User blog:Edwin Shade/There is No Limit To Googology

On the talk page for the Xi function on this Wikia, the first comment you will read on the page is this:

"I think this is where we start hitting the ceiling as to where googology can go.~FB100Z"

Later, the same person asked this question:

"So, then, we come to a Big Question: What's between computability theory and infinity?"

Sbiis Saibian gave a response, which is below.

"I don't think that's a question that can be answered. Robert Munafo makes the point that if we went beyond formalism and computability theory, we'd by definition have nothing to define. There probably is no hard limit that we are going to bump into in googology, but ... that will still all be within a formalism that we can't escape, and that will place a certain fundamental limit on what kinds of finite numbers we can express. The catch is, we'll probably never be able to describe the size of that box without barry paradoxes or worse. So some finite numbers will probably always be inaccessible to us. We know they must exist, but that's about all we can say without being self-contradictory. If it's true that we can't escape computability theory then it proves the point I was trying to make with my site: that saying we can "continue indefinitely" is not exactly true. Not at least in the sense that there is no fundamental limit"

Personally, I think we can continue indefinitely in making larger and larger numbers, for a reason I will explain in full.

In recognizing that there even exists a box, or limit to our current formal systems, we are recognizing our limitations, which paradoxically, has the effect that we are able to transcend those limitations with some ingenuity and construct even larger numbers. Suppose for instance, that a powerful system for producing numbers had been developed, let us call it system $$f$$, which is formally defined. We define a new number $$n$$ which is the largest number definable in a given number of symbols of system $$f$$. Clearly, we are diagnosing over the theory we have developed itself, and so we can transcend it. We could make a new theory which accounts for this transcendental function, $$f'$$, but then we could define yet another new number $$n'$$ as the largest number expressible in a given number of symbols of the theory $$f'$$. We can continue doing this indefinitely, even going as far as to expand upon the central axioms of $$f$$ itself and create a stronger theory, $$F$$, from which we can derive the number $$N$$, which perhaps instead of diagnolizing over the number of symbols in a theory can diagnolize over a given aspect of that theory, a particular kind of symbol or form.

Going past a given theory is as easy as the following four words: "There is something else." With the assertion that there is something beyond the system you're working in, you can always find a way past the formal while still sticking to formal rules, by defining a system that includes your formal system as a smaller portion of itself.