apeiron said:
What else is an axiom but a universal assumed to be true for the sake of rational argument that is then justified by a model's ability to predict/control reality?
That's what's called a
dogma.
Though axioms and dogmata are formally the same, they carry a different meaning. An axiom is not implied to be universally true by the author, it may simply be 'for sake of argument', or indeed, used to prove a contradiction.
In fact, the reductio ad absurdum technique of proving is proving that X is true (proving that ¬X is false) by not proving ¬¬X from the other axioms, but rather proving one of the other axioms false from ¬X in conjunction wit the other axioms.
¬X is introduced as an axiom then and most logics have been proven for that:
{¬X,B,C,D ... } |- ¬Y
is aequivalent to
{Y,B,C,D ... } |- X
We prove the former, which leads to the latter, thus we can use it to prove that X is true in our axiom scheme {Y,B,C,D ...} by showing that introducing the negation to it is able to prove one of the axioms false (a contradiction).
Edit: this relies on the assumption that {Y,B,C,D ... } is not inconsistent itself though, but that doesn't matter for the reductio ad absurdm. Because if it is inconsistent the principle of explosition dictates that
all statements are theorems, including X and ¬X so what we aim to prove is proven by default then.
¬X is then formally an axiom, but not a dogma. It is never claimed by the author to be universally, more often, claimed to be universally false.
However, that was beyond my argument, my argument was in response to your claim that you must make axioms to prove.
This is true, but even fundamental rules which all people live and swear by are axioms. For instance:
{x, x->y} |- y
Modus ponens, if x is true, and x implies y; y is a theorem.
That's an axiom, we can state that axiom, then we end up with:
(X |- x) |- (x in X)
If x is a theorem in X, then x is true in X.
But we already used theorem here, but it still goes on and on, you never hit a bottom.
So, you can either say that such fundamental tautologies as x = x, or the modens ponens are simply always true and not axioms, else you can't speak of logic any more (debatable position), which implies that you do NOT need to assume any thing at all to prove truth, as x = x is always true.
Or you can call them axioms too, and that means you can never formally prove any thing at all. As your your rules of proving from your axioms are also axioms.
We extract the global generalisations which allow us to predict the local particulars. That is the way the mind works (ideas => impressions). And the way philosophy and science work (generals => particulars, models => measurements).
Logic works a bit more abstract I suppose.
'You only understand a concept as soon as you can write it down in lambda calculus.'