1.9. Axioms and proofs

Statements

Realistic vs. axiomatic theories in mathematics and other sciences

An axiomatic theory is a formally given theory T = (τ, L, X) with an axioms list X, that means to define the class of its models, as that of all systems M interpreting the language L where all axioms (elements of X) are true, which will be denoted M⊨T.

This truth will usually be ensured for realistic theories of pure mathematics : arithmetic and set theory (though the realistic meaning of set theory will not always be clear depending on considered axioms). These theories will also admit non-standard models, making their realistic and axiomatic meanings effectively differ.
Outside pure mathematics, the truth of realistic theories may be dubious (questionable): non-mathematical statements over non-mathematical systems may be ambiguous (ill-defined), while the truth of theories of applied mathematics may be approximative, or speculative as the intended "real" systems may be unknown (contingent among other possible ones). There, a theory is called falsifiable if, in principle, the case of its falsity can be discovered by comparing its predictions (theorems) with observations. For example, biology is relative to a huge number of random choices silently accumulated by Nature on Earth during billions of years ; it has lots of "axioms" which are falsifiable and require a lot of empirical testing.
Non-realistic theories outside mathematics (not called "axiomatic" by lack of mathematical precision) would be works of fiction describing imaginary or possible future systems.

Provability

We say that A is provable in T, or a theorem of T, and write T ⊢ A if a proof of A in T exists. In practice, we only qualify as theorems the statements known as such, i.e. for which a proof is known. But synonyms for "theorem" are traditionally used according to their importance: a theorem is more important than a proposition; either of them may be deduced from an otherwise less important lemma, and easily implies an also less important corollary.

Logical validity

A proof of A using some axioms can also be seen as a proof of (conjunction of these axioms ⇒ A) without axiom, thus making this implication logically valid.

Refutability and consistency

⊢ (A ∧ ¬A) ⇔ 0
((T ⊢ A) ∧ (T ⊢ B)) ⇔ (T ⊢ A∧B)
((T ⊢ A) ∧ (T ⊢ ¬A)) ⇔ (T ⊢ 0).

Beyond the required quality of soundness of the proof theoretical part of a logical framework, more remarkable is its converse quality of completeness : that for any axiomatic theory it describes, any statement that is true in all models is provable. In other words, any unprovable statement is false somewhere, and any irrefutable statement is true somewhere. Thus, any consistent theory has existing models, but often a diversity of them, as any undecidable statement is true in some and false in others. Adding some chosen undecidable statements to axioms leads to different consistent theories which can «disagree» without conflict, all truly describing different existing systems. This resolves much of a priori divergence between Platonism and formalism while giving proper mathematical definiteness to the a priori intuitive concepts of "proof", "theorem" and "consistency".