[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.