Terence Tao is a brilliant mathematician whom I follow. He posts about a lot of very interesting mathematical topics. I found this neat post related to formals systems. The beginning of the article should hopefully be at least somewhat familiar based on what I have discussed, but it does get heavy pretty quickly. Nevertheless, check it out and enjoy!

# Definable subsets over (nonstandard) finite fields, and almost quantifier elimination

Advertisements

Seems like it has been nine months or so since you last posted. Are you still thinking about Tegmark’s 4-level multiverse hierarchy, formal systems, and such?

Yes, I am very fascinated by his idea.

I have been sincerely wondering what mathematical structure, if any, that is reality or at least isomorphic to reality. My starting point is to think of some structure that precedes all mathematical structures, something that includes all mathematical objects. (Reality is taken to mean at least some type of totality of objects.) And the path I’m on now suggests that I need to be looking into formal systems. I think it’s possible that reality is a formal system or a space of formal systems but I’m not sure what the grammar, axioms, and rules of inference are. The axioms might be the initial conditions, theorems (that which is justified by applying the rules of inference) could be events that really will happen, happened, or did happen, and grammar putting some type of limit on what can really happen.

I’ve been wondering and entertaining the same ideas. In some sense, the mathematical structure that would be our reality (or isomorphic to it) would have to include all the systems and structures we can think of.

I believe ZFC set theory is sufficiently robust that it can talk of formal systems so in a sense, some formal systems are capable of describing formal systems. So if reality is a formal system, then it is possible that it is sufficiently robust to talk about other formal systems. Hopefully it is able to thusly include all formal systems… but perhaps that is how we can identify which structure reality is (or is isomorphic to): start with finding an all-inclusive mathematical structure and see if that is isomorphic to reality. My hunt for an all-inclusive mathematical structure has led me to a point where it seems that a formal system is going to be the answer. And Tegmark even puts formal systems as the origin point for all his other structures on that road map of structures in that nice picture he has. I think that he might be looking to the top of the picture for a TOE but I rather think it has to be near the bottom.

So, what is your background?

I have an MA and BA in math…no Phd.