[hzhou321]

Infinite defined as not finite is meaningless. For example, infinity is not really a number in a conventional sense. What it is is not clear other than it is not finite.

A real number with infinite precision is equally meaningless.

(Above are not directly related to your comments. I simply like to summarize my thought).

Now to your comment. You can start counting from 0 by 2s, you'll never hit 1, but that doesn't show 1 is not countable. It only shows that 1 is not countable in this particular counting scheme. Yes, you can devise a counting scheme that never hits some numbers, doesn't really contribute to either proof or insight.

I assume you are familiar with the counting scheme of rational numbers, and in that scheme, it can hit any number within any (finite) precisions. Just as infinity, a number with infinite precision is unclear. You certainly can define it, but the definition will have infinite built-in, and it is not clear what meaning does such definition adds.

[raattgift]

Counting integers from 0 by 2s and never hitting 1 (or 3 or 5 ...) still results in a finite number of natural numbers counted in finite time, skipping a finite number of natural numbers at each step. How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?

Returning to my previous attempt, you could think of instead a successor function; for any finite natural number the immediately adjacent natural number can be found in finite time. For any real number, the immediately adjacent real number cannot be found in finite time because the step from one real number to the next is infinitesimally small.

All of these examples are "de-generalizations" of the mapping argument. Counting integers from 0 by 2s maps bijectively onto the natural numbers. The naturals map injectively and surjectively onto the reals; you exhaust all the naturals counting between 0.0 and 1.0, or 1.0 and 2.0, or even between 0.01 and 0.011.

"A real number with infinite precision is equally meaningless": uhm, integration of infinitesimals (dS, dV, ...) ?

[hzhou321]

> How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?

Here you sneaked the concept of reals in. Remember reals are defined on top of infinity. You can't have reals if we are still debating what infinity is. There are infinite amount of numbers between 2.0 and 4.0, in the same sense there are infinite amount of numbers in the natural set.

Your successor function defines any finite natural number, it does not define infinity. In the rational counting scheme, we can reach any number within any finite precision. A real number that is defined on the base of infinity precision requires infinity time to reach with the same counting scheme -- the same way infinity requires infinity time to reach by 1, 2, 3, ... So if you allow infinity time, the same way you allowed infinity in your definition of real, then all real numbers can be reached (including infinity time) by counting -- not that provide any meaning.

Calculus is based on taking limit -- that is assuming a finite precision, albeit arbitrary. Infinitesimals are still finite, not infinite. Otherwise, you cannot divide them.