondence with all the points in the line from 0 to 1 expressed as a
> decimal.
>
> 1 - 0.a1,a2,a3,a4,a5 ...
> 2 - 0.b1,b2,b3,b4,b5 ...
> 3 - 0.c1,c2,c3,c4,c5 ...
> 4 - 0.d1,d2,d3,d4.d5 ...
> .
> .
>
> The trouble is it doesn't work, there are decimals not included, for example,
> the point 0.A1,B2,C3,D4,E5 ... where A1 is any digit except a1, B2 is any
> digit except b2, C3 is any digit except c3 etc. This point differs in at
> least one decimal place with any point in our one to one scheme, we've used
> all the integers but there are still points remaining, so there must be more
> points on a line than integers.

There are an infinite number of points between 0 and 1. There are an infinite
number of points between 1 and 2. What about all the points between 0 and 2?
There is an infinite number, yes, but it is composed of the points from 0 to 1
and 1 to 2. Does this mean that the number of points between 0 and 2 is double
that of both?

No, it's just infinite. Does it really make sense to say that all infinities are
not equal? I mean, infinity is used in mathematics thanks to a symbol that
supposedly quantizes it, or at least allows us to represent the mathematical
concept of infinity. But is infinity ever a quantity? We can't really ever grasp
infinity anymore than we can grasp the size of [(10^100)^100]^100. The difference
is that such big numbers (positive) have a meaningful upper limit that we can use
to define its value. Infinity does not. So any number that does not have a
meaningful upper limit (or lower limit, if negative) is equal to every other
number that has no such limit. Infinity is infinity no matter in what context it
occurs.

If infinities are not equal (this infinity is greater than that infinity) then
that means that some infinities are even larger, and then some infinities are
even larger than that, ad infinitum. So then the ultimate infinity is the
infinity that is larger than all the rest -- the inifinitely infinite infinity.
But then what about an infinity even greater than that....

Blah, blah, blah....

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.