Quote:

Originally Posted by

**WhattayaBrian**
There are different sizes of infinity, but "Infinity * 2" both a) doesn't make sense mathematically as an operation, and b) if it did, it would be the same size as "Infinity * 1".

Your statement is correct, but just want to point out to everyone else that "Infinity * 2" having the same size (cardinality) as regular Infinity does not imply that they are equal.

Taking "Infinity" to mean the first/"smallest" infinite ordinal number, the cardinality of the natural numbers, expressions like "Infinity + 1" and "Infinity * 2" are perfectly well defined.

In set theory, the standard definition of the ordinal numbers are as follows:

0 is the empty set, { }.

1 is the set containing 0, {0} ( = { {} } )

2 is the set containing 0 and 1 {0, 1} ( = { {}, {{}} }

3 is the set containing 0, 1, and 2 {0, 1, 2}

etc.

In general, we define "n + 1" to be "the set containing n, and also every number inside of n." We also define < to mean that "A < B" if A is an element in the set of B (remember, numbers are just special kinds of sets in this system, and 3 really does contain 2, which is our justification for saying 2 < 3. You can think of the < sign as being equivalent to the ∈ or ⊂ sign if that makes you feel better - they're all equivalent on these particular sets.

So the "first infinity" is ω, the set of all the normal counting numbers, {0, 1, 2, 3, 4, ... }. ω is therefore both the set of all finite numbers and itself a number that is "bigger" than all of those numbers (in the sense that 3 < ω, 5 < ω, 100,000,000 < ω).

Since it's a number, we can +1 it to get the next number. Here goes:

ω + 1

= the set containing everything in ω (0, 1, 2, 3, .... ) and ω itself

= {0, 1, 2, 3, ... , ω}

which is a perfectly well defined number in our system. Since ω is a thing in ω + 1, then

ω < ω + 1

is a true statement - BUT ω and ω + 1 have the same cardinality (which is the bijection / bragging contest thing WhattayaBrian brought up earlier.) So in terms of cardinality, ω and ω + 1 have the same size (i.e., |ω| = |ω + 1| ) but at the same time, in terms of less than/greater than in terms of ordinality, ω < ω + 1.

If you extend "+ 1" into "+ n" in the sane way that gives you normal addition, and define multiplication in terms of this addition, it turns out that ω * 2 is still perfectly well defined number - it comes out to ω + ω, which is {0, 1, 2, 3, ... , ω, ω + 1, ω + 2, ω + 3, ... } which is greater than both ω and ω + 1 (and ω +2, etc) but is the same size as both of them.

so

ω < ω + 1 < ω + 500,000 < ω + ω = ω * 2

but

|ω| = |ω + 1| = |ω + 500,000| = |ω + ω| = |ω * 2| (the |x| means "the cardinality of x")

see

http://en.wikipedia.org/wiki/Ordinal_number and

http://en.wikipedia.org/wiki/Cardinal_number if your brain can still handle more.