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Using Madore's psi function, the ordinal \(\psi(\Omega_{\omega})\) is a large countable ordinal that is the proof theoretic ordinal of \(\Pi_1^1\)-\(\text{CA}_0\), a subsystem of second-order arithmetic.

The subcubic graphs, which are used in definition of SCG function, can be ordered so that we can make bijection between them and ordinals below \(\psi(\Omega_{\omega})\), as well as Buchholz hydras with \(\omega\) labels removed.[1]

It is the first ordinal \(\alpha\) for which \(g_{\alpha}(n)\) in the slow-growing hierarchy catches up with \(f_{\alpha}(n)\) the fast-growing hierarchy, under the most common usage.

Sources Edit

  1. Bird, Chris. Fast-Growing Hierarchy in terms of Bird's Array Notation

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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