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In the special case when φ<sub>0</sub>(α)=ω<sup>α</sup>
this family of functions is known as the '''Veblen hierarchy'''.
The function φ<sub>1</sub> is the same as the [[Epsilon numbers (mathematics)|ε function]]: φ<sub>1</sub>(α)= ε<sub>α</sub>.<ref>Stephen G. Simpson, ''Subsystems of Second-order Arithmetic'' (2009, p.387)</ref> If <math>\alpha < \beta \,,</math> then <math>\varphi_{\alpha}(\varphi_{\beta}(\gamma)) = \varphi_{\beta}(\gamma) \,.</math> From this and the fact that φ<sub>β</sub> is strictly increasing we get the ordering: <math>\varphi_\alpha(\beta) < \varphi_\gamma(\delta) </math> if and only if either (<math>\alpha = \gamma </math> and <math>\beta < \delta </math>) or (<math>\alpha < \gamma </math> and <math>\beta < \varphi_\gamma(\delta) </math>) or (<math>\alpha > \gamma </math> and <math>\varphi_\alpha(\beta) < \delta </math>).
=== Fundamental sequences for the Veblen hierarchy ===
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