I will just use this blog for testing random LaTeX stuff. This blog will likely grow in length over time xD
\(\textrm{四}\)
\begin{eqnarray*} r: J & \to & S \\ s & \mapsto & r(s) \\ \\ o: J & \to & T \\ s & \mapsto & o(s) \\ \\ h: J & \to & T \\ s & \mapsto & o(s) \\ \\ q: J & \to & \mathbb{N} \\ s & \mapsto & o(s) \\ \end{eqnarray*}
Let \(\mathfrak{C}\) denote the set of computable well-orderings on \(\mathbb{N}\). Let \(\leq_{\mathfrak{C}}\) be a well-ordering on it, defined as \(s \leq_{\mathfrak{C}} t \iff \neg (\text{otyp}(t) < \text{otyp}(s))\).
I define the map \(\prec: n \mapsto \prec_n\) as the unique order isomorphism \((\mathbb{N}, \leq) \to (\mathfrak{C}, \leq_{\mathfrak{C}})\).
Then, I define the predicate \(\text{WO}_k(s, t)\) (\(k, s, m \in \mathbb{N}\)) like so:
- If \(s \equiv t \equiv 0 \mod 2\), then \(\text{WO}_k(s, t) \iff s \prec_k t\).
- If \(s \equiv t \equiv 1 \mod 2\), then \(\text{WO}_k(s, t) \iff \text{WO}_{k+1}(\frac{s-1}{2}, \frac{t-1}{2})\)
- If \(s \equiv 0 \mod 2\) and \(t \equiv 1 \mod 2\), then \(\text{WO}_k(s, t)\) holds.
- If \(s \equiv 1 \mod 2\) and \(t \equiv 0 \mod 2\), then \(\text{WO}_k(s, t)\) does not hold.
Well-orderings[]
- \(s <_{\omega_1^{\text{CK}}} t \iff \text{WO}_0(s, t)\ \text{ holds}\)
For all cardinals \(\kappa\), regular cardinals \(\theta \geq \kappa\) and natural numbers \(n\), for all \(\mathcal{M}_1, \mathcal{M}_2, \cdots, \mathcal{M}_n\) so that for all \(1 \leq m \leq n\), \(\mathcal{M}_m \in C_\kappa(\mathcal{V}_m, \theta)\), there exist \(\mu_1, \mu_2, \cdots, \mu_n\) so that for all \(1 \leq m \leq n\), \(\mu_m \in D_\kappa(\mathcal{W}_m, \mathcal{M}_m)\) and there does not exist some \(p \leq n\) satisfying the following:
- \(\bigcup \limits_{q < p} \mu_q \in D_\kappa((\bigcup \limits_{q < p} \mathcal{M}_q), \emptyset)\)
- \(\bigcup \limits_{q < p} \mu_q\) has a well-founded ultrapower
- \(((\bigcup \limits_{q < p} \mathcal{M}_q), \in, (\bigcup \limits_{q < p} \mu_q)) \models "\text{For any sequence } \vec{X} = \langle X_\alpha: \alpha < \kappa \rangle \textrm{ of elements of } (\bigcup \limits_{q < p} \mu_q) \text{, the diagonal sequence } \triangle \vec{X} \in \bigcup \limits_{q < p} \mu_q"\)
\(茶\)