[net.math] the funny nature of infinity

aark (01/26/83)

Infinity is a funny thing.

Consider two infinite sets: the set of nonnegative integers and the set
of all integers.  The first set is a subset of the second, and one would
think that the first set is half as large, whatever that means, as the
second set.  Actually, the two sets are exactly the same size.  More
rigorously, the members of the two sets can be put into one-to-one
correspondence.

One would also intuitively think that there are many more rational
numbers (the set of all numbers that can be expressed as the ratio of
two integers) than there are nonnegative integers.  Wrong again: these
two sets are also the same size (can be put into one-to-one correspondence).

Infinity is a funny thing.

Alan R. Kaminsky
Bell Laboratories, Naperville, IL
...ihnp4!ihuxe!aark

arens@UCBKIM (01/27/83)

From: arens@UCBKIM (Yigal Arens)
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To: net-math@ucbvax


That's funny?  When you finally figure that all infinities are the same, you
realize (or are told...) that there REALLY ARE more real numbers than
rational numbers!!

leichter (01/27/83)

In fact:

Def:  A is INFINITE if and only if there is a proper subset B of A such that
A and B can be put in one-to-one correspondence.

(If you are used to some other definition of "infinite" you may find it en-
lightening to prove the two definitios equivalent.)
							-- Jerry
						decvax!yale-comix!leichter

ss (01/28/83)

	"....all infinities are the same..."

	THEY ARE NOT!!!!
	as an example the number of points in a line is infinity.
	The number of points in a given area is also infinity.

	It can be shown that the second "infinity"
	is larger than the first.

Sharad Singhal

stan (01/28/83)

Referring to the note by rabbit!ss (Sharad Singhal), the points
on a line can be put into one-to-one correspondence
with the points on a plane, or area.

The only proof I can remember is one that was in Sci. Am. many years ago.
Each point is represented by real-number coordinates expressed in
decimal expansion (I know some of you doubt the validity of that, but...).

To map points on the plane into points on the line, one forms the latter's
coordinate by alternately taking digits from each of the decimal expansions
that specify the former's coordinates.

To map points on the line into points on the plane, simply reverse
the process.

		Stan King			phone: 201-386-7433
		Bell Labs, Whippany, NJ		Cornet:  8+232-7433
		room 2A-111			uucp:	 floyd!stan

dmy (01/28/83)

The "number" of points in a line segment is the "same" transcendental "number"
as the number of points in the entire infinite plane, or for that matter,
in all of 3-space, or for that matter, in all of n-space for any n.

--dmy--

mat (01/29/83)

Sharad Singal states that the number of points in a line, and the number of
points in a square are both infinite, but that the second infinity is greater
than the first ... this does not agree with what I learned.  My understanding
is that both of these infinities belong to aleph 1, and that aleph null is reserved for denumerable infinities.  Can
someone set me straight ?
					-Mark Terribile

ss (01/31/83)

I confess. I goofed. Looking over my arguments about the difference
between the number of points in a line and an area I can now see the
mistake...
Thanks for setting me straight...
Sharad Singhal.