User blog:Edwin Shade/Turing Machines and a Problem by Vel! Posed Half a Decade Ago

Turing Machines
In one of my prior blog posts, I asked a question which roughly was this: "I've heard of two types of numbers so far, computable and uncomputable. Are there any types of numbers beyond uncomputable numbers ?" The user who responded said no because any number must either be computable or uncomputable.

I agree completely with this, but at the same time had an idea for a class of numbers that while still being uncomputable, would be more uncomputable in a way that I will explain.

Let us take the busy beaver function as an example of a uncomputable function. This is because it would take an infinite amount of time to determine the largest terminating turing machine with n-cards due to the fact that many configurations lead to infinite loops or go on forever. However, we can enumerate the time increments themselves and assign a natural number to them, therefore the total calculation time of this function is equal to the limit of the natural numbers, or $$\omega$$. $$\omega$$ is also the number of cells available on it's tape space. Because the tape space is infinite,

Next however, we can define a machine that is infinitely more powerful than a turing machine, which I call a Continuum machine. A continuum machine functions by operating on a tape space that contains the set of all real numbers, or $$\aleph_1$$. A finite number of cards is used to generate successive states of the continuum machine as in the turing machine, and so to some states loop or go on indefinitely, and other states terminate. To calculate what states terminate however one needs an $$\aleph_1$$ amount of time. Thus any number returned as a result of a function that calculate the highest terminating number of such a continuum machine would be beyond turing uncomputable numbers, and fall into their own uncomputable class.

As we can have $$\aleph_1$$ continuum machines, we can have $$\aleph_2$$ continuum machines, whose tape space contains a set of cells equal to the set of all complex numbers on a plane. These machines will require infinitely more time powerful than $$aleph_1$$ continuum machines to evaluate, thus putting them into a higher class of 'uncomputable' than prior continuum machines.

A Possible Way To Solve Vel!'s Hyperfactorial Digit Problem
In this blog post, user Vel! asked if there are any base-10 integers that are equal to the sums of the hyperfactorial of their digits, and if so what is it ?

I believe I have a method that could be used to quickly solve this problem.

[Will finish writing this section]