User:Emlightened/xkcd Notations/Psi's Letters

This is the first attempt of large continuous googology I've seen; it's continuous up to \(\omega^3\).

Capital letters are operators. The unary version is continuous in its variable, and the binary one is continuous in the left variable, but not the right,

\(\\aEb = a10^b \\Ea = 1Ea = 10^a\)

\(\\aF1 = Ea\) \\aFb = E(aF(b-1)) (b>1) \\Fa = 10^a (0\leq a<1) \\Fa = EF(a-1) (a\geq1\)

\(\\aG1 = Fa\) \\aGb = F(aG(b-1)) (b>1) \\Ga = 10^a (0\leq a<1) \\Ga = FG(a-1) (a\geq1\)

\(\\aH1 = Ga \\aHb = G(aH(b-1)) (b>1) \\Ha = 10^a (0\leq a<1) \\Ha = GH(a-1) (a\geq1\)

\(\\aJb = 10^a (0\leq a<1) \\aJb = ((a-1)Jb)J(b-1) (a\geq1,b>1) \\aJ1 = 10^a \\Ja = (2\times5^{a-\lfloor a\rfloor})J\lfloor a\rfloor\)

\(\\aK1 = Fa (0\leq a<2) \\aK1 = Ja (a>2) \\aKb = 10^a (a\leq1,b>1) \\aKb = ((a-1)Kb)K(b-1) (a>1,b>1) \\Ka = (2\times5^{a-\lfloor a\rfloor})K\lfloor a\rfloor (a>2) \\Ka = 1\kappa_1a (all a)\)

\(\\a\kappa_1b = aKb \\a\kappa_nb = 10^a (a\leq1) \\a\kappa_n1 = (10^{a-1})\kappa_{n-1}1 (11) \\a\kappa_n1 = (2\times5^{a-\lfloor a\rfloor})\kappa_{n-1}\lfloor a\rfloor(a\geq2,n>1) \\a\kappa_nb = ((a-1)\kappa_nb)\kappa_n(b-1) (a>1,b>1,n>1) \\\kappa_na = a\kappa_{n+1}1 \\La = \kappa_110^{a-1} (1\leq a<2) \\La = \kappa_{\lfloor a\rfloor}(2\times5^{a-\lfloor a\rfloor}) (a\geq2)\)

Unfortunately, I don't know how to get \(M\). It used the digits of the positive part of it's argument for collapsing across \(\omega^\omega\), but the post's gone now.

E,F,G,H J,K,L