User blog comment:Alemagno12/What if ill-defined numbers, numbers that end in an infinite loop, numbers that don't have a definition, etc, were valid, well-defined numbers?/@comment-30167082-20161231114508

Let me help you out.

Let's assume some ill-defined expression could be well-defined to some machine. Well, we would have to tell the machine, even a sentient one (a human), what is implied by the expression. For example, "complete and well-defined", "system of maths" etc. Otherwise, it would be well-defined to whom? A deity?

Moreover, googologists could reach nor get past them, since googology is confined to the infinite set of well-defined expressions. Also, how could one extend "a complete and well-defined system of mathematics"? We have to continue constructing larger and larger pools of infinite possibility which we have no possibility of understanding nor, eventually, improving. Consider "maximum number crafted in at most n-symbols of any language of thought, speech or expression".