User blog:Edwin Shade/The Most Powerful Array Notation Yet Devised

To preface the array notation I am about to introduce, I should mention I have not worked out fully a language with which trans-infinite ordinals may be constructed, (though I'm in the midst of doing so), but in the meantime will define functions with the supposition that such a language exists.

Let $$N(n)$$ be the lowest ordinal definable in n-symbols or less using the aforementioned language. $$N(n)$$ is to be read as: Nova of n. $$N(n)$$ may be inserted into the subscript of the fast-growing hierarchy, and as long as it possesses a proper fundamental sequence, (which it must if it is to be a proper ordinal for our purposes), $$f_{N(n)}(a)$$ has a finite value. Even trans-infinite values of $$N(n)$$ that extend past $$N(\omega)$$ such as $$N(\epsilon_0)$$ can be evaluated like so.

$$f_{N(\epsilon_0)}(3) =f_{N(\omega^{\omega})}(3) =f_{N(\omega^3)}(3) =f_{N({\omega^2}2+{\omega}2+3)}(3)$$

Here we do something clever, by moving 1 away from the 3 and treating it as if were outside the Nova function as thus:

$$f_{N({\omega^2}2+{\omega}2)+2)+1$$

we can go past $$N(\omega)$$. In fact, now that we can evaluate $$N(\alpha)$$, where $$\alpha$$ is any ordinal with a well-defined fundamental sequence.

Now let $$N(N(N(...(a)))$$ be notated as $$N(a,b)$$, where there are b N's in the first expression. This is the beginning of two-entry linear Nova arrays.

Again, when we encounter an expression such as $$f_{N(\omega,\epsilon_0)}(3)$$ where the single-argument linear Nova array is iterated a trans-infinite number of times we may simplify it as shown.

$$f_{N(\omega,\epsilon_0)}(3) =f_{N(\omega,{\omega^{\omega}})}(3) =f_{N(\omega,{\omega^3})}(3) =f_{N(\omega,({\omega^2}2+{\omega}2+3)}(3)$$

Now we place 1 from the three outside the Nova function, as we did before to get past $$N(\epsilon_0)$$, thus obtaining

$$f_{N(\omega,({\omega^2}2+{\omega}2+2)+1}(3)$$

It is with this technique that we can evaluate any two-entry linear Nova array where both entries are proper trans-infinite ordinals.

Triple-entry linear Nova arrays are defined similarly to how two-entry linear Nova arrays are defined, but with a slight difference in that they use fractal iteration, described like so $$N(a,b,c)=Z_c, Z_n=N(Z_{n-1},Z_{n-1}), Z_1=N(a,b)$$

For example, $$N(\omega,\omega,3)$$ would be equal to $$N(N(N(\omega,\omega),N(\omega,\omega)),N(N(\omega,\omega),N(\omega,\omega)))$$ according to the above definition.

We can now generalize the concept of n-entry linear Nova arrays by stating $$N(n_1,n_2,n_3,...,n_a)=Z_{n_a}, Z_m=N(Z_{m-1},Z_{m-1},Z_{m-1},...,Z_{m-1}) w/ n_a Z_{m-1}'s, Z_1=N(n_1,n_1,n_3,...,n_{a-1})$$.

Now let us define two-entry planar Nova arrays, $$N(\[ \left[ {\begin{array}{c}   a \\   b \\  \end{array} } \right] \])$$ to be equal to $$N(a,a,a,...,a)$$, where there are b a's. We can evaluate linear arrays with trans-infinitely many entries by implementing the "take one and move it outside the Nova function" rule, (from now on I'm calling this the TOAMIO rule), that we employed earlier. As for triple-entry planar Nova arrays, they are defined with fractal recursion the same way that triple-entry linear Nova arrays are.

Moving past planar arrays, there are cubic arrays, which naturally denote the amount of terms in a planar array; and choric arrays, which denote the amount of terms in a cubic array. In general, nth dimensional arrays consisting solely of the entry a with b copies of that entry can be notated $$N(a(n)b)$$. There can be trans-infinitely many dimensions provided we employ the TOAMIO rule.

We can go extremely far with trans-infinitely dimensional Nova arrays inputted into the subscript of the fast-growing hierarchy, (a 'small' number like $$f_{N(\omega,\omega)}(10^100)$$ already exceeds most if not all of the well-defined numbers on this Wikia, (and that includes BIGFOOT)), but we can go further.

It is time to introduce Supernova Array notation, which begins at single-entry linear Supernova arrays. $$SN(n)$$ is to be pronounced as: Supernova of n. $$SN(n)$$ is equal to the highest ordinal that may be constructed out with the language mentioned at the beginning of this blog post, but, that also includes Nova Notation.

We may define planar, cubic, choric, and trans-infinitely dimensional Supernova arrays the same way we did for Nova arrays.

Likewise, we may define an even higher form of array notation, which implements the same definition of the Supernova array notation, but with the Supernova array notation itself. Let this be called Hypernova Array Notation

After Hypernova Array notation, there is Quasar array notation, after Quasar array notation, there is Kugelblitz array notation, after Kugelblitz array notation, there is Wrinkle array notation, after Rift array notation, there is Big Rift array notation, after Big Rift array notation, there is Big Bang array notation, and after Big Bang array notation, there is Bing-Bang Zam level array notation !

Let us now define a number now based on Bing-Bang Zam level array notation. It is $$f_{BBZ(\omega)}(10^{100})$$, where $$BBZ(n)$$ refers to Bing-Bang Zam level array notation. Such a number is so big it deserves a big name, I think a "bababadalgharaghtakamminarronnkonnbronntonnerronntuonnthunntrovarrhounawnskawntoohoohoordenenthurnuk" will do.

Well, this is the extent of my array notation for now, but expect it to get much bigger in the future. Revisions and additions will be made accordingly.