06/06/2022, 04:00 PM

Unless I'm the one confused here, it might be, I suggest to NOT confuse circulation/omegation with FGHs.

The omega in omegation/circulation is NOT an ordinal. It is just notation sugar, it has nothing to do with the ordinal omega. Also the operation \(o(a,b)=\lim a\uparrow^n b\) is not even well defined, let alone taking superfunctions out of it.

Also the point of FGH is that they extend Ackermann-Goodstein \(a\uparrow^n b\) like functions from \(n\in\mathbb N\) to \(n\in {\bf On}\), but not to all the ordinals but only to sufficiently small ones, i.e. transfinite ordinal that must be countable \(|\alpha|\leq\aleph_0\), and are also recursively definable in some technical sense (or you can compute the fundamental sequences).

Also there is not a single way to extend it to transfinite ordinals, but multiple ways to do it, some more natural than others, and all the various ways depend fundamentally on a choice of a system fundamental sequences. A fundamental sequence is a system of choices about how to define it for limit ordinals, and to my limited knowledge, it amounts to an algorithm of diagonalization. In other words \(a\uparrow^\omega b\) has not really something to do with the idea of infinity or limit but it is defined using a trick, something like defining \(a\uparrow^\omega b=a\uparrow^b b\).

The omega in omegation/circulation is NOT an ordinal. It is just notation sugar, it has nothing to do with the ordinal omega. Also the operation \(o(a,b)=\lim a\uparrow^n b\) is not even well defined, let alone taking superfunctions out of it.

Also the point of FGH is that they extend Ackermann-Goodstein \(a\uparrow^n b\) like functions from \(n\in\mathbb N\) to \(n\in {\bf On}\), but not to all the ordinals but only to sufficiently small ones, i.e. transfinite ordinal that must be countable \(|\alpha|\leq\aleph_0\), and are also recursively definable in some technical sense (or you can compute the fundamental sequences).

Also there is not a single way to extend it to transfinite ordinals, but multiple ways to do it, some more natural than others, and all the various ways depend fundamentally on a choice of a system fundamental sequences. A fundamental sequence is a system of choices about how to define it for limit ordinals, and to my limited knowledge, it amounts to an algorithm of diagonalization. In other words \(a\uparrow^\omega b\) has not really something to do with the idea of infinity or limit but it is defined using a trick, something like defining \(a\uparrow^\omega b=a\uparrow^b b\).

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)