- Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929.
A first-order formula is called logically valid if it is true in every structure for its language. The completeness theorem shows that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula. The deduction is a finite object that can be verified by hand or computer. This relationship between truth and provability establishes a close link between model theory and proof theory in mathematical logic.
An important consequence of the completeness theorem is that it is possible to enumerate the logical consequences of any effective first-order theory, by enumerating all the correct deductions using axioms from the theory.
Gödel's incompleteness theorem, referring to a different meaning of completeness, shows that if any sufficiently strong effective theory of arithmetic is consistent then there is a formula (depending on the theory) which can neither be proven nor disproven within the theory. Nevertheless the completeness theorem applies to these theories, showing that any logical consequence of such a theory is provable from the theory.
There are numerous deductive systems for first-order logic, including systems of natural deduction and Hilbert-style systems. Common to all deductive systems is the notion of a formal deduction. This is a sequence (or, in some cases, a finite tree) of formulas with a specially-designated conclusion. The definition of a deduction is such that it is finite and that it is possible to verify algorithmically (by a computer, for example, or by hand) that a given collection of formulas is indeed a deduction.
A formula is logically valid if it is true in every structure for the language of the formula. To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system. A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense. Thus, in a sense, there is a different completeness theorem for each deductive system.
Statement and consequences
Gödel's completeness theorem says that a deductive system of first-order predicate calculus is "complete" in the sense that no additional inference rules are required to prove all the logically valid formulas. A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system. Taken together, these theorems imply that a formula is logically valid if and only if it is the conclusion of a formal deduction.
A more general version of Gödel's completeness theorem holds. It says that for any first-order theory T and any sentence S in the language of the theory, there is a formal deduction of S from T if and only if S is satisfied by every model of T. This more general theorem is used implicitly, for example, when a sentence is shown to be provable from the axioms of group theory by considering an arbitrary group and showing that the sentence is satisfied by that group.
The branch of mathematical logic that deals with what is true in different models is called model theory. The branch called proof theory studies what can be formally proved in particular formal systems. The completeness theorem establishes a fundamental connection between these two branches, giving a link between semantics and syntax. The completeness theorem should not, however, be misinterpreted as obliterating the difference between these two concepts; in fact Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. The name for the incompleteness theorem refers to another meaning of complete (see model theory - Using the compactness and completeness theorems). In particular, Gödel's completeness theorem deals with formulas that are logical consequences of a first-order theory, while the incompleteness theorem constructs formulas that are not logical consequences of certain theories.
An important consequence of the completeness theorem is that the set of logical consequences of an effective first-order theory is an enumerable set. The definition of logical consequence quantifies over all structures in a particular language, and thus does not give an immediate method for algorithmically testing whether a formula is logically valid. Moreover, a consequence of Gödel's incompleteness theorem is that the set of logically valid formulas is not decidable. But the completeness theorem implies that the set of consequences of an effective theory is enumerable; the algorithm is to first construct an enumeration of all formal deductions from the theory, and use this to produce an enumeration of their conclusions. The finitary, syntactic nature of formal deductions makes it possible to enumerate them.
Relationship to the compactness theorem
The completeness theorem and the compactness theorem are two cornerstones of first-order logic. While neither of these theorems can be proven in a completely effective manner, each one can be effectively obtained from the other.
The compactness theorem says that if a formula φ is a logical consequence of a (possibly infinite) set of formulas Γ then it is a logical consequence of a finite subset of Γ. This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set. This proof of the compactness theorem is originally due to Gödel.
Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.
The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA0 (a second-order variant of Peano arithmetic restricted to induction over Σ01 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma. In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of same cardinality, knowing that on countable sets, the ultrafilter lemma becomes equivalent to weak König's lemma. Also, although every consistent, countable first-order theory has a finite or countable model (as a consequence of the completeness theorem and the Löwenheim-Skolem theorem), it is known there are effective consistent first-order theories that do not have computable models.
Completeness in other logics
The completeness theorem is a central property of first-order logic that does not hold for all logics. Second-order logic, for example, does not have a completeness theorem for its standard semantics (but does have the completeness property for Henkin semantics), and the same is true of all higher-order logics. It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete. The set of logically-valid formulas in second-order logic is not enumerable.
Lindström's theorem states that first-order logic is the strongest (subject to certain constraints) logic satisfying both compactness and completeness.
Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.
In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof. Henkin's proof directly constructs a term model for any consistent first-order theory. James Margetson (2004) developed a computerized formal proof using the Isabelle theorem prover.  Other proofs are also known.
- Gödel, K (1929). Über die Vollständigkeit des Logikkalküls. Doctoral dissertation. University Of Vienna.. The first proof of the completeness theorem.
- Gödel, K (1930). "Die Vollständigkeit der Axiome des logischen Funktionenkalküls" (in German). Monatshefte für Mathematik 37 (1): 349–360. doi:10.1007/BF01696781. JFM 56.0046.04. The same material as the dissertation, except with briefer proofs, more succinct explanations, and omitting the lengthy introduction.
- Stanford Encyclopedia of Philosophy: "Kurt Gödel" -- by Juliette Kennedy.
- MacTutor biography: Kurt Gödel.
- Detlovs, Vilnis, and Podnieks, Karlis, "Introduction to mathematical logic."
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