If there are theories of everything for science, are there equations of everything for math?

Question

Some considerations:

• A related question is whether math is finite, and has been asked here before. Unlike science, math does not seem to be finite. As a dear friend of mine once retorted to me, "You just stop finding patterns at some point?" Mathematics contains the patterns of many a universe, including this one.
• I see two ways the question could appear to be possible. One, whereby certain axioms can be regarded as 'equations of everything' due to their nature of permitting deduction of all other patterns. Or two, at some point after sufficient mathematical development, results and conclusions are integrated via synthesis into a singular equation of patterns. I wonder then, what would a last equation of math be like?
• Unfortunately, the first approach does not seem to work. Strongly referring to the ramifications of Gödel's incompleteness theorems:

"...modern logic has had the tools ... to show that Russell was right to claim that the axioms of Principia Mathematica, and the Three Laws of Thought, are no more tautologous and no more self-evidently true than other truth-functional tautologies. All truth-functional tautologies are true for every combination of truth values for their constituent simple statements; they are all on an equal footing, for they are all demonstrably tautologous by the truth-table method. This important fact shows, however, that self-evidence—whether real or illusory or a cognitive function of meaning understanding—is completely unnecessary in logic. What really matters in a system of logic ... is that it be expressively complete (i.e., that every truth function is constructible with the symbols of the system) and that it be deductively complete in that every valid argument can be proved valid by means of its rules" —Introduction to Logic 15e, Irving Copi et al.

• Unfortunately, the second approach does not seem to work either. As mentioned in the first point, if patterns of math are infinite, there can never be a satisfactory synthesis.
• Thus then: Is all math doomed to never fit together? Or is this its very nature?

Answer 1

Once upon a time, set theorists began to tinker with, and then put some of their weird hopes in, things called nontrivial elementary embeddings. They're nontrivial because they're not "the identity": think of them as some function j(x), which just goes to x for many an input, but for critical points goes to something greater. Penelope Maddy describes the commencement of the procedure:

[Let us] imagine ourselves in an Ord-long process, generating an R at each stage, one for each ordinal. Suppose we step several ranks at a time, so that by step a, we are already to R_ya, for some y > a. We keep careful track of the structures at each stage by making copious notations on a clipboard, one scoresheet for every stage; we note down every detail of the structure we have just generated, along with every detail of the process that got us there...

Note that in the quoted passage, "R" refers to "ranks," which are similar to ordinals but which, in set theories without the axiom of choice/global well-ordering, play a parallel, not identical, role (see the Wikipedia entry on Scott's trick for some relevant details). Now, the ranks could be models of any strength, from the "smallest" model L (the constructible universe) up to the vague image of V in general. So it occurred to the practitioners of embedding theory to ask if it was possible to nontrivially embed V into itself.

For better or worse, things didn't work out so great. A certain Kenneth Kunen proved that trying to embed V into itself would lead to a violation of the Axiom of Choice (though see also analysis of how to abrogate the foundation axiom, in order to allow for the problematic embeddings, and more, to go through). I can't recall where I read it, I think it was in the work of a certain Kanamori, but the nonexistence of the embedding was given as the positive claim that j(V) does not equal V. (I would've expected the opposite, but I don't actually understand the inner workings of embedding theory.) So one good candidate for "the ultimate equation" seems to have been more or less ruled out; even if we readmitted the equation for some reason, it would be that weird reasoning that we'd be adverting to for our "ultimacy," yet just the same, then, we'd prompt ourselves to go on to yet another "ultimate candidate." At least, the zone above embedding theory is unknown; the theory tries to be all-encompassing by definition, and yet then we see that it breaks down (in fact, not just for V-into-itself, but V_a+2 into itself, for some limit ordinal a, and then embeddings of any higher (but not absolute) rank as such).

Some cases of category theory or type theory might strive for cosmic answers; if there is no set of all sets (or set of all other sets?), yet might there be a category of all (other) categories or a type of all (other) types? Or there's Corazza embedding theory, which involves applying category-theoretic techniques to the cosmic questions of set theory. Corazza's own "solution" to the problem of embedding V into itself is to abridge the axiom scheme of replacement, so that the specific stage of V's self-embedding as is implicated in the failure of the choice axiom, is ruled out. This has the dire side-effect of making V cofinal with the zeroth aleph, however, whereas cf(V) = V is widely considered to be more "likely."

So now could cf(V) = V be an "equation of everything"? By itself, it is not rich enough to teach us all that we wish to know in mathematics. Nevertheless, it might be thought of as expressing the theme of "everything," here: of the finality (or transfinality) proper, not only the cofinality, of the numerical cosmos.