A limit ordinal is a non-zero ordinal that doesn't have a predecessor[citation needed]. Formally, an ordinal \(\alpha>0\) is a limit ordinal iff \(\nexists\beta:\beta+1=\alpha\). The least limit ordinal is \(\omega\), some of the next limit ordinals are \(\omega\times2\) and \(\omega\times3\), and limit ordinals can informally be thought of as "multiples of \(\omega\)" due to how all limit ordinals are of the form \(\omega\times \beta\). Some author allows \(0\) to be a limit ordinal[citation needed], and hence a limit ordinal is sometimes called a non-zero limit ordinal. The class of limit ordinals is often denoted by \(\textrm{Lim}\).

Comparisons of sizes

An important fact about limit ordinals is that not all limit ordinals are of the form \(\beta+\omega\) where \(\beta\) is another limit ordinal. For example, \(\omega^2\) is a limit ordinal that can't be written as \(\beta+\omega\) where \(\beta\) is a limit ordinal \(<\omega^2\).

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · ordinal notation · transfinite induction
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)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert'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: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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