Cantor's ordinal

View full site to see MathJax equation

Cantor's ordinal \(\zeta_0\) (pronounced "zeta-zero", "zeta-null" or "zeta-nought") is a small countable ordinal, defined as the first fixed point of the function \(\alpha \mapsto \)\(\varepsilon\)\(_\alpha\).. Note that some sources give the name "\(\zeta_0\)" to other ordinals, for example Rathjen has used \(\zeta_0\) to denote the Feferman-Schutte ordinal, and Sbiis Saibian has reserved the name for an ordinal \(``\omega\uparrow\uparrow\uparrow\omega\!"\) for a sufficient - although not currently complete - extension of hyper operators to ordinals.

It is equal to \(\varphi(2,0)\) using the Veblen function, \(\psi(\Omega)\) using Madore's \(\psi\) function, and \(\psi_0(\Omega^2) = \psi_0(\psi_1(\psi_1(0)))\) using Buchholz's function.

Larger zeta ordinals

Similarly to the epsilon ordinals, larger zeta ordinals can be defined as larger fixed points of the map \(\alpha\mapsto\varepsilon_\alpha\). For example, \(\zeta_1\) is the next fixed point of this that is greater than \(\zeta_0\), \(\zeta_2\) is the next greater than \(\zeta_1\), etc.

Formally:

• \(\zeta_0=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\})\)
• \(\zeta_\alpha=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\land(\forall(\beta<\alpha)(\gamma>\zeta_\beta))\})\)
• One fundamental sequence for \(\zeta_1\) is \(\zeta_1[0]\)=\(\zeta_0+1\) and \(\zeta_1[n+1]=\varepsilon_{\zeta_1[n]}\), and, in general, \(\zeta_{\alpha+1}[0]\)=\(\zeta_{\alpha}+1\) and \(\zeta_{\alpha+1}[n+1]=\varepsilon_{\zeta_{\alpha+1}[n]}\).

Fixed points

Fixed points of the zeta function \(\lambda\alpha.\zeta_\alpha\) are sometimes called \(\eta\)-ordinals, and also are equivalent to ordinals of the form \(\varphi(3,\beta)\) where \(\varphi\) denotes the Veblen function.

See also

Ordinals, ordinal analysis, and set theory

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)