On 7/16/2013 8:06 AM, scott wrote:
>
> How do I better understand that all integers have a finite number of
> digits, but there are infinite number of them? That's the bit I'm
> struggling with.
>
> The way I see it, the number of positive integers with N or fewer digits
> is 10^N, so if N is finite then 10^N has got to be finite too?
>

Ahh I see.  This is one of many counter intuitive things which happens 
with infinities.  Perhaps the issue here is that there are two different 
ways that we tend to think about sizes of things at play here, and it's 
not always made clear which one we're talking about:

1) finite vs. infinite
2) bounded vs. unbounded

#1 here refers the the number of elements within a set, for instance the 
set {3,7,9} has three elements, the set of integers has infinitely many 
elements, and the set of reals has "even more infinitely" many elements 
in a certain well defined sense.

#2 instead relates to some notion of the "size" of the different 
elements within a set.  This is tricky because depending on what you're 
doing you might care about a different definition of what "size" is. 
For example with positive integers you might care about the number of 
digits, but with real numbers you might care about the magnitude of the 
number with relation to the < and > relations (which is certainly *not* 
the same as the number of digits in it!).  Often you just figure it out 
from context.

So this tells you a little bit about the concepts, but I think that some 
examples are useful in getting the mathematical intuitions.  For instance:

* If you have a set of integers, where the number of digits in each 
integer is bounded, then there are only finitely many integers in that 
set (as you said, at most 10^N).

* If you have a set of integers where the number of digits in each 
integer is unbounded, then there are infinitely many integers in that 
set.  For although each particular integer has only a finite number of 
digits, for any particular number N, you can always "look further" and 
find an integer with more than N digits.

I think this last one is probably where your confusion comes from. 
There is more that you can say about the relationships between the 
concepts of finite/infinite and bounded/unbounded, but hopefully the 
small bit I've covered this helps clarify things a little.

You can, by the way, give precise mathematical definitions of these 
concepts, but I've decided not to do so since I wasn't sure it would 
actually make the issue more clear.

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