User blog:Edwin Shade/The OFF Function

The fast-growing hierarchy is arguably among the most intuitive ways to make large numbers from tranfinite ordinals, and it does this by continually decreasing the ordinals in the subscript throughout the calculation, which by the theorem that an infinite chain of continuously decreasing ordinals cannot exist, ensures that all calculations involving the FGH with countable ordinals in the subscript eventually terminates. This however is not only way to make large numbers, as one may employ a construction of finite numbers that is isomorphic to the construction of transfinite's, using them as a mold.

Since the study and production of ordinals seems to have progressed significantly farther than that of array functions, it would be good to go about constructing finite's in the way we construct transfinite's, because that will get us much farther. Now I introduce my two-argument Ordinal-Form-Function,(OFF for short), which takes as it's first input a finite number, the base, and in the second entry a transfinite ordinal. An example is below.

$$[10,\varepsilon_0]$$

The base number is to treated as isomorphic to $$\omega$$, so that if the base number is 10 and the ordinal-form is $$\varepsilon_0$$, then the resulting number would be $$10\uparrow\uparrow 10$$, as $$\varepsilon_0=\omega\uparrow\uparrow\omega$$. Hence $$[10,\omega]$$ would simply equal $$10$$.

$$[10,\varepsilon_1]$$ is $$10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10\uparrow\uparrow 10+1}}}}}}}}}}$$, as $$\varepsilon_1=\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{.^{.^{.^{\omega^{\varepsilon_0+1}}}}}}}}}}$$, where there are $$\omega$$ number of $$\omega\text{'s}$$. As $$10\cong\omega$$, the stack of 10's is 10 high in $$[10,\varepsilon_1]$$.

It will continue...