Singularity Notes

(Acknowledgment: This post has inputs from Sanket of First principles fame.) When teaching about formal axiomatic systems, I'm usually asked a question something like, "But, how do you find the axioms in the first place?" What are axioms, you ask? To take a few steps back, reasoning processes based on logic and deduction are set in an "axiomatic" context. Axioms are the ground truths, based on which we set out to prove or refute theorems within the system. For example, "Given any two points, there can be only one line that passes through both of them" is an axiom of Eucledian geometry. When proving theorems within an axiomatic system, we don't question the truth of the axioms themselves. As long as the set of axioms are consistent among themselves (i.e. don't contradict one another) it is fine. Axioms are either self-evident or well-known truths, or somethings that are assumed to be true for the context. But then, the main question becomes ver