Consistency

From Logic

Template:Other

In logic, a consistent theory 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, 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 in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930. Stronger logics, such as second-order logic, are not complete.

A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by incompleteness theorems, 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. 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 is an axiom system for the natural numbers under addition. It is both consistent and complete.

Gödel's incompleteness theorems show that any sufficiently strong effective theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and Primitive recursive arithmetic (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 Failed to parse (Can't write to or create math temp directory): \\Phi

in first-order logic is consistent (written ConFailed to parse (Can't write to or create math temp directory): \\Phi

) if and only if there is no formula Failed to parse (Can't write to or create math temp directory): \\phi

such that Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi
and Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\lnot\\phi

. Otherwise Failed to parse (Can't write to or create math temp directory): \\Phi

is inconsistent and is written IncFailed to parse (Can't write to or create math temp directory): \\Phi

.

Failed to parse (Can't write to or create math temp directory): \\Phi

is said to be simply consistent if and only if for no formula Failed to parse (Can't write to or create math temp directory): \\phi
of Failed to parse (Can't write to or create math temp directory): \\Phi
are both Failed to parse (Can't write to or create math temp directory): \\phi
and the negation of Failed to parse (Can't write to or create math temp directory): \\phi
theorems of Failed to parse (Can't write to or create math temp directory): \\Phi

.

Failed to parse (Can't write to or create math temp directory): \\Phi

is said to be absolutely consistent or Post consistent if and only if at least one formula of Failed to parse (Can't write to or create math temp directory): \\Phi
is not a theorem of Failed to parse (Can't write to or create math temp directory): \\Phi

.

Failed to parse (Can't write to or create math temp directory): \\Phi

is said to be maximally consistent if and only if for every formula Failed to parse (Can't write to or create math temp directory): \\phi

, if Con Failed to parse (Can't write to or create math temp directory): \\Phi \\cup \\phi

 then  Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi

.

Failed to parse (Can't write to or create math temp directory): \\Phi

is said to contain witnesses if and only if for every formula of the form Failed to parse (Can't write to or create math temp directory): \\exists x \\phi
there exists a term Failed to parse (Can't write to or create math temp directory): t
such that Failed to parse (Can't write to or create math temp directory): (\\exists x \\phi \\to \\phi {t \\over x}) \\in \\Phi

. See First-order logic.

Basic results

1. The following are equivalent:

(a) IncFailed to parse (Can't write to or create math temp directory): \\Phi

(b) For all Failed to parse (Can't write to or create math temp directory): \\phi,\\; \\Phi \\vdash \\phi.

2. Every satisfiable set of formulas is consistent, where a set of formulas Failed to parse (Can't write to or create math temp directory): \\Phi

is satisfiable if and only if there exists a model Failed to parse (Can't write to or create math temp directory): \\mathfrak{I}
such that Failed to parse (Can't write to or create math temp directory): \\mathfrak{I} \\vDash \\Phi 

.

3. For all Failed to parse (Can't write to or create math temp directory): \\Phi

and Failed to parse (Can't write to or create math temp directory): \\phi

(a) if not Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi , then ConFailed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\lnot\\phi\\}

(b) if Con Failed to parse (Can't write to or create math temp directory): \\Phi

and Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi

, then ConFailed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\phi\\}

(c) if Con Failed to parse (Can't write to or create math temp directory): \\Phi , then ConFailed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\phi\\}

or ConFailed to parse (Can't write to or create math temp directory):  \\Phi \\cup \\{\\lnot \\phi\\}

.

4. Let Failed to parse (Can't write to or create math temp directory): \\Phi

be a maximally consistent set of formulas and contain witnesses.  For all Failed to parse (Can't write to or create math temp directory): \\phi
and Failed to parse (Can't write to or create math temp directory):  \\psi 

(a) if Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi , then Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi ,

(b) either Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi

or Failed to parse (Can't write to or create math temp directory): \\lnot \\phi \\in \\Phi

,

(c) Failed to parse (Can't write to or create math temp directory): (\\phi \\or \\psi) \\in \\Phi

if and only if Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi
or Failed to parse (Can't write to or create math temp directory): \\psi \\in \\Phi

,

(d) if Failed to parse (Can't write to or create math temp directory): (\\phi\\to\\psi) \\in \\Phi

and Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi 

, then Failed to parse (Can't write to or create math temp directory): \\psi \\in \\Phi ,

(e) Failed to parse (Can't write to or create math temp directory): \\exists x \\phi \\in \\Phi

if and only if there is a term Failed to parse (Can't write to or create math temp directory): t
such that Failed to parse (Can't write to or create math temp directory): \\phi{t \\over x}\\in\\Phi

.

Henkin's theorem

Let Failed to parse (Can't write to or create math temp directory): \\Phi

be a maximally consistent set of formulas containing witnesses.

Define a binary relation on the set of S-terms Failed to parse (Can't write to or create math temp directory): t_0 \\sim t_1 \\!

if and only if Failed to parse (Can't write to or create math temp directory): \\; t_0 = t_1 \\in \\Phi

and let Failed to parse (Can't write to or create math temp directory): \\overline t \\!

denote the equivalence class of terms containing Failed to parse (Can't write to or create math temp directory): t \\!

and let Failed to parse (Can't write to or create math temp directory): T_{\\Phi} := \\{ \\; \\overline t \\; |\\; t \\in T^S \\}

where Failed to parse (Can't write to or create math temp directory): T^S \\!
is the set of terms based on the symbol set Failed to parse (Can't write to or create math temp directory): S \\!

.

Define the S-structure Failed to parse (Can't write to or create math temp directory): \\mathfrak T_{\\Phi}

over Failed to parse (Can't write to or create math temp directory):  T_{\\Phi} \\!
the term-structure corresponding to Failed to parse (Can't write to or create math temp directory): \\Phi
by:

(1) For Failed to parse (Can't write to or create math temp directory): n -ary Failed to parse (Can't write to or create math temp directory): R \\in S , Failed to parse (Can't write to or create math temp directory): R^{\\mathfrak T_{\\Phi}} \\overline {t_0} \\ldots \\overline {t_{n-1}}

if and only if Failed to parse (Can't write to or create math temp directory): \\; R t_0 \\ldots t_{n-1} \\in \\Phi

,

(2) For Failed to parse (Can't write to or create math temp directory): n -ary Failed to parse (Can't write to or create math temp directory): f \\in S , Failed to parse (Can't write to or create math temp directory): f^{\\mathfrak T_{\\Phi}} (\\overline {t_0} \\ldots \\overline {t_{n-1}}) := \\overline {f t_0 \\ldots t_{n-1}} ,

(3) For Failed to parse (Can't write to or create math temp directory): c \\in S , Failed to parse (Can't write to or create math temp directory): c^{\\mathfrak T_{\\Phi}}:= \\overline c .

Let Failed to parse (Can't write to or create math temp directory): \\mathfrak I_{\\Phi} := (\\mathfrak T_{\\Phi},\\beta_{\\Phi})

be the term interpretation associated with Failed to parse (Can't write to or create math temp directory): \\Phi

, where Failed to parse (Can't write to or create math temp directory): \\beta _{\\Phi} (x) := \\bar x .

For all Failed to parse (Can't write to or create math temp directory): \\phi

if and only if Failed to parse (Can't write to or create math temp directory):  \\; \\phi \\in \\Phi

Sketch of proof

There are several things to verify. First, that Failed to parse (Can't write to or create math temp directory): \\sim

is an equivalence relation.  Then, it needs to be verified that (1), (2), and (3) are well defined.  This falls out of the fact that Failed to parse (Can't write to or create math temp directory): \\sim
is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of Failed to parse (Can't write to or create math temp directory):  t_0, \\ldots ,t_{n-1} 
class representatives.  Finally, Failed to parse (Can't write to or create math temp directory):  \\mathfrak I_{\\Phi} \\vDash \\Phi 
can be verified by induction on formulas.

References

• The Cambridge Dictionary of Philosophy, consistency
• H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic
• Jevons, W.S., Elementary Lessons in Logic, 1870