On 7/17/2013 12:44 AM, scott wrote:
>
> I guess the problem is I'm thinking of each "integer" as a set of
> digits. This set has a finite yet unbounded number of elements. But then
> if you make a set of all possible "integers" you don't get a set with a
> finite yet unbounded number of elements, you get a set with an infinite
> number of elements.

It is a strange concept, and it's not for nothing that some people 
reject it entirely.  I don't really have an explanation which will make 
it less strange, except to mention that mathematically it might be best 
to think of "infinite" as meaning "not finite".

Let me explain a bit more.  If a set S has a finite number of elements, 
then there is some integer n such that S has n elements (this is pretty 
much the definition of "finite").  Therefore, if you can show that no 
matter which number n you pick, S will always have more then n elements, 
they s is not finite, and therefore infinite.

You probably already understand this reasoning, if which case I doubt it 
did much to make it "feel" like it makes sense.  The real strange thing 
that's going on is that you're grouping all the integers into a single 
set.  This is sort of the difference between unbounded and infinite in 
this case (i.e. with positive integers).  The individual integers in 
this set increase without bound, but it's the while set of all of them 
that's infinite.  With relation to the above definition of "finite" you 
can think that for any particular integer i, there's some n such that i 
<= n (obviously), as soon as you consider *all* the integers as a single 
thing this stops being true.

I don't know if this ever starts to make sense so much as you just start 
to get used to it.

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