User blog:Fejfo/Function levels

This is inspired by https://googology.wikia.org/wiki/User_blog:Jacquesbailhache/Simmons_Ordinal_Notation. I had this idea more than a year ago but couldn't get it to work (or at least not very strong). But seeing an "example" made it much easier to extend it (even though I had to add serveral new defintions)

Range of symbols

 * \( \alpha, \beta \in \rm{Ord}\)
 * \( n,m \in \mathbb N\)
 * \( F_\alpha = \) class of all (total) functions of level \( \alpha \)
 * \( f_\alpha \) the standard function of level \( \alpha \)
 * \( g_\alpha \) any function of level \( \alpha \)
 * \( G_\alpha \subset F_\alpha \)
 * \( \beta_0, \beta_1, \cdots \) any series of ordinals such that \(\forall i\in \mathbb N \exists n \in \mathbb N : \beta_{i+1}=\beta_i[n]\)
 * \( \omega \) the smallest limit ordinal
 * \( g \) any function of a level greater than 0
 * \( x \) any argument in the domain of \( g \)

Notation convetion
we interprete \( g_\alpha g_\beta \) polymorphically as \( \sup \{ g_\alpha f_\alpha[n] g_\beta : n\in \mathbb N : \beta < \alpha[n] \}\)
 * \( (\alpha+1)[n] = \alpha \)
 * A lambda calculus like notation is used for function application. Ie: \( f a_n \cdots a_2 a_1 = f(a_n)\cdots(a_2)(a_1)\).
 * This is extended to series of arguments: \( f \beta_0 \beta_1 \cdots \beta_n = f(\beta_0)(\beta_1)\cdots(\beta_n)
 * (Polymorphism) If \( \forall n \in \mathbb N : \beta \neq \alpha[n] \) and \( \beta < \alpha \) (ie \( g_\beta \) isn't in the domain of \( g_\alpha \), but it is a lower level function)

Level

 * \( F_0 = \rm{Ord}\)
 * \( \alpha \) non-zero: \( F_\alpha = \{ f : F_{\alpha[n]} \mapsto F_{\alpha[n]} : n \in \mathbb N \}\)

Supermums of higher levels

 * \( \alpha \) non-zero:\( \sup G_\alpha g_{\alpha[n]} = \sup \{ g_\alpha g_{\alpha[n]} : g \in G_\alpha \}\)

Transfinite iteration

 * \( g^0 x = x \)
 * \( g^{\alpha+1} = g (g^\alpha x) \)
 * \( g^\alpha x = \sup \{ g^{\alpha[n]} x : n \in \mathbb N \} \)

Fundamental sequences
WIP

Standard functions
 * \( f_0 = \omega \)
 * \( f_1 \alpha = \alpha +1 \)
 * \( f_{\beta_0+1} g_{\beta_0} g_{\beta_1} \cdots g_{\beta_n} \alpha  =  g_{\beta_0}^\alpha g_{\beta_1} \cdots g_{\beta_n} f_0 \) Where \( \beta_0 \ge 1 \) and \( \beta_n = 1 \)
 * \( f_\alpha g_\beta = \sup \{ f_{\alpha[n]} g_\beta : n \in \mathbb N : \alpha[n] > \beta \} \) Where \( \alpha \) is a limit ordinal and \( \beta = \alpha[n] \) for some \( n \in \mathbb N \), the expression in the superemum may have to be interpreted polymorphically

Final function

 * \( h \alpha = f_{1+\alpha} f_0 \)