Are the reals a "subset" of the class of ordinals

Question

I am not sure if it's even correct to use subset in this context but I'm sure it gets the point across. I just want to know if the class of ordinals includes non-integer elements like $4.5$, $\pi$, $e$ among others. And if yes, can we combine the transfinite ordinals with non-integer constants like $\omega + \frac{1}{4}$ or $2\pi\cdot\omega$ for example?

Answer 1

Ordinal numbers arise when one tries to construct natural numbers within the axioms of the set theory. Basically, you start from 0 as an empty set, and then you define operation of "adding 1" in a language of simple operations on sets. As a consequence, you note that, using this definition, nothing stops you from "adding 1" to the whole set of naturals, and there you go.

But reals cannot be obtained from repeated "addition of 1" to naturals, so no, they aren't included in ordinals. Instead, one first constructs rationals from naturals and then uses Dedekind cuts (or any alternative approach).

However, within the funny number system called surreal numbers, one can define both ordinals and reals and give a meaning to expressions like $2\pi \cdot \omega$, $\frac{\omega}{2}$ and so on. I highly recommend "Surreal Numbers" by Knuth if you're interested in this topic.

As John Conway wrote about them: