Consistency

For other uses, seeConsistency (disambiguation).

Inclassicaldeductive logic,aconsistenttheoryis one that does not lead to a logicalcontradiction.A theory$T$in aformal systemsatisfying theprinciple of explosionis consistent if there is noformula$\varphi$such that both$\varphi$and its negation$\lnot \varphi$are elements of the set of consequences of$T$.Let$A$be a set ofclosed sentences(informally "axioms" ) and$\langle A\rangle$the set of closed sentences provable from$A$under some (specified, possibly implicitly) formal deductive system. The set of axioms$A$isconsistentwhen there is no formula$\varphi$such that$\varphi \in \langle A\rangle$and$\lnot \varphi \in \langle A\rangle$.A theory in a formal system in general is consistent if it proves everything; a consistent theory may have a provable pair of a sentence and its negation. A consistent theory is asyntacticnotion, whosesemanticcounterpart is asatisfiable theory.A theory is satisfiable if it has amodel,i.e., there exists aninterpretationunder which allaxiomsin the theory are true.This is whatconsistentmeant in traditionalAristotelian logic,although in contemporary mathematical logic the termsatisfiableis used instead.

In asound formal system,every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductivelogic,the logic is calledcomplete.The completeness of thesentential calculuswas proved byPaul Bernaysin 1918andEmil Postin 1921,while the completeness ofpredicate calculuswas proved byKurt Gödelin 1930,and consistency proofs for arithmetics restricted with respect to theinduction axiom schemawere proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).Stronger logics, such assecond-order logic,are not complete.

Aconsistency proofis amathematical proofthat a particular theory is consistent.The early development of mathematicalproof theorywas driven by the desire to provide finitary consistency proofs for all of mathematics as part ofHilbert's program.Hilbert's program was strongly impacted by theincompleteness theorems,which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).

Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. Thecut-elimination(or equivalently thenormalizationof theunderlying calculusif there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.

Consistency and completeness in arithmetic and set theory

In theories of arithmetic, such asPeano arithmetic,there is an intricate relationship between the consistency of the theory and itscompleteness.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 arithmeticis an axiom system for the natural numbers under addition. It is both consistent and complete.

Gödel's incompleteness theoremsshow that any sufficiently strongrecursively enumerabletheory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories ofPeano arithmetic(PA) andprimitive recursive arithmetic(PRA), but not toPresburger arithmetic.

Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it doesnotprove 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, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such asZermelo–Fraenkel set theory(ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed.

Because consistency of ZF is not provable in ZF, the weaker notionrelative consistencyis interesting in set theory (and in other sufficiently expressive axiomatic systems). IfTis atheoryandAis an additionalaxiom,T+Ais said to be consistent relative toT(or simply thatAis consistent withT) if it can be proved that ifTis consistent thenT+Ais consistent. If bothAand ¬Aare consistent withT,thenAis said to beindependentofT.

First-order logic

Notation

In the following context ofmathematical logic,theturnstile symbol$\vdash$means "provable from". That is,$a\vdash b$reads:bis provable froma(in some specified formal system).

Definition

• A set offormulas$\Phi$in first-order logic isconsistent(written$\operatorname {Con} \Phi$) if there is no formula$\varphi$such that$\Phi \vdash \varphi$and$\Phi \vdash \lnot \varphi$.Otherwise$\Phi$isinconsistent(written$\operatorname {Inc} \Phi$).
• $\Phi$is said to besimply consistentif for no formula$\varphi$of$\Phi$,both$\varphi$and thenegationof$\varphi$are theorems of$\Phi$.
• $\Phi$is said to beabsolutely consistentorPost consistentif at least one formula in the language of$\Phi$is not a theorem of$\Phi$.
• $\Phi$is said to bemaximally consistentif$\Phi$is consistent and for every formula$\varphi$,$\operatorname {Con} (\Phi \cup \{\varphi \})$implies$\varphi \in \Phi$.
• $\Phi$is said tocontain witnessesif for every formula of the form$\exists x\,\varphi$there exists aterm$t$such that$(\exists x\,\varphi \to \varphi {t \over x})\in \Phi$,where$\varphi {t \over x}$denotes thesubstitutionof each$x$in$\varphi$by a$t$;see alsoFirst-order logic.

Basic results

1. The following are equivalent:
   1. $\operatorname {Inc} \Phi$
   2. For all$\varphi,\;\Phi \vdash \varphi.$
2. Every satisfiable set of formulas is consistent, where a set of formulas$\Phi$is satisfiable if and only if there exists a model${\mathfrak {I}}$such that${\mathfrak {I}}\vDash \Phi$.
3. For all$\Phi$and$\varphi$:
   1. if not$\Phi \vdash \varphi$,then$\operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)$;
   2. if$\operatorname {Con} \Phi$and$\Phi \vdash \varphi$,then$\operatorname {Con} \left(\Phi \cup \{\varphi \}\right)$;
   3. if$\operatorname {Con} \Phi$,then$\operatorname {Con} \left(\Phi \cup \{\varphi \}\right)$or$\operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)$.
4. Let$\Phi$be a maximally consistent set of formulas and suppose it containswitnesses.For all$\varphi$and$\psi$:
   1. if$\Phi \vdash \varphi$,then$\varphi \in \Phi$,
   2. either$\varphi \in \Phi$or$\lnot \varphi \in \Phi$,
   3. $(\varphi \lor \psi )\in \Phi$if and only if$\varphi \in \Phi$or$\psi \in \Phi$,
   4. if$(\varphi \to \psi )\in \Phi$and$\varphi \in \Phi$,then$\psi \in \Phi$,
   5. $\exists x\,\varphi \in \Phi$if and only if there is a term$t$such that$\varphi {t \over x}\in \Phi$.

Henkin's theorem

Let$S$be aset of symbols.Let$\Phi$be a maximally consistent set of$S$-formulas containingwitnesses.

Define anequivalence relation$\sim$on the set of$S$-terms by$t_{0}\sim t_{1}$if$\;t_{0}\equiv t_{1}\in \Phi$,where$\equiv$denotesequality.Let${\overline {t}}$denote theequivalence classof terms containing$t$;and let$T_{\Phi }:=\{\;{\overline {t}}\mid t\in T^{S}\}$where$T^{S}$is the set of terms based on the set of symbols$S$.

Define the$S$-structure${\mathfrak {T}}_{\Phi }$over$T_{\Phi }$,also called theterm-structurecorresponding to$\Phi$,by:

1. for each$n$-ary relation symbol$R\in S$,define$R^{{\mathfrak {T}}_{\Phi }}{\overline {t_{0}}}\ldots {\overline {t_{n-1}}}$if$\;Rt_{0}\ldots t_{n-1}\in \Phi;$
2. for each$n$-ary function symbol$f\in S$,define$f^{{\mathfrak {T}}_{\Phi }}({\overline {t_{0}}}\ldots {\overline {t_{n-1}}}):={\overline {ft_{0}\ldots t_{n-1}}};$
3. for each constant symbol$c\in S$,define$c^{{\mathfrak {T}}_{\Phi }}:={\overline {c}}.$

Define a variable assignment$\beta _{\Phi }$by$\beta _{\Phi }(x):={\bar {x}}$for each variable$x$.Let${\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })$be theterminterpretationassociated with$\Phi$.

Then for each$S$-formula$\varphi$:

${\mathfrak {I}}_{\Phi }\vDash \varphi$if and only if$\;\varphi \in \Phi.$

Sketch of proof

There are several things to verify. First, that$\sim$is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that$\sim$is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of$t_{0},\ldots,t_{n-1}$class representatives. Finally,${\mathfrak {I}}_{\Phi }\vDash \varphi$can be verified by induction on formulas.

Model theory

InZFC set theorywith classicalfirst-order logic,aninconsistenttheory$T$is one such that there exists a closed sentence$\varphi$such that$T$contains both$\varphi$and its negation$\varphi '$.Aconsistenttheory is one such that the followinglogically equivalentconditions hold

1. $\{\varphi,\varphi '\}\not \subseteq T$
2. $\varphi '\not \in T\lor \varphi \not \in T$

See also

• Cognitive dissonance
• Equiconsistency
• Hilbert's problems
• Hilbert's second problem
• Jan Łukasiewicz
• Paraconsistent logic
• ω-consistency
• Gentzen's consistency proof
• Proof by contradiction

References

• Gödel, Kurt (1 December 1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I".Monatshefte für Mathematik und Physik.38(1): 173–198.doi:10.1007/BF01700692.
• Kleene, Stephen(1952).Introduction to Metamathematics.New York: North-Holland.ISBN0-7204-2103-9.10th impression 1991.
• Reichenbach, Hans(1947).Elements of Symbolic Logic.New York: Dover.ISBN0-486-24004-5.
• Tarski, Alfred(1946).Introduction to Logic and to the Methodology of Deductive Sciences(Second ed.). New York: Dover.ISBN0-486-28462-X.
• van Heijenoort, Jean(1967).From Frege to Gödel: A Source Book in Mathematical Logic.Cambridge, MA: Harvard University Press.ISBN0-674-32449-8.(pbk.)
• "Consistency".The Cambridge Dictionary of Philosophy.
• Ebbinghaus, H. D.; Flum, J.; Thomas, W.Mathematical Logic.
• Jevons, W. S. (1870).Elementary Lessons in Logic.

External links

• Mortensen, Chris (2017)."Inconsistent Mathematics".Stanford Encyclopedia of Philosophy.