this got kinda long, so I'm not rereading it. prepare for spelling and especially grammar errors.
>>12290741axioms are just the starting definitions by which everything else is derived in some mathematical area. If you have trouble understanding that, just think about an axiomatic system as the kind of rigorous math that professional mathematicians deal with that you don't see in say highschool.
godels theorems are proofs in mathematical logic. they don't actually say anything about physics or consciousness(so there is nothing to test). They are however really special in the sense that they are a kind of meta result about the limitations of axiomatic systems and thus of math. That's why cooks sometimes like spin up something crazy about that incorporates them rather than some random math theorem.
the key point for penrose is that one of the theorems says that within one axiomatic system i.e. one set of definitions from which we derive all other proofable statements, there will be statements that are both truth given our axioms and also not derivable from these axioms. This means that there are statements within our chosen system that are true, but we cannot proof that they are true! (this kind of proofability also sort of corresponds in some sense to computability)
Now penrose thinks that humans have a way around this and can just "see" certain statements to be true(yeah that's vague, but so is he). Now our current laws of physics are described by these kind of systems, but humans(according to penrose) are doing something that goes beyond what can be done with theses systems. Therefore there must be new physics.
The wavefunction collapse is something where there is debate about whether there needs to be a physical mechanism or not. Penrose thinks we need one. So he put the two things together. Basically putting consciousness into this gap of our understanding of quantum theory.
