User blog:Mush9/KING function

Introduction
In this blog post, I will describe the KING function and a number stemming from it: HIGH KING. It is built as an extension of the FOOT function, but cannot absolutely promise to be much, if at all, stronger than it.

Essentially, we will formalise the language of orbor; suppose that every object written in it is an orble; suppose that a superset of all sets satisfying some particular orble \(\text{a}\) is an orbleverse and the orbleverse of some particular orble \(\text{a}\); suppose that the orbleverse of some particular orble \(\text{a}\) is an orbinal number iff the orbleverse of some particular orble \(\text{a}\) is a Von Neumann Ordinal; suppose that, conversely, some particular orble \(\text{a}\) is an orbinal expression iff the orbleverse of some particular orble \(\text{a}\) is an orbinal number.

Orb
Since we have defined all of our terms in the introduction, we will introduce a new object - \(\text{Orb})\. We need not concern ourselves with the orbinal expression of \(\text{Orb}\).

\(\text{Orb})\ is the largest orbinal number - a superset of orbinal numbers, including, as its last element, \(\text{Orb})\ itself.

Notice that we do not call \(\text{Orb})\ the largest orbleverse. It should be pretty obvious why: if we try to find the least orbinal number satisfying a particular orble, we can simply go through the elements of \(\text{Orb}\); we should not search through every orbleverse since we are only concerned with sets that can be mapped to numbers - so that we can find a number for googology.

Note: We will not find the least orbinal satisfying a particular orble. We look for all orbinals satisfying a particular orble. Of course, this means that there are an infinite set of infinite sets - there is an infinite amount of orbleverses which are infinite in length! Luckily, we can set sorts of limits in the language of orbor or precede each FOST or FOOT expression with something resembling every orbinal number smaller than the smallest orbinal number such that - this essentially will give an orbinal number equal to the smallest orbinal number which is talked about, with only a few more symbols to type it.

Orbor
While the ideas surrounding the language of orbor have been defined, the language itself has not.

Orbor is essentially an extension of the language of FOOT, which will presumably allow more, but not necessarily. We start by giving FOOT the following:
 * an infinite set of variables
 * \(\text{Orb}\)
 * \(a \in b\): true iff a is an element of b;
 * \(a = b\): true iff a is equal to b;
 * \(a ∧ b\): true iff both a and b are true;
 * \(¬ (a)\): true iff a is not true;
 * \(\exists a (b)\): true iff there is an element of \(\text{Orb}\), \(\textit{a}\), such that it satisfies b;
 * \(Ⅎ(a)\): true;

Finishing?
Soon.