Recall that $\omega_1$ is defined as: $$\omega_1 = \{\text{Set of all countable ordinals}\}$$ and the cantor diagonal set is defined as: $$T = \{ s \in S: s \not\in f(s) \}$$ where $f : S \to \mathcal{P}(S)$ not surjective We note from this proof due to Carl Mummert that $\omega_1$ is uncoun...

This is based on Ex. 6.4.6 in Stillwell's "Real Numbers." Using previous exercises, it was established that one can construct for any countable ordinal $\gamma$ disjoint half-open intervals $[a_{\alpha}, a_{\alpha+1})$ for all $\alpha\lt\gamma$, with the properties $a_{\alpha}\lt a_{\beta}$ iff ...

This is a great question! There are a large uncountable number of these orders. Consider as a background order the long rational line, the order $\mathcal{Q}=([0,1)\cap\mathbb{Q})\cdot\omega_1$. For each $A\subset\omega_1$, let $\mathcal{Q}_A$ be the suborder obtained by keeping the first point...

Ok, so maybe that wasn't the best way of phrasing the question, but I think it's specific enough. Let me explain myself a bit more below in case I am wrong. So I'm assuming (although I've never checked) that the irrational numbers are defined as simply all reals that are not rational. I'm askin...

I recently came upon the following theorem which was attributed to J. Conway: For each $A\subset \omega_1$, let $\Phi(A)$ be a linear ordering of type $\sum_{\alpha<\omega_1} \tau_\alpha$, where $\tau_\alpha$ is $\eta$ (the order-type of the rationals) whenever $\alpha\notin A$, and $1+\eta$ whe...