Rathjen's \(\Psi\) function

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Not to be confused with Rathjen's psi function.

Rathjen's \(\Psi\) function based on the least weakly compact cardinal is an ordinal collapsing function. A weakly compact cardinal can be defined as a cardinal \(\mathcal{K}\) such that it is \(\Pi_1^1\)-indescribable. He uses this to diagonalise over the weakly Mahlo hierarchy.

Definition

The functions \(M^{\alpha}\), \(C(\alpha,\pi)\), \(\Xi(\alpha)\), and \(\Psi^{\xi}_{\pi}(\alpha)\) are defined using simultaneous recursion in the following way:

\(M^0=\mathcal K\cap\text{Lim}\), where \(\textrm{Lim}\) denotes the class of limit ordinals.

For \(\alpha>0\), \(M^\alpha\) is the set of \(\pi<\mathcal K\) such that \(\pi\) satisfies these 3 conditions:

1. \(C(\alpha,\pi)\cap\mathcal K = \pi\)
2. \(\forall(\xi\in C(\alpha,\pi) \cap \alpha)(M^{\xi} \text{ is stationary in }\pi)\)
3. \(\alpha\in C(\alpha,\pi)\)

\(C(\alpha,\beta)\) is the closure of \(\beta\cup\{0,\mathcal K\}\) under:

• addition,
• \((\xi,\eta)\mapsto\,\)\(\varphi\)\((\xi,\eta)\),
• \(\xi\mapsto\Omega_\xi\) given \(\xi<\mathcal K\),
• \(\xi\mapsto\Xi(\xi)\) given \(\xi<\alpha\),
• \((\xi,\pi,\delta)\mapsto\Psi^\xi_\pi(\delta)\) given \(\xi\le\delta<\alpha\).

\(\Xi(\alpha)=\min(M^\alpha \cup\{\mathcal K\})\).

For \(\xi\le\alpha\), \(\Psi^\xi_\pi(\alpha)=\min(\{\rho\in M^\xi\cap\pi\,\colon C(\alpha,\rho) \cap \pi = \rho \land (\pi,\alpha)\in C(\alpha,\rho)^2\} \cup \{\pi\})\).

Ordinal notation

Rathjen created an ordinal notation associated to \(\Psi\). Readers should be careful that googologists tend to "simplify" Rathjen's OCFs being unaware of the issue that such a "simplification" might not admit an ordinal notation associated to them, because the notion of an ordinal notation is quite difficult for the majority of them.

Sources

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)