Thanks for this answer. You helped me think about things a little deeper. I also read through the recent guide on how to self-study logic and learned a couple m

Thanks for this answer. You helped me think about things a little deeper. I also read through the recent guide on how to self-study logic and learned a couple more things on the subject. If I'm understanding things right it's that proving things with logic means you are showing that something is consistent within a system and that is all you are definitely able to say from that. Not whether something is 'true' in a wider sense outside of that system. And all axioms are assumptions, so all our proofs are saying is that 'assuming that these things are the case then we can show the following result is commensurate with those assumptions and completely internally consistent'.