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In [[ordinal]] analysis, a '''normal function''' is a function \(f: \text{On} \mapsto \text{On}\) that is strictly increasing and continuous. That is, \(\alpha < \beta \Rightarrow f(\alpha) < f(\beta)\) and \(\alpha \in \text{Lim} \Rightarrow f(\alpha) = \sup\{f(\beta) : \beta < \alpha\}\).
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In [[ordinal]] analysis, a '''normal function''' is a function \(f: \text{On} \to \text{On}\) that is strictly increasing and continuous. That is, \(\alpha < \beta \Rightarrow f(\alpha) < f(\beta)\) and \(\alpha \in \text{Lim} \Rightarrow f(\alpha) = \sup\{f(\beta) : \beta < \alpha\}\), where \(\text{Lim}\) denotes the [[class (set theory)|class]] of non-zero limit ordinals.
   
 
A trivial example of a normal function is the identity function \(f(\alpha)=\alpha\). Less trivial examples include functions such as \(f(\alpha)=1+\alpha\) or \(f(\alpha)=\omega^\alpha\).
 
A trivial example of a normal function is the identity function \(f(\alpha)=\alpha\). Less trivial examples include functions such as \(f(\alpha)=1+\alpha\) or \(f(\alpha)=\omega^\alpha\).

Latest revision as of 22:59, March 25, 2020

In ordinal analysis, a normal function is a function \(f: \text{On} \to \text{On}\) that is strictly increasing and continuous. That is, \(\alpha < \beta \Rightarrow f(\alpha) < f(\beta)\) and \(\alpha \in \text{Lim} \Rightarrow f(\alpha) = \sup\{f(\beta) : \beta < \alpha\}\), where \(\text{Lim}\) denotes the class of non-zero limit ordinals.

A trivial example of a normal function is the identity function \(f(\alpha)=\alpha\). Less trivial examples include functions such as \(f(\alpha)=1+\alpha\) or \(f(\alpha)=\omega^\alpha\).

Normal functions are subject to an important property given by the fixed-point lemma, which states that a normal function has arbitrarily large fixed points. The derivative \(f'\) of an ordinal function \(f\) is defined as the enumerating function of the fixed points of \(f\), so we can say that every normal function has a derivative. Not only this, but the derivative of a normal function is itself a normal function — so every normal function also has a second derivative, third derivative, etc. Even further, the procedure can be iterated transfinitely (by taking the common fixed points of lower derivatives) and we will always have normal functions. This forms the basis of the Veblen hierarchy, which is created from increasing derivatives of the function \(f(\alpha)=\omega^\alpha\).

See also Edit

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence · ordinal collapsing function · transfinite induction
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\omega_\alpha^\text{CK}\) · \(\Pi_n\)-reflecting ordinal · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal collapsing functions Madore · Buchholz · Jäger
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · Class (set theory)
Other concepts: Veblen function · absolute infinity

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