Next Steps

Next Steps#

Here’s a bunch of stuff I’ve come across that seems vaguely related to prods, along with some unfinished thoughts about topics that are mostly beyond my paygrade.

Other Number Systems#

  • \(p\)-adics is a number system that makes no sense but makes pretty pictures. If mathematicians can get away with studying this nonsense, surely they can do something with prods…

  • Surreal numbers is pretty fun: you literally define numbers as games! This was a very inspiring example for me. I’m also very envious because not only does surreal numbers recover rational and real numbers, it goes beyond and produces new numbers (hence the name). My biggest disappointment is that prods can, so far, only deal with natural numbers.

Infinite Prod#

As mentioned above, I’m sad that prods are merely countable. So how to make them infinite?

In set theory, you apparently get infinite sets by taking the limit of an increasing sequence of sets. So what if you took the limit of the following increasing sequence?

sequence of prods

If that’s allowed, then take an arbitrary function \(f: \mathbb{N} \to \mathbb{N}\) and encode it like so:

sequence of f values

If (infinite) prods can encode arbitrary functions, there’s uncountably many of them. They seem well-defined, but I don’t know how to work with them because I can’t just code them up in python any more. But maybe there’s some fun stuff here.

Logic#

  • One of the most important insights in defining prods was to make them recursive. This was directly inspired by reading about Godel numbering, which recursively represents logical formulae as numbers through exponentiation. Godel numbering is the key ingredient to the incompleteness theorems, so can they also be proven for productive arithmetic? If true, would there be any reason to care?

  • We’ve seen that productive Heyting algebras are subtly different from the divisibility lattices. Does this give rise to meaningful differences in the internal logics?

Independence#

I’ve claimed that addition can’t be defined productively. I have a hunch that \(\sqcup\) can’t be defined additively. Would be nice to prove those claims. More generally, I reckon a model theoretic analysis on the productive axioms (plus induction schema of course) would be useful.

Number Theory#

Do prods simplify the proof of Fermat’s Last Theorem? I wrote down a version of this once, but can’t remember where I put it.

What about the Riemmann hypothesis? I wouldn’t be much of a crackpot internet number theorist if I didn’t claim to have solved it, would I?

But seriously, what can prods do for number theory? Blind platonic faith tells me there has to be something.

Lean#

I had hoped to type up all of the proofs of this book into lean to show that my handwaving was not mere crankery. But unfortunately lean turned out to be harder to learn than I expected, at least for the case of endless nested inductions that prods seems to produce.

If you do know lean, it would be much appreciated if you could help me write up the proofs. You can find my pathetic attempts here.

You#

One of the wonderful things about math is how often unexpected connections pop up between things that really have no right being so similar. I don’t know where the next connection might be, but maybe you do. So why not try?

I’m hoping this book can become a collaborative space for productive numbers. Please share your thoughts and suggestions in the github discussion for how to better spread the love.

That is all, my friends. Go forth and productivize.