A statement A is called logically valid, which is written ⊢ A, if it is provable without
using any axiom (thus a theorem of any theory regardless of axioms). Then A is true
in every system by virtue of the logical framework. The simplest ones are the tautologies; others will be given in 1.10.
A proof of A from some axioms can also be seen as a proof of (conjunction of these axioms
⇒ A) without axiom,
thus showing the logical validity of this implication.
Again in T, a refutation of A is a proof of ¬A. If one exists
(T ⊢ ¬A), the statement A is called refutable (in T).
A statement is called decidable (in T) if it is either provable or refutable.
A theory T is called contradictory or inconsistent if T ⊢ 0, otherwise it is called consistent. If a statement is both provable and refutable in T then all are, because it means that T is inconsistent, independently of the chosen statement:
⊢ (A ∧ ¬A) ⇔ 0
((T ⊢ A) ∧ (T ⊢ B))
⇔ (T ⊢ A∧B)
((T ⊢ A) ∧ (T ⊢ ¬A)) ⇔ (T ⊢ 0).
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. For this, a list of axioms is chosen as a set of statements which, for some reason, are considered known as true on the intended system(s). Such a theory is said to be "true" if its axioms are indeed true there.
The truth of realistic theories is usually well ensured in pure mathematics, but those of other fields may be 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 system(s) may be unknown (contingent among alternative possibilities, that are equally possible thus mathematically "existing"). A theory beyond pure mathematics is called falsifiable if, in principle, the case of its falsity can be discovered by finding a mismatch between its predictions (theorems) and 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.An axiomatic theory is a theory given formally with an axioms list that means to define the 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. Outside pure mathematics, non-realistic theories (not well called axiomatic, as the use of axioms may lose clarity) would be works of fiction describing imaginary or possible future systems.
For example Euclidean geometry first came as a realistic theory of applied mathematics (for its role of first theory of physics), but is now understood as an axiomatic theory of pure mathematics in a landscape of diverse equally legitimate geometries, while the real physical space is more accurately described by the non-Euclidean geometry of General Relativity.This quality of first-order logic, with its ability to express all mathematics (any logical framework beyond it can anyway be developed from set theory, itself expressible in first-order logic), gives it a central importance in the foundations of mathematics, and dissolves much of a priori disagreements between realism (Platonism) and formalism : any consistent axiomatic theory describes some mathematical reality (model) that exists, but is generally not unique, as an unlimited diversity of models may remain. So, for a realistic theory to not be an axiomatic theory, requires to conceive its intended (range of) system(s) otherwise than by the criteria of mathematical existence and satisfaction of given first-order statements.
The proof of the completeness theorem (given in 4.6) will consist in building a model of any consistent first-order axiomatic theory, as follows. The infinite set of all ground terms with operation symbols derived from the theory (those of its language plus others coming from its existence axioms), is turned into a model by progressively defining each predicate symbol over each combination of values of its arguments there, while keeping consistency. As this construction just involves the data of a theory and the set ℕ of natural numbers, the validity of this theorem only depends on the axiom of infinity, that is the existence of ℕ as an actual infinity.
Provability (with a fixed language) can also be expressed as a formula of first-order arithmetic (theory with the only
type "natural number", constants 0,1, and operation of addition and multiplication), stating the
existence of a number which encodes a proof, related to the free variable which encodes
a statement (candidate theorem), depending on a predicate that encodes the axioms list.
This is independent of the formalism of proof encoding, in the sense that all formulas
expressing valid solutions to encode "provability" are provably equivalent to each other,
under the ordinary axioms of first-order arithmetic which will be sketched in 4.3 and 4.4.
However, these axioms do not suffice to make these formulas decidable for every value
of the free variable. To give full definiteness (Boolean values) to these formulas and any
other formulas of arithmetic, requires to interpret arithmetic not axiomatically but realistically,
in its ideal model called the standard model of arithmetic, intuitively described as
the true set ℕ of exactly all really finite natural numbers, as will be clarified in
Part 4.
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 |