Consistency proof

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Consistency proof

In logic

Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, a consistent theory

Theory (mathematical logic)

In mathematical logic, a theory is a set of sentence s in a formal language. For example, a first-order theory is a set of first-order logic sentences....
is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic

Term logic

In philosophy, term logic, also known as traditional logic, is a loose name for the way of doing logic that began with Aristotle, and that was dominant until the advent of modern predicate logic in the late nineteenth century....
, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete. The completeness of sentential calculus was proved by Paul Bernays

Paul Bernays

Paul Isaac Bernays was a Switzerland mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics....
in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel

Kurt Gödel

Kurt G?del was an Austrian-United States logician, mathematician and philosopher. One of the most significant logicians of all time, G?del made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A....
in 1930.

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Encyclopedia

In logic

Logic

Theory (mathematical logic)

Term logic

Paul Bernays

Kurt Gödel

Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
, are not complete.

A consistency proof is a mathematical proof

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
that a particular theory is consistent. The early development of mathematical proof theory

Proof theory

Proof theory is a branch of mathematical logic that represents Mathematical proofs as formal mathematical objects, facilitating their analysis by mathematical techniques....
was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program

Hilbert's program

Hilbert's program, formulated by Germans mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent....
. Hilbert's program was strongly impacted by incompleteness theorems

Gödel's incompleteness theorems

In mathematical logic, G?del's incompleteness theorems, proved by Kurt G?del in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest....
, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).

Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.

Consistency and completeness in arithmetic

In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness

Completeness

In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields....
. A theory is complete if, for every formula φ in its language, at least one of φ or ¬ φ is a logical consequence of the theory.

Presburger arithmetic

Presburger arithmetic

Presburger arithmetic is the first-order predicate calculus of the natural numbers with addition, named in honor of Mojzesz Presburger, who introduced it in 1929....
is an axiom system for the natural numbers under addition. It is both consistent and complete.

Gödel's incompleteness theorems

Gödel's incompleteness theorems

Primitive recursive arithmetic

Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Thoralf Skolem as a formalization of his finitist conception of the foundations of mathematics, and it is widely agreed that all reasoning of PRA is finitist....
(PRA), but not to Presburger arithmetic.

Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective theories that can describe a strong enough fragment of arithmetic – including set theories such as Zermelo–Frankel set theory. These set theories cannot prove their own Gödel sentences - provided that they are consistent, which is generally believed.

Formulas

A set of formulas in first-order logic is consistent (written Con) if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
there is no formula such that and . Otherwise is inconsistent and is written Inc.

is said to be simply consistent if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
for no formula of are both and the negation

Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
of theorems of .

is said to be absolutely consistent or Post consistent if and only if at least one formula of is not a theorem of .

is said to be maximally consistent if and only if for every formula , if Con then .

is said to contain witnesses if and only if for every formula of the form there exists a term such that . See First-order logic

First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
.

Basic results

1. The following are equivalent:

(a) Inc

(b) For all

2. Every satisfiable set of formulas is consistent, where a set of formulas is satisfiable if and only if there exists a model such that .

3. For all and :

(a) if not , then Con;

(b) if Con and , then Con;

(c) if Con , then Con or Con.

4. Let be a maximally consistent set of formulas and contain witnesses. For all and :

(a) if , then ,

(b) either or ,

(c) if and only if or ,

(d) if and , then ,

(e) if and only if there is a term such that .

Henkin's theorem

Let be a maximally consistent set of formulas containing witnesses.

Define a binary relation on the set of S-terms if and only if ; and let denote the equivalence class of terms containing ; and let where is the set of terms based on the symbol set .

Define the S-structure over the term-structure corresponding to by:

(1) For -ary , if and only if ,

(2) For -ary , ,

(3) For , .

Let be the term interpretation associated with , where .

For all , if and only if .

Sketch of proof

There are several things to verify. First, that is an equivalence relation

Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of class representatives. Finally, can be verified by induction on formulas.

Consistency proof

Consistency and completeness in arithmetic

Formulas

Basic results

Henkin's theorem

Sketch of proof

See also