The standard real numbers do not contain any infinitesimals.
There are some nonstandard models of the real numbers that do contain infinitesimals, but they have problems. They are not
complete. They are full of holes. They contain Cauchy sequences that do not converge.
If a number field contains even a single infinitesimal, it's not
Archimedean. And any non-
Archimedean number field must necessarily be incomplete
For that reason, the hyperreals (and also the surreals) are not a satisfactory model of the continuum. If we know anything about the continuum, it's that it has no holes. The hyperreals superficially satisfy some people's intuitions of how the real numbers must be; but in fact the hyperreals are much more problematic than people realize.
Then someone mentioned the intuitionist line. As the hyperreals have the reals PLUS a lot of infinitesimals, the intuitionist line contains the reals MINUS all the noncomputable numbers.
In mathematical intuitionism or constructivism, a real number exists when its decimal expression can be produced by an algorithm. For example $$\pi$$ is computable as witnessed by the many algorithms you can find like Leibniz's series.
Since there are only countably many algorithms, the constructive real line has a lot of holes in it. The intermediate value theorem is false. The graph of a continuous function can pass right through the $$x$$-axis without intersecting it. What kind of continuum is that??
So here's this interesting thing:
* The intuitionist line is incomplete because it's too restrictive. Not enough points.
* The hyperreal line is incomplete because it's too permissive. It's got all the reals plus a cloud of infinitesimals around each real. It's got too many points.
* The standard reals are complete because they aren't too restrictive, and they aren't too permissive. They standard reals are "just right," like
Goldilocks.