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The omega fixed point is a small uncountable ordinal. When referred to as a cardinal, it is also called the aleph fixed point. It is defined as the first fixed point of the normal function \(\alpha \mapsto \omega_\alpha\), which is defined like so:

  • \(\omega_0 = \omega\)
  • \(\omega_{\alpha + 1} = \min\{x \in \text{On} : |x| > |\omega_\alpha|\}\) (the smallest ordinal with cardinality greater than \(\omega_\alpha\))
  • \(\omega_\alpha = \sup\{\beta < \alpha : \omega_\beta\}\) for limit ordinals \(\alpha\) (the limit of all smaller members in the hierarchy)

The omega fixed point is most relevant to googology through ordinal collapsing functions. It can itself be expressed as \(\psi_I(0)\), using the Buchholz psi function and the first inaccessible cardinal.

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
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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