Stephen Wolfram on axiomatic systems? I was watching a video where Wolfram was discussing the development of Mathematics, he said something along the lines of:
" There is a whole universe of possible mathematics. I was curious about this question for logic for example. We always think of logic as being this absolute thing. But in fact, it is just a particular axiom system that lives in the space of all possible axiom systems..." 
He goes on to say that depending upon how we enumerate this space, logic is the 50 thousandth posssible axiom system. 
I was wondering, how would we enumerate possible axiom systems in order to state that logic could be the fifty thousandth possible axiom system?
Thanks.
https://www.youtube.com/watch?v=RlMMeqO7wOI
 A: The basic premise is that you can put a number to anything. Even to human life (think at insurances). Whether that number means something depends on how well you have thought your scheme out. Once you have put a number to anything, you can order things numerically. That's it.
Now concerning axioms: axioms are strings of text. A set of axioms is a single axiom which is the conjunction of all the axioms in the set (I assume it is finite set).What you have now is a string of characters.  Order the strings of the different axioms sets (representing different logics) by increasing length. The shortest come first. Order strings of the same length in alphabetical order, according to an (extended) alphabet which includes letters, numbers, mathematical signs, ect, that appear in your strings. You are done. There may be other smarter ways to do it, but that is the idea. You can read all Wolfram ideas about this in "A new kind of science" page 772 onward online  for example. This whole area of research is called "A new kind of science" (NKS) or "WolframScience". 
