Statements or formulas having like tautologies
the property of being true in every system by virtue of the logical framework are called logically valid ;
they are necessary theorems of any theory
regardless of axioms where they need not appear. More logically valid formulas 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 establishing the logical validity of this implication. But, beyond tautologies, logically valid statements with some size may be only provable by proofs with indescribably larger size.
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 for those of 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 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 very many "axioms" which are falsifiable and keep requiring a lot of empirical testing.
An axiomatic theory is a theory given formally with an axioms list that means
to define the selection (range) of models, as the class of all systems
(interpreting the given language) where all these axioms are true, rejecting others
where some axiom is false. By this definition, the truth of axioms in any model holds
automatically. This way, any consistent axiomatic theory describes a mathematical
reality (model) that exists, but is generally not unique, as an unlimited diversity of
models may remain. In particular, any undecidable formula will be true in some
models and false in other models (which are equally real and legitimate interpretations).
Different consistent theories may «disagree» without conflict, by
being all true descriptions of different systems, that may equally
«exist» in a mathematical sense without any issue of «where they
In fields outside pure mathematics, non-realistic theories (to not say axiomatic theories, as the use and
truth 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 for its role of first physical theory, but is now understood axiomatically, as one in a landscape of diverse geometries that are equally legitimate for pure mathematics, while the real physical space is more accurately described by the non-Euclidean geometry of General Relativity.
The proof of the completeness theorem, first expressed as the existence of a model of any consistent first-order theory, goes by constructing such models out of the infinite set of all ground expressions in a language constructed from the theory (the language of the theory plus more symbols extracted from its axioms). As the set of all ground expressions in a language can itself be constructed from this language together with 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, sufficient for all theories (ignoring the diversity of infinities in set theory).However, these are only theoretical properties, assuming a computer with unlimited (potentially infinite) available time and resources, able to find proofs of any size. Not only the precise size of a proof may depend on the particular formalism, but even some relatively simple formulas may only have «existing» proofs that «cannot be found» in practice as they would be too long, even bigger than the number of atoms in the visible physical Universe (as illustrated by Gödel's speed-up theorem). Within limited resources, there may be no way to distinguish whether a formula is truly unprovable or a proof has only not yet been found.
To include their case, the universal concept of provability (existence of a proof) has to be defined in the abstract. Namely, it can be expressed as a formula of first-order arithmetic (the first-order theory of natural numbers with operations of addition and multiplication), made of one existential quantifier that is unbounded in the sense of arithmetic (∃ℕ p, ) where p is an encoding of the proof, and inside is a formula where all quantifiers are bounded, i.e. with finite range (∀x < (...), ...), expressing a verification of this proof.However, once given an arithmetical formula known to be a correct expression of the provability predicate (while all such formulas are provably equivalent to each other), it still needs to be interpreted. This involves the concept of the realistic truth in first-order arithmetic. This is the ideally meant interpretation of arithmetic: the interpretation of statements of first-order arithmetic in «the true set ℕ of all, and only all, really finite natural numbers», called the standard model of arithmetic.
|Set theory and
Foundations of mathematics
(general table of contents)
| 1. First
foundations of mathematics
(detailed table of contents)
(Each subsection roughly uses those left and up)
| 1.1. Introduction
to the foundations of mathematics
1.2. Variables, sets, functions and operations
Intuitive representation and abstraction
Platonism vs Formalism
|1.3. Form of theories: notions, objects, meta-objects||
Realistic vs. axiomatic theories
| 1.4. Structures of mathematical systems
1.5. Expressions and definable structures
in Model theory
The time of interpretation
The metaphor of the usual time
The infinite time between models
| 1.6. Logical connectives
1.7. Classes in set theory
The Berry paradox
|1.8. Binders in set theory||1.D. Time
in Set theory
Expansion of the set theoretical universe
Can a set contain itself ?
The relative sense of open quantifiers1.E. Interpretation of classes
Classes in an expanding universe
| 1.10. Formalization
of set theory
1.11. Set generation principle
Justifying the set generation principle1.F. Concepts of truth in mathematics
Alternative logical frameworks
⇨ 2. Set theory (continued)