Definition[]
Let \(\mathfrak{C}\) denote the set of computable well-orderings on \(\mathbb{N}\). Let \(\mathfrak{C}^{\mathbb{N}}\) denote the set \(\{\textrm{code}(s): s \in \mathfrak{C}\}\), where \(\textrm{code}\) is the function that assigns a well-ordering to the minimal code of a Turing machine which computes it. Then, for \(n \in \mathfrak{C}^{\mathbb{N}}\), \(f(n)\) is the unique computable well-order which is computed by the Turing machine with code \(n\). \(\prec_n\) is the unique \(s \in \mathfrak{F}\) so that \(\forall m (m \in \mathfrak{F} \rightarrow f^{-1}(s) \leq f^{-1}(m))\), where \(\mathfrak{F} = \{s \in \mathfrak{C}: \forall m (m < n \rightarrow f^{-1}(\prec_m) < f^{-1}(s))\}\).
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 \(\text{WO}_k(s, t) \iff \frac{s}{2} \prec_k \frac{t}{2}\).
- 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.
\(s \vartriangleleft_{\omega_1^{\text{CK}}}\) is equivalent to \(\text{WO}_0(s, t)\). I will likely create a user page with further well-orderings.