- #1

Feynstein100

- 162

- 16

I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?

It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?

And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems. Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?