Every statement, of course, is derivable from axioms because any statement can be made an axiom.Not sure why exactly, it seems pretty obvious on second glance. Any statement can be logically proved in some system. What it does highlight is the importance of choosing one's axioms well.
If one takes as an axiom that 'meat is murder', for instance, then many of the tenets of veganism follow naturally from that. That a vegan diet is innately more healthy than a caveman diet has nothing to do with the meat is murder axiom. It is another axiom in the vegan paradigm, since we know that it is not backed up by research.
Derivability does not equal truth.
Godel's incompleteness theorems showed that there are always statements consistent with a system (set of axioms) that cannot be proved in that system. It is obviously true that kung-fu movies are way cooler than romantic comedies, but how does one go about proving that? I guess that has to be an axiom.
Of course, we don't live in a strict mathematical world, but in the real world. Is there a difference? Well, if one buys into the platonic ideal, and Roger Penrose convinced me to do that, then mathematical proofs are discoveries not constructs. An unveiling of reality rather than an artificial device.
Hold on, I will scare it up in his own words.
Still there? Ok great:
When one 'sees' a mathematical truth one breaks through into this (Plato's) world of ideas, and makes direct contact with it ('accessible via the intellect'). I have described this 'seeing' in relation to Godel's thereom, but it is the essence of mathematical understanding. When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. .... The mental images that each one has, when making this Platonic contact , might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world! (The Emperor's New Mind, 1989, pp. 554)(Roger Penrose is total ninja)
So while there's no such thing as an ideal circle, only by imagining one can we come upon greater truths that do describe reality. Or something along those lines. What is my point? Well I think the same can be said about bacon. It is a truth, and when paleoticians eat bacon, each one has a direct route to truth! The mental images might be different but each one is in contact with the same existing Platonic world, which has tons of bacon, of course.
A glimpse of eternal truth