Do surreals prove reals are countable?

Question

If the surreal number $\epsilon = 1/\omega$ is the lower bound of the difference between any two real numbers (since it is smaller than any real number), and there are a countable number of these infinitesimals in any interval (since $1/\omega∗ω=1$), how are the reals in any interval not countable?

Answer 1

The mistake is your claim

there are a countable number of these infinitesimals in any interval (since ${1\over \omega}\cdot\omega=1$).

You seem to be conflating the surreal number $\omega$ and the countable ordinal $\omega$. These are very different objects$^*$!

To get an initial hint of why your idea breaks down, consider the surreal number $\sqrt{\omega}$. It is infinite (= bigger than every natural number) but $<\omega$, and indeed $<{\omega\over n}$ for every natural number $n$. This means that we have $$0<{1\over\omega}<{2\over\omega}<{3\over\omega}<...<{\sqrt{\omega}\over\omega}={1\over\sqrt{\omega}}<{2\over\sqrt{\omega}}<{3\over\sqrt{\omega}}<...<{1\over 3}<{1\over 2}<{1\over 1}=1.$$ We can "add ${1\over\omega}$ to itself infinitely many times" and still wind up infinitesimal.

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$^*$OK fine, they're the same object, but they're playing very different roles. E.g. there's no ordinal $\omega-1$, but there absolutely is a surreal number $\omega-1$, and so forth. The point is that in general you can't transfer ideas about ordinals to ideas about surreals, even about the surreals-which-are-ordinals. (And even when you can it takes work.)