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The basic premise of the U function is:

U(n) is the limit of the n-th extension of my array notation (counting Extension - as the first, etc.)

The UNAN ruler breaks each extension up, it basically amounts to splitting it up some point in the middle of it. THis can be done as many times as you want to get as precise a number as you want.

Note, in the following tables, I'm only going up to the end of extension 16 or U(17) as I don't really have it fully planned out after that. And it starts at U(-1) = 1 and U(0) = 10.

n = 0 Ruler

U value My array notation expression (10[x]10) FGH ordinal or expression (f_a(n))
-1 number 1 0
0 0 1
1 0,1 \(\omega\)
2 0,0,1 \(\omega^2\)
3 0{1}1 \(\omega^\omega\)
4 0{2}1 \(\omega^{\omega^2}\)
5 0{0,1}1 \(\omega^{\omega^\omega}\)
6 0{0{1}1}1 \(\omega^{\omega^{\omega^\omega}}\)
7 0{0,11}1 \(\varepsilon_0\)
8 0{0,10,11}1 \(\zeta_0 = \psi(\Omega)\)
9 0{0{1}11}1 \(\varphi(\omega,0) = \psi(\Omega^\omega)\)
10 0{0{0,11},11}1 \(\Gamma_0 = \psi(\Omega^\Omega)\)
11 0{0{0{1}11}11}1 \(\psi(\Omega^{\Omega^{\Omega^\omega}})\)
12 0{0,21}1 \(\psi(\varepsilon_{\Omega+1})\)
13 0{0,31}1 \(\psi(\varepsilon_{\Omega_2+1})\)
14 0{0,0,11}1 \(\psi(\Omega_\omega)\)
15 0{0,0,111}1 \(\psi(\Omega_\Omega)\)
16 0{0,0,0,111}1 \(\psi(\Omega_{\Omega_\omega})\)
17 0{0/0/1}1 \(\psi(\psi_I(0))\)

n = 1 Ruler

U value My array notation expression (10[x]10) FGH ordinal or expression (f_a(n))
-1 number 1 0
-0.5 number 2 f_0(f_0(n))
0 0 1
0.5 1 2
1 0,1 \(\omega\)
1.5 0,2 \(\omega2\)
2 0,0,1 \(\omega^2\)
2.5 0,0,0,1 \(\omega^3\)
3 0{1}1 \(\omega^\omega\)
3.5 0{1}0{1}1 \(\omega^{\omega2}\)
4 0{2}1 \(\omega^{\omega^2}\)
4.5 0{3}1 \(\omega^{\omega^3}\)
5 0{0,1}1 \(\omega^{\omega^\omega}\)
5.5 0{0,0,1}1 \(\omega^{\omega^{\omega^2}}\)
6 0{0{1}1}1 \(\omega^{\omega^{\omega^\omega}}\)
6.5 0{0{0,1}1}1 \(\omega^{\omega^{\omega^{\omega^\omega}}}\)
7 0{0,11}1 \(\varepsilon_0\)
7.5 0{0,10{0,11}1}1 \(\varepsilon_{\varepsilon_0}\)
8 0{0,10,11}1 \(\zeta_0 = \psi(\Omega)\)
8.5 0{0,10,10,11}1 \(\varphi(3,0) = \psi(\Omega^2)\)
9 0{0{1}11}1 \(\varphi(\omega,0) = \psi(\Omega^\omega)\)
9.5 0{0{0{0,11}11}1 \(\varphi(\varepsilon_0,0) = \psi(\Omega^{\varepsilon_0})\)
10 0{0{0,11},11}1 \(\Gamma_0 = \psi(\Omega^\Omega)\)
10.5 0{0{0,10,11}11}1 \(\psi(\Omega^{\Omega^\Omega})\)
11 0{0{0{1}11}11}1 \(\psi(\Omega^{\Omega^{\Omega^\omega}})\)
11.5 0{0{0{0,11}1}1}1}1 \(\psi(\Omega^{\Omega^{\Omega^\Omega}}\)
12 0{0,21}1 \(\psi(\varepsilon_{\Omega+1})\)
12.5 0{0{0,21}21}1 \(\psi(\Omega_2^{\Omega_2})\)
13 0{0,31}1 \(\psi(\varepsilon_{\Omega_2+1})\)
13.5 0{0,41}1 \(\psi(\varepsilon_{\Omega_3+1})\)
14 0{0,0,11}1 \(\psi(\Omega_\omega)\)
14.5 0{0,0{0,11}11}1 \(\psi(\Omega_{\varepsilon_0})\)
15 0{0,0,111}1 \(\psi(\Omega_\Omega)\)
15.5 0{0,0,211}1 \(\psi(\Omega_{\Omega_2})\)
16 0{0,0,0,111}1 \(\psi(\Omega_{\Omega_\omega})\)
16.5 0{0,0,0,1111}1 \(\psi(\Omega_{\Omega_\Omega})\)
17 0{0/0/1}1 \(\psi(\psi_i(0))\)

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