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==== Conditions for canonicalness ====
The notations thus defined have the property that whenever they nest <math>\psi</math> functions, the arguments of the "inner" <math>\psi</math> function are always less than those of the "outer" one (this is a consequence of the fact that the <math>\Omega</math>-pieces of <math>\alpha</math>, where <math>\alpha</math> is the largest possible such that <math>\psi(\alpha)=\delta</math> for some <math>\varepsilon</math>-number <math>\delta</math>, are all less than <math>\delta</math>, as we have shown above). For example, <math>\psi(\psi(\Omega)+1)</math> does not occur as a notation: it is a well-defined expression (and it is equal to <math>\psi(\Omega) = \zeta_0</math> since <math>\psi</math> is constant between <math>\zeta_0</math> and <math>\Omega</math>), but it is not a notation produced by the inductive algorithm we have outlined.
Canonicalness can be checked recursively: an expression is canonical if and only if it is either the iterated Cantor normal form of an ordinal less than <math>\varepsilon_0</math>, or an iterated base <math>\delta</math> representation all of whose pieces are canonical, for some <math>\delta=\psi(\alpha)</math> where <math>\alpha</math> is itself written in iterated base <math>\Omega</math> representation all of whose pieces are canonical and less than <math>\delta</math>. The order is checked by lexicographic verification at all levels (keeping in mind that <math>\Omega</math> is greater than any expression obtained by <math>\psi</math>, and for canonical values the greater <math>\psi</math> always trumps the lesser or even arbitrary sums, products and exponentials of the lesser).
For example, <math>\psi(\Omega^{\omega+1}\,\psi(\Omega) + \psi(\Omega^\omega)^{\psi(\Omega^2)}42)^{\psi(1729)\,\omega}</math> is a canonical notation for an ordinal which is less than the Feferman–Schütte ordinal: it can be written using the Veblen functions as <math>\varphi_1(\varphi_{\omega+1}(\varphi_2(0)) + \varphi_\omega(0)^{\varphi_3(0)}42)^{\varphi_1(1729)\,\omega}</math>.
Concerning the order, one might point out that <math>\psi(\Omega^\Omega)</math> (the Feferman–Schütte ordinal) is much more than <math>\psi(\Omega^{\psi(\Omega)}) = \varphi_{\varphi_2(0)}(0)</math> (because <math>\Omega</math> is greater than <math>\psi</math> of anything), and <math>\psi(\Omega^{\psi(\Omega)}) = \varphi_{\varphi_2(0)}(0)</math> is itself much more than <math>\psi(\Omega)^{\psi(\Omega)} = \varphi_2(0)^{\varphi_2(0)}</math> (because <math>\Omega^{\psi(\Omega)}</math> is greater than <math>\Omega</math>, so any sum-product-or-exponential expression involving <math>\psi(\Omega)</math> and smaller value will remain less than <math>\psi(\Omega^\Omega)</math>). In fact, <math>\psi(\Omega)^{\psi(\Omega)}</math> is already less than <math>\psi(\Omega+1)</math>.
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