User blog:Ubersketch/Hierarchy for natural numbers

Let \(O\) be some fundamental sequence system, and \(n\) be some natural number.

\(S^{O}_n\) is the set of all natural numbers lesser than or equal to some \(s_{\alpha}(m)\) where m is lesser than or equal to n, and \(\alpha\) is a string in \(O\) using n or less symbols.

Analogously:

\(H^{O}_n\) is the set of all natural numbers lesser than or equal to some \(h_{\alpha}(m)\) where m is lesser than or equal to n, and \(\alpha\) is a string in \(O\) using n or less symbols.

\(F^{O}_n\) is the set of all natural numbers lesser than or equal to some \(f_{\alpha}(m)\) where m is lesser than or equal to n, and \(\alpha\) is a string in \(O\) using n or less symbols.

This lends itself to a possible well-order, which may or may not actually work.

Let \(O-\alpha\) designate \(O\) without symbol \(\alpha\), and \(N(S^{O}_n)\) be the smallest natural number m not in \(S^{O}_n\)

\(\alpha\>\beta)\ iff \(N(S^{O-\alpha}_4)<N(S^{O-\beta}_4)\)

Of course this only works if every string has a unique fundamental sequence, and also the first 4 elements of a given string's fundamental sequence are distinct.