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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.[1] 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[]

  1. Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).

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
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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