A philosophical question about an hypothetical theorem/equation of everything

Question

Preamble
I'm not a mathematician. I'm just curious. Please forgive my pseudo formalism. Please allow me, a non mathematician, to have just questions.

Definition
A mathematical theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms.

Questions

Let say from a theorem Tn, a theorem Tm can be deducted (Tn < Tm)

1. How far can we possibly go like that to the left and right?
   

I mean is there a T0 and a Tp such as: T0< ... < Tm < Tm+1 < .. <Tp ?

2. If we consider that more than one theorem can be deducted directly from Tm and denote this set (of deducted theorems) by {Tr(m+1), .., Ts(m+1)}: Is this sub set finite and any subsets for any element Tm+1?

3. And so on and so forth with the element of any subset (recursive approach)

In others words
As the above represent a tree, is there a root, and are the number of branches, sub-branches, etc.. finite, leading to leaf (are they then axioms?)

I'm looking at all the possible kind of branches: not only those of type T, but also of any type (T, U, ..).

4. Again, if we denote the set of all these types {.., T, U, ...}, is this set likely to be finite such it exist two elements A and Z such as A<..<T<U<..<Z?

While I tried poorly to express myself using a pseudo mathematical approach, my question is in fact:

5. Is there a possible super theorem (like in physics, an attempt finding the theory of everything), or a couple of them, from which all theorems can be deducted?

Comments

6. Can the existence of such super theorem(s) be concluded as existing just by using the mathematical induction principle ?

7. a) Could the mathematics we know today be sufficient to express a super theorem?
   b) Could/should any formalism be created to describe such super theorem ?

Final words (blabla that you may want to ignore)

I guess there may be some non sense in my questions due to my lack of skills, and that few logical assertions would provide me with the right an answer. I'm looking for these assertions.

Assuming mathematics is all about discovering (and not to inventing - hard to say though if invent is not discovering then), I suppose this open the door to questioning the existence of a super theorem.

Finally, in literature, by creating a random list of words, one can generate any books, articles, etc.. that have and will ever been written. This random generator doesn't constitute by itself a theorem (or does it somehow), but is it possible to do that in mathematics ?

Isn't mathematics a suite of logical concepts, having probably for initial ingredients the definition of the number sets and operations. Mixing then these numbers and operation to create groups, arranging group elements into series or whatever else, series leading to geometry or whatever else, etc.

Wondering is a random theorem generator is then possible to create, taking the base ingredients, classifying them, until a super theorem.

Compared to physics (top down approach), mathematics sounds doing the opposite (bottom up approach). This may be a reason why a super theorem doesn't exists or if it does, will never be found ?

Drawing the trees of the existing theorem, can one deduct missing one?

Answer 1

1. There is no leftmost theorem (= theorem from which no othre theorem can be deducted). For example if $\Phi$ is a theorem, then we can deduce $\neg\neg\Phi$. Or if $\forall n\ge a\colon \Psi(a)$ is a theorem then we can deduce from it that $\forall n\ge a+1\colon \Psi(a)$ is a theorem.

2. makes little sense as according to your notation Tm is deducted from Tm+1. At any rate, it is possible to derive infinitely many theorems from a single theorem in one step, for example by specialization of a general statement: $\forall n\in\Bbb N\colon \Phi(n)$ leads directly to $\Phi(42)$ and $\Phi(666)$ and ...

3. is not even a question

Remark: The "derivation dependency" is not a tree as it is not always the case that one theorem is derived from a single theoram. For example, to derive a theorem $\Phi\land \Psi$ one might require both $\Phi$ and $\Psi$. Additionally, theorems may be proved from very different theorems. For example a proof of $\Phi\lor \Psi$ might be from$\Phi$, another might be from $\Psi$, and another even from neither!

4. ???

5. That would be the conjunction of all axioms into a single axiom. However, these depend on the theory. A good starting point would be ZFC for set theory. Another however: many axiom systems involve infintely many axioms (so-called axiom schemes), so are not combinable into a single formula by $\land$.

6. No. It can be shown that certain theories are not finitely axiomatizable

7. Irrelevant because of 6

Answer 2

Preamble: This is not an answer.

I find your post infinitely more interesting than the immense amount of rubbish posted-per minute on this site, mostly by students who have little understanding of their subject and seek quick solutions for their useless homework. It is really refreshing to encounter original and interesting questions, even if a bunch of eager Ph.D students think it is not sufficiently "rigorous" and down-vote it, hopping on to earn their next MSE bounty by solving -- for the $n$'th time -- a standard homework problem.

As for your some of your questions.

1. I would say that there is no bound on how far you can go to the left, unless you decide at some point to disallow the introduction of new concepts, axioms and definitions in mathematics.

2. For a similar reason, I think that the set of theorems whose proof relies on some particular theorem has the potential to grow ad-infinitum.

About the random generator. I believe this has to do with the definition of a language. In principle, we could use Logic to code every single mathematical theorem into strictly rule-obeying strings of symbols, which would be considered as theorems. Then, we could go ahead and generate lots of random strings of symbols, which just as in the literary example you gave, would have the potential to contain all theorems that were ever proved and that ever will be. There is only one problem: What makes a string of symbols in the language a "theorem"? This problem is equivalent to it's literary counterpart: what makes a string of symbols a novel? a poem? Now these questions are not in the mathematical domain. Mathematics cannot answer this question. So strictly speaking -- yes, you could generate all possible strings that were ever written or will ever be, but it has no consequence, so it seems to me.