On the sound of sounding ridiculous, but in the line of "There are no stupid quetsions": Is there a way to express $\omega_1$ (and in general $\omega_k$ with $k >= 1$ as a Conway game (that is $
I can express $\omega_1 = < {i}_{i\in\mathbb{R}} | >$, which is essential the same as $< f: \mathbb{R} \to \mathbb{R} |>$ where $f$ is increasing (${i}_{i\in\mathbb{R}} = f(i)$), which isn't exactly a Conway game notation. I think this also borders on an idea of Gonshor, to express the surreals as maps from initial segments of ordinals to a two-element set). In general, I have replaced the sequence $<0,1,2, ...|>$ from one of the forms of $\omega_0$ by an increasing "sequence" where the index is a positive real number.
Questions are:
- The above question(s) ?
- Is my idea correct or at leat in the right direction ?
- Are there any other ways to do this ? (of course this is an open question if my idea is wrong in the first place).
And yes, I realize my question is little bit broad and I haven't an idea which model you should be work in (ZF, ZFC, NBG, etc), nor do I know how the answer varies with the choice of the model).