User blog:Emlightened/Hypernomials (Γ₀)

Hypernomials

Define \(H\) to be the least set such that \(\{(X\rightarrow0),(X\rightarrow1),(X\rightarrow X)\}\subseteq H\), and \(H\) is closed under addition and \(X\uparrow^fg\rightarrow h\) for functions \(f\), \(g\) and \(h\) of \(X\). \(H\) is the set of hypernomial functions. The order type of the hypernomial functions, under eventual domination, is \(\Gamma_0\).

We can use hypernomial functions to create a simple array notation, comprable to SuperJedi224's X-Sequence notation.

Hypernomials
Define Hypernomial Array Notation as follows:

\(\\c\langle 0\rangle n'\#=c\langle 0\rangle n+x \\X\langle a\rangle 1\#=x \\c\langle a'\rangle k'\#=c\langle a\rangle c\langle a'\rangle k \\X\langle A\rangle k\#=(X\langle A\#\rangle k)' \\n\langle A\rangle k\#=n\langle A\#\rangle k\)

Where:


 * \(A\) and \(B\) are limits; they are not of the form \(a+1\), for any \(a\).
 * \(a\) and \(b\) are anything.
 * \(n\) and \(k\) are nonnegative integers.
 * \(c\) is a nonnegative integer, or \(X\).
 * \(x\) is the base of the entire array (the leftmost number).
 * \(\langle a\rangle\) is right-associative.
 * \(+\) is treated as right-associative.
 * \(X\) does not have a value, but could be compared to \(\omega\).

And \(\#\) is repeatedly appended to the array until it is reduced to a single number.

The limit of hypernomials is \(\Gamma_0\). If no \(X\) is involved, then the limit is under \(\omega^\omega\omega\) on the Hardy Hierarchy.

I'm too lazy to write out a full analysis right now, but \(a\langle\lambda\rangle a\) is roughly equal to \(f_\lambda(a)\), with the \(X\)'s in the first expression being replaced by \(\omega\)'s in the second, with \(X\langle1+\alpha\rangle\omega\beta\approx\varphi(\alpha,\beta)\).