About: Omega fixed point   Sponge Permalink

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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: * \(\omega_0 = \omega\) * \(\omega_{\alpha + 1} = \min\{x \in ext{On} : |x| > |\omega_\alpha|\}\) (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)

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  • Omega fixed point
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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: * \(\omega_0 = \omega\) * \(\omega_{\alpha + 1} = \min\{x \in ext{On} : |x| > |\omega_\alpha|\}\) (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)
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dbkwik:googology/p...iPageUsesTemplate
abstract
  • 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: * \(\omega_0 = \omega\) * \(\omega_{\alpha + 1} = \min\{x \in ext{On} : |x| > |\omega_\alpha|\}\) (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.
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