Bashicu matrix system

Bashicu matrix system is a large number notation, invented by BashicuHyudora in 2014, that has not currently been proven to be well-founded. It is an extension of primitive sequence system and pair sequence system. 1-row matrices reach \(\varepsilon_0\), and 2-row \(\psi_0(\Omega_\omega)\) with respect to Buchholz's function, in the fast-growing hierarchy. We do not currently know how strong 3-row matrices are (an analysis that said that \(\begin{pmatrix}0&1 \\ 0&1 \\ 0&1 \\ 0&1\end{pmatrix}\) corresponded to the collapse of the least gap ordinal was shown to be incorrect.) It is widely believed that it is well-founded, but there are no proofs. The matrices can also be used to represent ordinals using fundamental sequences.

Definition
The definition given here is the one of BM4, the latest version. A matrix is an array of non-negative integers, like \(\big({}^0_0\;{}^1_1\;{}^2_1\big)\). A matrix \(\Big({}^{{}^{a_{00}}_{a_{01}}}_{{}^{a_{02}}_{\vdots}}\;{}^{{}^{a_{10}}_{a_{11}}}_{{}^{a_{12}}_{\vdots}}\;{}^{{}^{a_{20}}_{a_{21}}}_{{}^{a_{22}}_{\vdots}}\;{}^{{}^{\dots}_{\dots}}_{{}^{\dots}_{\ddots}}\;\Big)\) is abbreviated as (a00,a01,a02,...)(a10,a11,a12,...)(a20,a21,a22,...)... in many places, such as in the definition below.

We introduce some terminology: And here are the rules: I also let FSn(S) be G + B0 + B1 + ... + Bn-1. Then define a function Trans(M), defined as follows: Since Trans is defined by transfinite recursion along the set of all BM4 matrices, it is only well-defined if BM4 is well-founded.
 * Let S, G, B, M denote any matrices.
 * Let Si denote the (i+1)th column of S.
 * Let Si, j denote the (i+1)th element of the (j+1)th column of \(S\).
 * Let S + T for two matrices S and T be S concatenated to T.
 * Let |S| be the length of S, that is, the number of columns in it.
 * 1) [n] = n
 * 2) (S + (0))[n] = S[n2]
 * 3) If neither rule 1 or 2 applies, apply the rules below, in the order they are written.
 * 4) Let pk(m) be defined as the highest x such that Sk,x < Sk,m and that ak-1,m(x) = 1, if k>0. If x = 0, the part after the "and" does not apply.
 * 5) Let ak,m(t) be defined as returning 0 if m = 0, 1 if there exists a c > 0 such that pkc(m) = t, and 0 otherwise.
 * 6) Let z be the highest t such that Sundefined > 0.
 * 7) Let r, the bad root, be pt(|S|-1).
 * 8) Let B, the bad part, be Sr + Sr+1 + Sr+2 + ... + Sundefined.
 * 9) Let G, the good part, be S0 + S1 + S2 + ... + Sr-1.
 * 10) Let &Delta;b be Sb, - Sb,r if b < z, else 0.
 * 11) Let Bi, jk be Bi, j + ai,r+j(r) &times; &Delta;i &times; k.
 * 12) Let Bik be (B0,ik,B1,ik,B2,ik,...).
 * 13) Let Bk be B0k + B1k + B2k + ....
 * 14) Finally, let S[n] = (G + B0 + B1 + ... + Bn)[n2].
 * Trans = 0
 * Trans(S + (0, ..., 0)) = Trans(S) + 1, for any length of the column (0, ..., 0) consisting only of entry 0
 * Otherwise, Trans(M) = sup{Trans(FSi(M)) | i a natural number}

Analysis
This is a table of matrices and values of Trans for BM4. All &psi; in this table is Extended Buchholz, unless specified.

Primitive sequence system
\( \begin{eqnarray*} &=& 0 &=& 0 \\ (0) &=& 1 &=& \psi_0(0) \\ (0 \; 0) &=& 2 &=& \psi_0(0)\cdot 2 \\ (0 \; 0 \; 0) &=& 3 &=& \psi_0(0)\cdot 3 \\ (0 \; 1) &=& \omega &=& \psi_0(1) \\ (0 \; 1 \; 0) &=& \omega+1 &=& \psi_0(1)+\psi_0(0) \\ (0 \; 1 \; 0 \; 1) &=& \omega\cdot 2 &=& \psi_0(1)\cdot 2 \\ (0 \; 1 \; 0 \; 1 \; 0 \; 1) &=& \omega\cdot 3 &=& \psi_0(1)\cdot 3 \\ (0 \; 1 \; 1) &=& \omega^2 &=& \psi_0(2) \\ (0 \; 1 \; 1 \; 0) &=& \omega^2+1 &=& \psi_0(2)+\psi_0(0) \\ (0 \; 1 \; 1 \; 0 \; 1) &=& \omega^2+\omega &=& \psi_0(2)+\psi_0(1) \\ (0 \; 1 \; 1 \; 0 \; 1 \; 1) &=& \omega^2\cdot 2 &=& \psi_0(2)\cdot 2 \\ (0 \; 1 \; 1 \; 1) &=& \omega^3 &=& \psi_0(3) \\ (0 \; 1 \; 1 \; 1 \; 1) &=& \omega^4 &=& \psi_0(4) \\ (0 \; 1 \; 2) &=& \omega^\omega &=& \psi_0(\omega) \\ (0 \; 1 \; 2 \; 0) &=& \omega^\omega+1 &=& \psi_0(\omega)+\psi_0(0) \\ (0 \; 1 \; 2 \; 0 \; 1) &=& \omega^\omega+\omega &=& \psi_0(\omega)+\psi_0(1) \\ (0 \; 1 \; 2 \; 0 \; 1 \; 2) &=& \omega^\omega\cdot 2 &=& \psi_0(\omega)\cdot 2 \\ (0 \; 1 \; 2 \; 1) &=& \omega^{\omega+1} &=& \psi_0(\omega+1) \\ (0 \; 1 \; 2 \; 1 \; 1) &=& \omega^{\omega+2} &=& \psi_0(\omega+2) \\ (0 \; 1 \; 2 \; 1 \; 2) &=& \omega^{\omega\cdot 2} &=& \psi_0(\omega\cdot 2) \\ (0 \; 1 \; 2 \; 2) &=& \omega^{\omega^2} &=& \psi_0(\psi_0(2)) \\ (0 \; 1 \; 2 \; 2 \; 2) &=& \omega^{\omega^3} &=& \psi_0(\psi_0(3)) \\ (0 \; 1 \; 2 \; 3) &=& \omega^{\omega^\omega} &=& \psi_0(\psi_0(\omega)) \\ (0 \; 1 \; 2 \; 3 \; 2) &=& \omega^{\omega^{\omega+1}} &=& \psi_0(\psi_0(\omega+1)) \\ (0 \; 1 \; 2 \; 3 \; 2 \; 3) &=& \omega^{\omega^{\omega\cdot 2}} &=& \psi_0(\psi_0(\omega\cdot 2)) \\ (0 \; 1 \; 2 \; 3 \; 3) &=& \omega^{\omega^{\omega^2}} &=& \psi_0(\psi_0(\psi_0(2))) \\ (0 \; 1 \; 2 \; 3 \; 4) &=& \omega^{\omega^{\omega^\omega}} &=& \psi_0(\psi_0(\psi_0(\omega))) \\ (0 \; 1 \; 2 \; 3 \; 4 \; 4) &=& \omega^{\omega^{\omega^{\omega^2}}} &=& \psi_0(\psi_0(\psi_0(\psi_0(2)))) \\ (0 \; 1 \; 2 \; 3 \; 4 \; 5) &=& \omega^{\omega^{\omega^{\omega^\omega}}} &=& \psi_0(\psi_0(\psi_0(\psi_0(\omega)))) \\ (0 \; 1 \; 2 \; 3 \; 4 \; 5 \; 6) &=& \omega^{\omega^{\omega^{\omega^{\omega^\omega}}}} &=& \psi_0(\psi_0(\psi_0(\psi_0(\psi_0(\omega))))) \\ \end{eqnarray*} \)

To &Gamma;0
\( \begin{eqnarray*} \big({}^0_0\;{}^1_1\big) &=& \varepsilon_0 &=& \psi_0(\Omega) \\ \big({}^0_0\;{}^1_1\;{}^0_0\big) &=& \varepsilon_0+1 &=& \psi_0(\Omega)+\psi_0(0) \\ \big({}^0_0\;{}^1_1\;{}^0_0\;{}^1_0\big) &=& \varepsilon_0+\omega &=& \psi_0(\Omega)+\psi_0(1) \\ \big({}^0_0\;{}^1_1\;{}^0_0\;{}^1_0\;{}^2_0\big) &=& \varepsilon_0+\omega^\omega &=& \psi_0(\Omega)+\psi_0(\omega) \\ \big({}^0_0\;{}^1_1\;{}^0_0\;{}^1_1\big) &=& \varepsilon_0\cdot 2 &=& \psi_0(\Omega)\cdot 2 \\ \big({}^0_0\;{}^1_1\;{}^1_0\big) &=& \omega^{\varepsilon_0+1} &=& \psi_0(\Omega+1) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^0_0\;{}^1_1\;{}^1_0\big) &=& \omega^{\varepsilon_0+1}\cdot 2 &=& \psi_0(\Omega+1)\cdot 2 \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^1_0\big) &=& \omega^{\varepsilon_0+2} &=& \psi_0(\Omega+2) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\big) &=& \omega^{\varepsilon_0+\omega} &=& \psi_0(\Omega+\omega) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^2_0\big) &=& \omega^{\varepsilon_0+\omega^2} &=& \psi_0(\Omega+\psi_0(2)) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_0\big) &=& \omega^{\varepsilon_0+\omega^\omega} &=& \psi_0(\Omega+\psi_0(\omega)) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\big) &=& \omega^{\varepsilon_0\cdot 2} &=& \psi_0(\Omega+\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^2_0\;{}^3_1\big) &=& \omega^{\varepsilon_0\cdot 3} &=& \psi_0(\Omega+\psi_0(\Omega)\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^3_0\big) &=& \omega^{\omega^{\varepsilon_0+1}} &=& \psi_0(\Omega+\psi_0(\Omega+1)) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^3_0\;{}^4_0\big) &=& \omega^{\omega^{\varepsilon_0+\omega}} &=& \psi_0(\Omega+\psi_0(\Omega+\omega)) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^3_0\;{}^4_1\big) &=& \omega^{\omega^{\varepsilon_0\cdot 2}} &=& \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega))) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^3_0\;{}^4_1\;{}^4_0\big) &=& \omega^{\omega^{\omega^{\varepsilon_0+1}}} &=& \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega+1))) \\ \big({}^0_0\;{}^1_1\;{}^1_0\;{}^2_0\;{}^3_1\;{}^3_0\;{}^4_1\;{}^4_0\;{}^5_1\big) &=& \omega^{\omega^{\omega^{\varepsilon_0\cdot 2}}} &=& \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega)))) \\ \big({}^0_0\;{}^1_1\;{}^1_1\big) &=& \varepsilon_1 &=& \psi_0(\Omega\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\big) &=& \omega^{\varepsilon_1+1} &=& \psi_0(\Omega\cdot 2+1) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_0\big) &=& \omega^{\varepsilon_1+\omega} &=& \psi_0(\Omega\cdot 2+\omega) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\big) &=& \omega^{\varepsilon_1+\varepsilon_0} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_0\big) &=& \omega^{\varepsilon_1+\omega^{\varepsilon_0+1}} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega+1)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_0\;{}^3_1\big) &=& \omega^{\varepsilon_1+\omega^{\varepsilon_0\cdot 2}} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega+\psi_0(\Omega))) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\big) &=& \omega^{\varepsilon_1\cdot 2} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega\cdot 2)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\;{}^2_0\big) &=& \omega^{\omega^{\varepsilon_1+1}} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega\cdot 2+1)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\;{}^2_0\;{}^3_1\big) &=& \omega^{\omega^{\varepsilon_1+\varepsilon_0}} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega\cdot 2+\psi_0(\Omega))) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\;{}^2_0\;{}^3_1\;{}^3_1\big) &=& \omega^{\omega^{\varepsilon_1\cdot 2}} &=& \psi_0(\Omega\cdot 2+\psi_0(\Omega\cdot 2+\psi_0(\Omega\cdot 2))) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_1\big) &=& \varepsilon_2 &=& \psi_0(\Omega\cdot 3) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_1\;{}^1_0\big) &=& \omega^{\varepsilon_2+1} &=& \psi_0(\Omega\cdot 3+1) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\big) &=& \omega^{\varepsilon_2+\varepsilon_1} &=& \psi_0(\Omega\cdot 3+\psi_0(\Omega\cdot 2)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_1\;{}^1_0\;{}^2_1\;{}^2_1\;{}^2_1\big) &=& \omega^{\varepsilon_2\cdot 2} &=& \psi_0(\Omega\cdot 3+\psi_0(\Omega\cdot 3)) \\ \big({}^0_0\;{}^1_1\;{}^1_1\;{}^1_1\;{}^1_1\big) &=& \varepsilon_3 &=& \psi_0(\Omega\cdot 4) \\\ \big({}^0_0\;{}^1_1\;{}^2_0\big) &=& \varepsilon_\omega &=& \psi_0(\Omega\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^1_0\big) &=& \omega^{\varepsilon_\omega+1} &=& \psi_0(\Omega\cdot\omega+1) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^1_0\;{}^2_1\big) &=& \omega^{\varepsilon_\omega+\varepsilon_0} &=& \psi_0(\Omega\cdot\omega+\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^1_0\;{}^2_1\;{}^3_0\big) &=& \omega^{\varepsilon_\omega\cdot 2} &=& \psi_0(\Omega\cdot\omega+\psi_0(\Omega\cdot\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^1_1\big) &=& \varepsilon_{\omega+1} &=& \psi_0(\Omega\cdot(\omega+1)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^1_1\;{}^2_0\big) &=& \varepsilon_{\omega\cdot 2} &=& \psi_0(\Omega\cdot\omega\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^2_0\big) &=& \varepsilon_{\omega^2} &=& \psi_0(\Omega\cdot\psi_0(2)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^3_0\big) &=& \varepsilon_{\omega^\omega} &=& \psi_0(\Omega\cdot\psi_0(\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^3_1\big) &=& \varepsilon_{\varepsilon_0} &=& \psi_0(\Omega\cdot\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^3_1\;{}^3_1\big) &=& \varepsilon_{\varepsilon_1} &=& \psi_0(\Omega\cdot\psi_0(\Omega\cdot 2)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^3_1\;{}^4_0\big) &=& \varepsilon_{\varepsilon_\omega} &=& \psi_0(\Omega\cdot\psi_0(\Omega\cdot\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_0\;{}^3_1\;{}^4_0\;{}^5_1\big) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} &=& \psi_0(\Omega\cdot\psi_0(\Omega\cdot\psi_0(\Omega))) \\ \big({}^0_0\;{}^1_1\;{}^2_1\big) &=& \zeta_0 &=& \psi_0(\Omega^2) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^1_1\big) &=& \varepsilon_{\zeta_0+1} &=& \psi_0(\Omega^2+\Omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^1_1\;{}^2_1\big) &=& \zeta_1 &=& \psi_0(\Omega^2\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_0\big) &=& \zeta_\omega &=& \psi_0(\Omega^2\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_0\;{}^3_0\big) &=& \zeta_{\omega^\omega} &=& \psi_0(\Omega^2\cdot\psi(\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_0\;{}^3_1\big) &=& \zeta_{\varepsilon_0} &=& \psi_0(\Omega^2\cdot\psi(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_0\;{}^3_1\;{}^4_1\big) &=& \zeta_{\zeta_0} &=& \psi_0(\Omega^2\cdot\psi(\Omega^2)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_1\big) &=& \eta_0 = \varphi_3(0) &=& \psi_0(\Omega^3) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_1\;{}^2_0\big) &=& \eta_\omega &=& \psi_0(\Omega^3\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^2_1\;{}^2_1\big) &=& \varphi_4(0) &=& \psi_0(\Omega^4) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\big) &=& \varphi_\omega(0) &=& \psi_0(\Omega^\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^1_0\;{}^2_1\;{}^3_1\;{}^4_0\big) &=& \omega^{\varphi_\omega(0)\cdot 2} &=& \psi_0(\Omega^\omega+\psi_0(\Omega^\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^1_1\big) &=& \varepsilon_{\varphi_\omega(0)+1} &=& \psi_0(\Omega^\omega+\Omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^1_1\;{}^2_1\;{}^3_0\big) &=& \varphi_\omega(1) &=& \psi_0(\Omega^\omega\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^2_0\big) &=& \varphi_\omega(\omega) &=& \psi_0(\Omega^\omega\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^2_0\;{}^3_1\big) &=& \varphi_\omega(\varepsilon_0) &=& \psi_0(\Omega^\omega\cdot\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^2_0\;{}^3_1\;{}^4_0\big) &=& \varphi_\omega(\varphi_\omega(0)) &=& \psi_0(\Omega^\omega\cdot\psi_0(\Omega^\omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^2_1\big) &=& \varphi_{\omega+1}(0) &=& \psi_0(\Omega^{\omega+1}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^2_1\;{}^3_0\big) &=& \varphi_{\omega\cdot 2}(0) &=& \psi_0(\Omega^{\omega\cdot 2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^3_0\big) &=& \varphi_{\omega^2}(0) &=& \psi_0(\Omega^{\psi_0(2)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_0\big) &=& \varphi_{\omega^\omega}(0) &=& \psi_0(\Omega^{\psi_0(\omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\big) &=& \varphi_{\varepsilon_0}(0) &=& \psi_0(\Omega^{\psi_0(\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\big) &=& \varphi_{\zeta_0}(0) &=& \psi_0(\Omega^{\psi_0(\Omega^2)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_0\big) &=& \varphi_{\varphi_\omega(0)}(0) &=& \psi_0(\Omega^{\psi_0(\Omega^\omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_0\;{}^7_1\big) &=& \varphi_{\varphi_{\varepsilon_0}(0)}(0) &=& \psi_0(\Omega^{\psi_0(\Omega^{\psi_0(\Omega)})}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_0\;{}^7_1\;{}^8_1\big) &=& \varphi_{\varphi_{\zeta_0}(0)}(0) &=& \psi_0(\Omega^{\psi_0(\Omega^{\psi_0(\Omega^2)})}) \\ \end{eqnarray*} \)

To &psi;0(&Omega;2)
\( \begin{eqnarray*} \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\big) &=& \Gamma_0 &=& \psi_0(\Omega^\Omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^1_1\big) &=& \varepsilon_{\Gamma_0+1} &=& \psi_0(\Omega^\Omega+\Omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^1_1\;{}^2_1\big) &=& \zeta_{\Gamma_0+1} &=& \psi_0(\Omega^\Omega+\Omega^2) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^1_1\;{}^2_1\;{}^3_0\big) &=& \varphi_\omega(\Gamma_0+1) &=& \psi_0(\Omega^\Omega+\Omega^\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^1_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \varphi_{\Gamma_0}(1) &=& \psi_0(\Omega^\Omega+\Omega^{\psi_0(\Omega^\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^1_1\;{}^2_1\;{}^3_1\big) &=& \Gamma_1 &=& \psi_0(\Omega^\Omega\cdot 2) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_0\big) &=& \Gamma_\omega &=& \psi_0(\Omega^\Omega\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_0\;{}^3_1\big) &=& \Gamma_{\varepsilon_0} &=& \psi_0(\Omega^\Omega\cdot\psi_0(\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_0\;{}^3_1\;{}^4_1\big) &=& \Gamma_{\zeta_0} &=& \psi_0(\Omega^\Omega\cdot\psi_0(\Omega^2)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_0\;{}^3_1\;{}^4_1\;{}^5_1\big) &=& \Gamma_{\Gamma_0} &=& \psi_0(\Omega^\Omega\cdot\psi_0(\Omega^\Omega)) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\big) &=& \varphi(1,1,0) &=& \psi_0(\Omega^{\Omega+1}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^2_0\big) &=& \varphi(1,1,\omega) &=& \psi_0(\Omega^{\Omega+1}\cdot\omega) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^2_1\big) &=& \varphi(1,2,0) &=& \psi_0(\Omega^{\Omega+2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\big) &=& \varphi(1,\omega,0) &=& \psi_0(\Omega^{\Omega+\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\;{}^4_1\big) &=& \varphi(1,\varepsilon_0,0) &=& \psi_0(\Omega^{\Omega+\psi(\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\big) &=& \varphi(1,\zeta_0,0) &=& \psi_0(\Omega^{\Omega+\psi(\Omega^2)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \varphi(1,\Gamma_0,0) &=& \psi_0(\Omega^{\Omega+\psi(\Omega^\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\;{}^5_1\big) &=& \varphi(1,\varphi(1,1,0),0) &=& \psi_0(\Omega^{\Omega+\psi(\Omega^{\Omega+1})}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_1\big) &=& \varphi(2,0,0) &=& \psi_0(\Omega^{\Omega\cdot 2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^2_1\big) &=& \varphi(2,1,0) &=& \psi_0(\Omega^{\Omega\cdot 2+1}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_0\big) &=& \varphi(2,\omega,0) &=& \psi_0(\Omega^{\Omega\cdot 2+\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^2_1\;{}^3_1\big) &=& \varphi(3,0,0) &=& \psi_0(\Omega^{\Omega\cdot 3}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_0\big) &=& \varphi(\omega,0,0) &=& \psi_0(\Omega^{\Omega\cdot\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\big) &=& \varphi(\varepsilon_0,0,0) &=& \psi_0(\Omega^{\Omega\cdot\psi_0(\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\;{}^5_1\big) &=& \varphi(\zeta_0,0,0) &=& \psi_0(\Omega^{\Omega\cdot\psi_0(\Omega^2)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \varphi(\Gamma_0,0,0) &=& \psi_0(\Omega^{\Omega\cdot\psi_0(\Omega^\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\big) &=& \varphi(1,0,0,0) &=& \psi_0(\Omega^{\Omega^2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\big) &=& \varphi(1,0,1,0) &=& \psi_0(\Omega^{\Omega^2+1}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_0\big) &=& \varphi(1,0,\omega,0) &=& \psi_0(\Omega^{\Omega^2+\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\big) &=& \varphi(1,1,0,0) &=& \psi_0(\Omega^{\Omega^2+\Omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_0\big) &=& \varphi(1,\omega,0,0) &=& \psi_0(\Omega^{\Omega^2+\Omega\cdot\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\big) &=& \varphi(1,\varepsilon_0,0,0) &=& \psi_0(\Omega^{\Omega^2+\Omega\cdot\psi_0(\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \varphi(1,\Gamma_0,0,0) &=& \psi_0(\Omega^{\Omega^2+\Omega\cdot\psi_0(\Omega^\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\;{}^6_1\big) &=& \varphi(1,\varphi(1,0,0,0),0,0) &=& \psi_0(\Omega^{\Omega^2+\Omega\cdot\psi_0(\Omega^{\Omega^2})}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_1\big) &=& \varphi(2,0,0,0) &=& \psi_0(\Omega^{\Omega^2\cdot 2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_0\big) &=& \varphi(\omega,0,0,0) &=& \psi_0(\Omega^{\Omega^2\cdot\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_0\;{}^4_1\big) &=& \varphi(\varepsilon_0,0,0,0) &=& \psi_0(\Omega^{\Omega^2\cdot\psi_0(\Omega)}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\;{}^6_1\big) &=& \varphi(\varphi(1,0,0,0),0,0,0) &=& \psi_0(\Omega^{\Omega^2\cdot\psi_0(\Omega^{\Omega^2})}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_1\big) &=& \varphi(1,0,0,0,0) &=& \psi_0(\Omega^{\Omega^3}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_1\big) &=& \varphi(2,0,0,0,0) &=& \psi_0(\Omega^{\Omega^3\cdot 2}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_1\;{}^3_0\big) &=& \varphi(\omega,0,0,0,0) &=& \psi_0(\Omega^{\Omega^3\cdot\omega}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^3_1\;{}^3_1\;{}^3_1\big) &=& \varphi(1,0,0,0,0,0) &=& \psi_0(\Omega^{\Omega^4}) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^\omega}) &=& \varphi\left({}^1_\omega\right) &\color{white}=& \text{(Small Veblen ordinal)} \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^\omega}\cdot 2) &=& \varphi\left({}^1_\omega\;{}^1_0\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^2_1\big) &=& \psi_0(\Omega^{\Omega^\omega+1}) &=& \varphi\left({}^1_\omega\;{}^1_1\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^2_1\;{}^3_1\big) &=& \psi_0(\Omega^{\Omega^\omega+\Omega}) &=& \varphi\left({}^1_\omega\;{}^1_2\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^2_1\;{}^3_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^\omega\cdot 2}) &=& \varphi\left({}^2_\omega\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_0\big) &=& \psi_0(\Omega^{\Omega^\omega\cdot\omega}) &=& \varphi\left({}^\omega_\omega\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_0\;{}^4_1\big) &=& \psi_0(\Omega^{\Omega^\omega\cdot\varepsilon_0}) &=& \varphi\left({}^{\varepsilon_0}_\omega\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \psi_0(\Omega^{\Omega^\omega\cdot\Gamma_0}) &=& \varphi\left({}^{\Gamma_0}_\omega\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_0\;{}^4_1\;{}^5_1\;{}^6_1\;{}^7_0\big) &=& \psi_0(\Omega^{\Omega^\omega\cdot\psi_0(\Omega^{\Omega^\omega})}) &=& \varphi\left({}^{\varphi\left({}^1_\omega\right)}_\omega\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_1\big) &=& \psi_0(\Omega^{\Omega^{\omega+1}}) &=& \varphi\left({}^1_{\omega+1}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_1\;{}^3_1\big) &=& \psi_0(\Omega^{\Omega^{\omega+2}}) &=& \varphi\left({}^1_{\omega+2}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^3_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^{\omega\cdot 2}}) &=& \varphi\left({}^1_{\omega\cdot 2}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^{\omega^2}}) &=& \varphi\left({}^1_{\omega^2}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_0\big) &=& \psi_0(\Omega^{\Omega^{\omega^\omega}}) &=& \varphi\left({}^1_{\omega^\omega}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\big) &=& \psi_0(\Omega^{\Omega^{\varepsilon_0}}) &=& \varphi\left({}^1_{\varepsilon_0}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\;{}^6_1\;{}^7_1\big) &=& \psi_0(\Omega^{\Omega^{\Gamma_0}}) &=& \varphi\left({}^1_{\Gamma_0}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\;{}^6_1\;{}^7_1\;{}^7_1\big) &=& \psi_0(\Omega^{\Omega^{\varphi(1,0,0,0)}}) &=& \varphi\left({}^1_{\varphi(1,0,0,0)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\;{}^6_1\;{}^7_1\;{}^8_0\big) &=& \psi_0(\Omega^{\Omega^{\psi_0(\Omega^{\Omega^\omega})}}) &=& \varphi\left({}^1_{\varphi\left({}^1_\omega\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\;{}^6_1\;{}^7_1\;{}^8_0\;{}^9_1\big) &=& \psi_0(\Omega^{\Omega^{\psi_0(\Omega^{\Omega^{\varepsilon_0}})}}) &=& \varphi\left({}^1_{\varphi\left({}^1_{\varepsilon_0}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_0\;{}^5_1\;{}^6_1\;{}^7_1\;{}^8_0\;{}^9_1\;{}^{10}_1\;{}^{11}_1\big) &=& \psi_0(\Omega^{\Omega^{\psi_0(\Omega^{\Omega^{\Gamma_0}})}}) &=& \varphi\left({}^1_{\varphi\left({}^1_{\Gamma_0}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega}}) &=& \varphi\left({}^1_{\left(1,0\right)}\right)\;{}^* && \text{(Large Veblen ordinal)} \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^3_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega}\cdot\omega}) &=& \varphi\left({}^\omega_{\left(1,0\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^3_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega+1}}) &=& \varphi\left({}^1_{\left(1,1\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^3_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega+\omega}}) &=& \varphi\left({}^1_{\left(1,\omega\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^3_1\;{}^4_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega\cdot 2}}) &=& \varphi\left({}^1_{\left(2,0\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^4_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega\cdot\omega}}) &=& \varphi\left({}^1_{\left(\omega,0\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^4_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega^2}}) &=& \varphi\left({}^1_{\left(1,0,0\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega^\omega}}) &=& \varphi\left({}^1_{\left({}^1_\omega\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega^\Omega}}) &=& \varphi\left({}^1_{\left({}^1_{\left(1,0\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^5_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega\cdot\omega}}}) &=& \varphi\left({}^1_{\left({}^1_{\left(\omega,0\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^5_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^2}}}) &=& \varphi\left({}^1_{\left({}^1_{\left(1,0,0\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^6_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^\omega}}}) &=& \varphi\left({}^1_{\left({}^1_{\left({}^1_\omega\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^6_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}) &=& \varphi\left({}^1_{\left({}^1_{\left({}^1_{(1,0)}\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^6_1\;{}^7_0\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\omega}}}}) &=& \varphi\left({}^1_{\left({}^1_{\left({}^1_{\left({}^1_\omega\right)}\right)}\right)}\right) \\ \big({}^0_0\;{}^1_1\;{}^2_1\;{}^3_1\;{}^4_1\;{}^5_1\;{}^6_1\;{}^7_1\big) &=& \psi_0(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}) &=& \varphi\left({}^1_{\left({}^1_{\left({}^1_{\left({}^1_{(1,0)}\right)}\right)}\right)}\right) \\ \end{eqnarray*} \)

&ast; Dimensional Veblen, definition here

To &psi;0(&Omega;3)
\( \big({}^0_0\;{}^1_1\;{}^2_2\big) &=& \psi_0(\Omega_2) &\color{white}=& \text{(Bachmann-Howard ordinal)} \\ \big({}^0_0\;{}^1_1\;{}^2_2\;{}^0_0\big) &=& \psi_0(\Omega_2)+1\\ \)