User blog:LittlePeng9/Formalizing googology - naming schemes

For quite a while we have had around here users which were complaining about the low level of formality when it comes to many things on the wiki (I admit I was one of them, though perhaps not the most notorious one). Today I wanted to propose a long-term project whose goal would be to formalize most (perhaps all) of googology in order to put it on par with other branches of mathematics.

In this post I would like to start off this project by outlining a formal foundation to one of the more popular parts of googology, especially among newcomers - the theory of naming schemes. It turns out that it can be explained in the language of algebraic geometry, a widely accepted branch of mathematics.

Algebraic geometry has its own notion of a scheme, and not surprisingly, those are closely related to naming schemes. Recall that a scheme is a topological space together with a sheaf of rings which is locally isomorphic to a spectrum of some ring. Of course, in googology, since we usually deal with integers, we will be mostly interested in integral schemes, but as we will quickly see, there is nothing stopping us from considering, for example, rational points over other schemes.

The simplest nontrivial scheme is, of course, the spectrum of a field \(k\), which is denoted by \(\newcommand{\Spec}{\operatorname{Spec}}\Spec k\). Those are directly related to one of the naming schemes due to Sbiis Saibian. As he explains on this page, the suffix -speck appended to a number n gives us a number m/1010. Clearly, this operation can be applied not only to single numbers, but full naming schemes too. Specifically, given a naming scheme \(N\) which corresponds to the scheme \(X\), we can construct the naming scheme \(N-\text{speck}\) which corresponds to the product \(X\times\Spec k\). Because \(\Spec k\) has one point, we get that the rationa points of the two schemes are in an obvious correspondence, as we would expect from naming schemes.

As you can see, there is a lot of potential in this method, and I do believe that the entire theory of naming schemes can be formalized this way, and by using other tools from algebraic geometry, possibly even other parts of googology. I am looking forward to your thoughts on this approach.