eternal notation for fat infinities

so i've devised this array function that creates cardinals/ordinals i guess

1. you can choose between a cardinal and an ordinal by prefixing the array with "C" or "O". for example: C[0, 1] = $\aleph_0$

2. one array entry represents the number input. example: O[7] = 7 as an ordinal

3. two entries represent different things based on prefix. example: C[0, 1] = $\aleph_0$ and O[0, 1] = $\omega$

those are just simple examples. let's get cooking.

in ordinal arrays, we can add ordinals beyond omega to the entries to represent higher omegas. examples are O[ $\omega+2$ ] = $\omega_{2}$ and O[ $\omega^2$ ] = $\omega_{\omega_{\omega_{...}}}$

we may run into a problem however, so we can use omega-one to remove cases of nested omegas. O[ $\omega_1 + {\omega_1}2$ ] = $\omega_{{\omega_1}2}$

in cardinality, it's meaningless in one entry arrays, but can be used in two+ entry arrays to make sets of aleph numbers and the aleph numbers after those. an example is C[ $\omega+1$ , 1] = $\aleph_{\omega+1}$

oh forgot to mention, in cardinality the 1st entry represents the aleph-number, the 2nd entry represents the nested alephs. C[a, b+1] = $\aleph_{C[a, b]}$

you can add .5 to the end of the first entry in cardinality to represent a beth number, an extra pair of square brackets next to the second filled with a string of 0s and 1s to indicate which are beths and which aren't. $C[3.5, 4[1011]] ~ = ~ \beth_{\aleph_{\beth_{\beth_3}}}$

this string of 0s and 1s can be put into exponential terms, 0^a representing a constant a-long string of 0s before moving on, same for 1^a. thus the infinite string of beth's would be called C[n.5, C[0, 1][1^{C[0, 1]}]]

we know what we can do with omega-one in the ordinal prefix, but what about cardinal? cardinal omega-one represents the inaccessible cardinals. C[ $\omega_1$ ] = $\aleph_0$ , since aleph-null is the smallest inaccessible cardinal, but we can push further, through the aleph numbers, using C[ $\omega_1 + K$ ]- NO! YOU CAN'T MASH TOGETHER CARDINALS AND ORD- haha omega-one + kappa go brr we don't care that cardinals and ordinals don't click together like that, it's all about the notation at this point.

actually, scratch that. we don't care about the aleph-numbers, we already have notation for those. all of those. C[0, 0, 1] is the pity limit of alephs. we also have a pity limit of beths. we're skipping straight to the strongly inaccessible. C[ $\omega_1$ ] = $\theta$ , and C[ $\omega_1 + n + 1$ ] = the smallest cardinal strongly inaccessible from C[ $\omega_1 + n$ ].