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. This is the halting condition of a proof searching algorithm, processed by an abstract "computer" with unlimited (infinite) available time and resources, trying every possible piece of proof until it eventually finds a valid proof of the given statement.
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.Any good proof theory needs of course to be sound, which means only "proving" always true statements : provability implies truth in every model (where all axioms are true).
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.
⊢ (A ∧ ¬A) ⇔ 0
((T ⊢ A) ∧ (T ⊢ B))
⇔ (T ⊢ A∧B)
((T ⊢ A) ∧ (T ⊢ ¬A)) ⇔ (T ⊢ 0).
An axiomatic theory is a theory given formally with an axioms list that means to define its range of models, as the class of all systems, interpreting the given language, where all axioms are true (rejecting those where some axiom is false). This makes automatic the truth of axioms in any model. Non-realistic theories outside mathematics (not called "axiomatic" by lack of mathematical form) would be works of fiction describing imaginary or possible future systems.
A realistic theory is a theory involved to describe either a fixed system or the systems from a range, seen as given from some independent reality. Its given axioms are statements which, for some reason, are considered known as true on all these systems. Such a theory is true if all its axioms are indeed true there. In other words, these systems are models, qualified as standard for contrast with other, unintended models of that theory taken axiomatically.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.
Euclidean geometry was first conceived as a realistic theory of applied mathematics (for its role of first theory of physics), then became understood as an axiomatic theory of pure mathematics among diverse other, equally legitimate geometries in a mathematical sense; while the real physical space is more accurately described by the non-Euclidean geometry of General Relativity.Set theory and Foundations of Mathematics | |
1. First
foundations of mathematics |
⇨ 1.10. Quantifiers |
Philosophical aspects | |
2. Set theory - 3. Algebra - 4. Arithmetic | 5. Second-order foundations |