User blog:Deedlit11/Ordinal Notations III: Collapsing Higher Cardinalities

Here we define notations for ordinals up to the proof theoretic ordinal for \(\Pi^1_1 - \text{TR}_0\).

\(\theta\) function up to \(\theta(\Omega_\omega,0)\)
We have already defined collapsing functions that take ordinals of cardinality \(\Omega\) to large countable ordinals. To get larger and larger countable ordinals, then, we merely need to define larger ordinals of cardinality \(\Omega\). Our notations already define ordinals up to \(\varepsilon_{\Omega+1}\);  We can define larger uncountable ordinals using the Veblen function, Extended Veblen function, or the Schutte Klammersymbolen. But our strongest notation for countable ordinals has been the collapsing functions themselves, so why not define a function that collapses to large ordinals of cardinality \(\Omega\)? So we add \(\Omega_2\), the next higher cardinal, to our notation, and collapse ordinals of cardinality \(\Omega_2\) to ordinals of cardinality \(\Omega\), which in turn collapse to large ordinals. Then we can add \(\Omega_3\), \(\Omega_4\), etc. So we get the following version of the \(\theta\) function. (again due to Feferman)

\(C_0 (\alpha, \beta) = \beta \cup \lbrace \Omega_\nu \rbrace, \nu \le \omega \)

\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \theta(\eta, \gamma) | \gamma, \delta, \eta \in C_n (\alpha, \beta); \eta < \alpha \rbrace \)

\(C (\alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n (\alpha, \beta) \)

\(In (\alpha) = \lbrace \beta | \beta \notin C (\alpha, \beta) \rbrace \)

\(\theta (\alpha, \beta) = \) the \(\beta\)th ordinal in \(In (\alpha)\)

The ordinal \(\theta(\Omega_\omega, 0)\) is the proof-theoretic ordinal for \(\Pi^1_1 - \text{CA}_0\). The ordinal \(\theta(\varepsilon_{\Omega_\omega + 1}, 0) \) is the Takeuti-Feferman-Buchholz ordinal.

\(\psi\) function up to \(\psi_0(\Omega_\omega)\)
Wilfried Buchholz that, using the \(\theta\) function, one could generate a set of ordinals of the same order type as the full system by using +, \(\varphi\), and \(\alpha, \nu \mapsto \theta(\alpha, \Omega_\nu)\). Using this idea, he defined the following simplified version of the \(\theta\) function:

\(C_0 (\nu, \alpha) = \Omega_\nu

\(C_{n+1} (\nu, \alpha) = \lbrace \beta + \gamma, \varphi(\beta, \gamma), \psi_\mu(\delta) | \beta, \gamma, \delta \in C_n (\alpha); \delta < \alpha; \nu \le \mu \le \omega \rbrace \)

\(C (\nu, \alpha) = \bigcup_{n = 1}^{\infty} C_n (\nu,\alpha) \)

\(\psi_\nu (\alpha) = \min \lbrace \beta | \beta \notin C(\nu,\alpha) \rbrace \)

Note that, unlike the \(\theta\) function, it is not enough to merely add the larger cardinals \(\Omega_\nu\) to the notation;  one must define additional functions \(\psi_\nu\) for \(\nu > 0\) in order to get collapsing at the higher stages. For \(\theta\), we automatically get collapsing at higher stages because \(\theta(\alpha, \beta)\) can have arbitrary cardinality.

Buchholz proved that \(\psi_0 (\varepsilon_{\Omega_\nu + 1}) = \theta (\varepsilon_{\Omega_\nu + 1}, 0)\) for \(1 \le \nu \le \omega\). It follows that \(\psi_0 (\Omega_{\omega}) = \theta (\Omega_{\omega}) = \) the proof theoretic ordinal of \(\Pi^1_1 - \text{CA}_0\).

\(\vartheta\) function up to \(\vartheta(\Omega_\omega)\)
A more modern version is the \(\vartheta\) function, defined as:

\(C_0 (\nu, \alpha, \beta) = \beta \cup \Omega_\nu

\(C_{n+1} (\nu, \alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \vartheta_\mu(\eta) | \gamma, \delta, \eta \in C_n (\alpha, \beta); \eta < \alpha; \nu \le \mu \le \omega \rbrace \)

\(C (\nu, \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n (\nu, \alpha, \beta) \)

\(\vartheta_\nu (\alpha) = \min (\lbrace \beta < \Omega_{\nu + 1} | C(\nu, \alpha, \beta) \cap \Omega_{n+1} \subseteq \beta \wedge \alpha \in C(\alpha, \beta) \rbrace \cup {\Omega_{n+1}) \)

Like before, the \(\vartheta_\nu\) functions differ from the \(\theta\) and \(\psi\) functions in that, rather than stabilizing at certain ordinals, it skips over those ordinals and continues increasing. So the \(\vartheta_\nu\) functions are 1-1 functions for each \(nu\).

We again have \(\vartheta_0 (\varepsilon_{\Omega_\nu + 1}) = \theta (\varepsilon_{\Omega_\nu + 1}, 0)\) for \(1 \le \nu \le \omega\) and \(\vartheta_0 (\Omega_{\omega}) = \theta (\Omega_{\omega}) = \) the proof theoretic ordinal of \(\Pi^1_1 - \text{CA}_0\).