If you're having math problems, I feel bad for you, son...

[napoleon_in_rags]

20. Yeah, it guarantees the size N is infinite, not that any number in N is infinite.

So you can construct the natural numbers with 1, and with the successor function S(x) = x+1, so that calling the successor function recursively, with 1 as an input, any FINITE amount of times produces any natural number. For instance:

5 = S(S(S(S(1))))

But that being done a FINITE amount of times is critical, that's what makes the set countable. If you enumerate the naturals, every natural number will be enumerated in a finite amount of time, even though the set itself is infinite. So the number

2135307957079005346450789 is a natural, because it will be enumerated after 2135307957079005346450789 steps...A large number, but still finite. That's the quality they all have, even though the number of them is infinite. Its an infinite amount of finite sized things.

Now on the other hand, consider counting the decimals. Would could start with the range -1 to 1 with one decimal place of precision, so our first 21 numbers are -1.0, -0.9, -0.8, ... 0.9, 1.0 then our next 201 numbers would be -2 to 2 with two decimals, -2.00, -1.99, -1.98, ... 1.99, 2.00, then -3 to 3 with 3 decimals, and so on, so that any decimal number with a finite integer part and finite decimal part would be counted within a finite amount of time. So when does pi, or 0.333... get counted under this system? Never. They both have infinite decimal places, so our counting system never enumerates them in finite time. They are uncountable in this system, which is to say that don't have a finite definition. They go on forever.

However if we express 0.333... as a 1/3, than it is countable, as a rational. And if we express pi as a computer program that enumerates all its digits, its countable, by mapping natural numbers to to the binary of representations of all such computer programs. So generally when we say a number is countable, we mean its countable in some system of numeric representation, which is the same as saying it's position on the number line or complex plane can be specified within a finite amount of time. It can be described precisely with a finite amount of information.

So what about these other numbers which are truly uncountable? Not countable in any system? Well, thinking of them in decimal form, they would have a decimal part that goes on infinitely, but with no pattern at all like 0.333.. or that can be described with a computer program, like pi. They would be totally random numbers, requiring infinite information to express them. Because they have no finite definition, I could never specify the location of such a number on the number line to another person, because doing so would take an infinite amount of time.

So the specific value of any particular uncountable number must remain ill defined in the language of math.

Once you see that, you can see that when you take a number like 1/3 from the system of numeric representation where it is well defined and try to express it in a system of representation where it requires an infinite definition like the decimals, and thus becomes uncountable in that system its also ill defined within that system. Just like when you try to express pi as a rational (or even a decimal) its poorly defined. Its like e, you can express it as an infinite sum of rationals, but that doesn't make it a rational, just like an infinite sum of integers isn't an integer. Those infinite sums are just a recipe for good approximations within a more limited system of numeric representation, just like 0.333... is a recipe for good approximation within a system too limited to express the actual number 1/3. The actual number is out there, transcendent of our little system, with a finite definition only in a higher system of representation. Without discussing that finite definition, we have no basis for argument. The number we are arguing over is ill defined.