You need to define Logic: I am making it dependent on Thought, and say that part of the Thought process which deals with assumptions, conclusions and priortising. Why do I need an axiom for this
Assumption making process is part of logic
So is inference making process, I dont get your question
(Okay, let's get one thing straight: I am not a gatekeeper of anything. I do not speak for the community. Even if I wanted the question back, which I'm not sure I do or I don't, I couldn't make that call. I could put it up to a vote, at best.)
Listen, you are not going to change my mind on this one. Yes, it is an real, honest question. Yes, it does not belong on MSE in the form you have presented it and discussed it with me.
1) Google 'philosophy forums' and see if you have any luck there. Philosophy is somewhat optional, although I think it is the best fit; it is really "forum" that is the keyword.
2) You are going to have to put in much, much more work than you already have if you want your work to get read by academics, and even more if by mathematicians.
You CANNOT CANNOT CANNOT talk about wanting to understand Godel in the same breath as saying you have some new ideas about the foundations of mathematics
This is 100% guaranteed way that you will never be taken seriously.