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|Section 1='''Basics:''' [[cardinal|cardinal numbers]] {{*}} [[normal function]] {{*}} [[ordinal notation]] {{*}} [[ordinal|ordinal numbers]] {{*}} [[fundamental sequence]] {{*}} [[ordinal collapsing function]]
 
|Section 1='''Basics:''' [[cardinal|cardinal numbers]] {{*}} [[normal function]] {{*}} [[ordinal notation]] {{*}} [[ordinal|ordinal numbers]] {{*}} [[fundamental sequence]] {{*}} [[ordinal collapsing function]]
 
|Section 2='''Theories:''' [[Presburger arithmetic]] {{*}} [[Peano arithmetic]] {{*}} [[Kripke–Platek set theory|KP]] {{*}} [[second-order arithmetic]] {{*}} [[Zermelo–Fraenkel set theory|ZFC]]
 
|Section 2='''Theories:''' [[Presburger arithmetic]] {{*}} [[Peano arithmetic]] {{*}} [[Kripke–Platek set theory|KP]] {{*}} [[second-order arithmetic]] {{*}} [[Zermelo–Fraenkel set theory|ZFC]]
|Section 3='''Countable ordinals:''' {{mathlink|omega|\omega}} {{*}} {{mathlink|ε₀|\varepsilon_0}} {{*}} {{mathlink|Cantor's ordinal|\zeta_0}} {{*}} {{mathlink|Feferman–Schütte ordinal|\Gamma_0}} {{*}} {{mathlink|Ackermann ordinal|\vartheta(\Omega^3)}} {{*}} {{mathlink|small Veblen ordinal|\vartheta(\Omega^\omega)}} {{*}} {{mathlink|large Veblen ordinal|\vartheta(\Omega^\Omega)}} {{*}} [[Bachmann-Howard ordinal|\(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\)]] {{*}} {{mathlink|Ψ(Ω_ω)|\psi(\Omega_\omega)}} {{*}} {{mathlink|Takeuti-Feferman-Buchholz ordinal|\psi(\varepsilon_{\Omega_\omega + 1})}} {{*}} {{mathlink|Ψ(ψᵢ(0))|\psi(\psi_I(0))}}‎ {{*}} {{mathlink|omega one chess|\omega_1^\mathfrak{Ch}<nowiki/>}} {{*}} {{mathlink|Church-Kleene ordinal|{{wck}}}} {{*}} [[infinite time Turing machine|\(\lambda,\zeta,\Sigma,\gamma\)]] {{*}} [[List of countable ordinals]]
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|Section 3='''Countable ordinals:''' {{mathlink|omega|\omega}} {{*}} {{mathlink|ε₀|\varepsilon_0}} {{*}} {{mathlink|Cantor's ordinal|\zeta_0}} {{*}} {{mathlink|Feferman–Schütte ordinal|\Gamma_0}} {{*}} {{mathlink|Ackermann ordinal|\vartheta(\Omega^3)}} {{*}} {{mathlink|small Veblen ordinal|\vartheta(\Omega^\omega)}} {{*}} {{mathlink|large Veblen ordinal|\vartheta(\Omega^\Omega)}} {{*}} [[Bachmann-Howard ordinal|\(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\)]] {{*}} {{mathlink|Ψ(Ω_ω)|\psi(\Omega_\omega)}} {{*}} {{mathlink|Takeuti-Feferman-Buchholz ordinal|\psi(\varepsilon_{\Omega_\omega + 1})}} {{*}} {{mathlink|Ψ(ψᵢ(0))|\psi(\psi_I(0))}}‎ {{*}} {{mathlink|omega one chess|\omega_1^\mathfrak{Ch}<nowiki/>}} {{*}} {{mathlink|Church-Kleene ordinal|{{wck}}}} {{*}} [[Admissible ordinal|\(\omega_\alpha^\text{CK}\)]] {{*}} [[infinite time Turing machine|\(\lambda,\zeta,\Sigma,\gamma\)]] {{*}} [[List of countable ordinals]]
 
|Section 4='''Ordinal hierarchies:''' [[Fast-growing hierarchy]] {{*}} [[Slow-growing hierarchy]] {{*}} [[Hardy hierarchy]] {{*}} [[Middle-growing hierarchy]] {{*}} [[N-growing hierarchy]]
 
|Section 4='''Ordinal hierarchies:''' [[Fast-growing hierarchy]] {{*}} [[Slow-growing hierarchy]] {{*}} [[Hardy hierarchy]] {{*}} [[Middle-growing hierarchy]] {{*}} [[N-growing hierarchy]]
 
|Section 5='''Uncountable cardinals:''' {{mathlink|first uncountable ordinal|\omega_1}} {{*}} [[omega fixed point]] {{*}} [[inaccessible cardinal|inaccessible cardinal \(I\)]] {{*}} [[Mahlo cardinal|Mahlo cardinal \(M\)]] {{*}} [[weakly compact cardinal|weakly compact cardinal \(K\)]] {{*}} [[indescribable cardinal]] {{*}} [[rank-into-rank cardinal]] {{*}} [http://cantorsattic.info/ more...]
 
|Section 5='''Uncountable cardinals:''' {{mathlink|first uncountable ordinal|\omega_1}} {{*}} [[omega fixed point]] {{*}} [[inaccessible cardinal|inaccessible cardinal \(I\)]] {{*}} [[Mahlo cardinal|Mahlo cardinal \(M\)]] {{*}} [[weakly compact cardinal|weakly compact cardinal \(K\)]] {{*}} [[indescribable cardinal]] {{*}} [[rank-into-rank cardinal]] {{*}} [http://cantorsattic.info/ more...]

Latest revision as of 08:18, January 16, 2020

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence · ordinal collapsing function
Theories: Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\omega_\alpha^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · 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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