Ordinal collapsing function: Difference between revisions

Revision as of 16:31, 7 April 2011

Revision as of 22:28, 12 April 2011

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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 conseequenceconsequence 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).