On 7/19/2013 2:32 PM, Stephen wrote:
>
> Damn! that was what I was going to do but thought that it was too big a
> leap to say that infinity * infinity = infinity. From what I had written
> before.
> So out of (a passing) interest what is the difference?
>
> PS the passing is my little joke. ;-)
>

I'm trying to think of an answer which won't make you regret asking that 
question :)

For some definitions of infinity there isn't a difference and 2*infinity 
= infinity*2 = infinity

The most common definition of infinity for which there *is* a difference 
can be thought of as reasoning about (possibly infinitely long) 
sequences of things.  In order to not get into the confusing details 
about this, let me just say that it turns out that if you follow this 
logic through you find that you're allowed to take a sequence, A, and 
create a new sequence by replacing each entry in A with another sequence B.

Let me give an example.  I'll denote each element in a sequence tieh the 
character @.  You can represent a finite number by a sequence of a 
certain length, so 1 is "@", 2 is "@@", 3 is "@@@", 7 is "@@@@@@@" etc. 
  So let's say we take the sequence for 3:

@@@

and replace each element in it by the sequence for 2, the we get:

(@@)(@@)(@@)

Where I've put parentheses in for readability.  Writing it without the 
parens you get:

@@@@@@

Which is the sequence of 6.  And what a coincidence 2*3 = 6!  In fact, 
this strange process works exactly like the multiplication you've come 
to know and love for all finite numbers.  Since it works just like 
multiplication, we'll just say that this is what we mean by multiplying 
two of these sequences together (if it walks like a duck and it quacks 
like a duck).  Here's a slightly more precise definition:

If A and B are sequences, then by "A*B" we mean the sequence which would 
result by replacing each element in B by the sequence A.

By this definition:

2*3 = (@@)*(@@@) = (@@)(@@)(@@) = @@@@@@ = 6
3*2 = (@@@)*(@@) = (@@@)(@@@) = @@@@@@ = 6

So we get 2*3 = 3*2 = 6 like you'd expect.  But notice that the 
parentheses were different in that middle step.  This is where 
everything goes to hell when you allow infinite sequences.

Let me give an example.  Say you've got your basic infinite sequence 
(which I'll write as "w" since writing "infinity" get confusing when you 
have more than one infinity).  So:

w = @@@@@...

Where the "..." just means "goes on forever".  I won't force you to 
endure a mathematically precise definition of that.  So with our 
previous definition of multiplication:

2*w = (@@)(@@)(@@)... = @@@@@@... = w
w*2 = (@@@@...)(@@@@...) = @@@@...@@@@... = now you may be confused

I don't have a way to make that w*2 feel like it's not a little crazy, 
but the best I can do is restate it in plain English:

2*w = "count in forever in multiple of two"
w*2 = "count forever, then when you're done, do it again."

So while counting forever by multiples of two is basically the same as 
counting forever normally, counting forever and then doing it again is 
another thing entirely.

Provided you're willing to accept that any of this makes sense at all, 
that's what the difference is (for that definition of "number" of 
course, which surprisingly is a relatively common one).

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