More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals α<sub>β</sub>, provided that all but a finite number of them are zero. Notice that if such a sequence of ordinals is chosen from those less than an uncountable [[regular cardinal]] κ, then the sequence may be encoded as a single ordinal less than κ<sup>κ</sup>. So one is defining a function φ from κ<sup>κ</sup> into κ.
The definition can be given as follows: let <u>α</u> be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) ''which ends in zero'' (i.e., such that α<sub>0</sub>=0), and let <u>α</u>[0↦γ0@γ] denote the same function where the final 0 has been replaced by γ. Then γ↦φ(<u>α</u>[0↦γ0@γ]) is defined as the function enumerating the common fixed points of all functions ξ↦φ(<u>β</u>) where <u>β</u> ranges over all sequences which are obtained by decreasing the smallest-indexed nonzero value of <u>α</u> and replacing some smaller-indexed value with the indeterminate ξ (i.e., <u>β</u>=<u>α</u>[ι<sub>0</sub>↦ζ@ζ,ι↦ξι@ξ] meaning that for the smallest index ι<sub>0</sub> such that α<sub>ι<sub>0</sub></sub> is nonzero the latter has been replaced by some value ζ<α<sub>ι<sub>0</sub></sub> and that for some smaller index ι<ι<sub>0</sub>, the value α<sub>ι</sub>=0 has been replaced with ξ).
For example, if <u>α</u>=(ω↦1ω@1) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(ω↦1ω@1) is the smallest fixed point of all the functions ξ↦φ(ξ,0,...,0) with finitely many final zeroes (it is also the limit of the φ(1,0,...,0) with finitely many zeroes, the small Veblen ordinal).
The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., which cannot be reached "from below" using the Veblen function of transfinitely many variables) is sometimes known as the [[Large Veblen ordinal|"large" Veblen ordinal]].
