Consistency




For other uses, see Consistency (disambiguation).

In classical deductive logic, a consistent theory is one that does not contain a contradiction.[1][2] The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T{displaystyle T}T is consistent if and only if there is no formula φ{displaystyle varphi }varphi such that both φ{displaystyle varphi }varphi and its negation ¬φ{displaystyle lnot varphi }{displaystyle lnot varphi } are elements of the set T{displaystyle T}T. Let A{displaystyle A}A be a set of closed sentences (informally "axioms") and ⟨A⟩{displaystyle langle Arangle }{displaystyle langle Arangle } the set of closed sentences provable from A{displaystyle A}A under some (specified, possibly implicitly) formal deductive system. The set of axioms A{displaystyle A}A is consistent when ⟨A⟩{displaystyle langle Arangle }langle Arangle is.[3]


If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete.[citation needed] The completeness of the sentential calculus was proved by Paul Bernays in 1918[citation needed][4] and Emil Post in 1921,[5] while the completeness of predicate calculus was proved by Kurt Gödel in 1930,[6] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).[7] Stronger logics, such as second-order logic, are not complete.


A consistency proof is a mathematical proof that a particular theory is consistent.[8] 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.




Contents






  • 1 Consistency and completeness in arithmetic and set theory


  • 2 First-order logic


    • 2.1 Notation


    • 2.2 Definition


    • 2.3 Basic results


    • 2.4 Henkin's theorem


    • 2.5 Sketch of proof




  • 3 Model theory


  • 4 See also


  • 5 Footnotes


  • 6 References


  • 7 External links





Consistency and completeness in arithmetic and set theory


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 recursively enumerable 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 recursively enumerable 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, 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 as Zermelo–Fraenkel set theory. 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 notion relative consistency is interesting in set theory (and in other sufficiently expressive axiomatic systems). If T is a theory and A is an additional axiom, T + A is said to be consistent relative to T (or simply that A is consistent with T) if it can be proved that
if T is consistent then T + A is consistent. If both A and ¬A are consistent with T, then A is said to be independent of T.



First-order logic



Notation


⊢{displaystyle vdash }vdash (Turnstile symbol) in the following context of mathematical logic, means "provable from". That is, a⊢b{displaystyle avdash b}{displaystyle avdash b} reads: b is provable from a (in some specified formal system). See List of logic symbols. In other cases, the turnstile symbol may mean implies; permits the derivation of. See: List of mathematical symbols.



Definition


  • A set of formulas Φ{displaystyle Phi }Phi in first-order logic is consistent (written Con⁡Φ{displaystyle operatorname {Con} Phi }{displaystyle operatorname {Con} Phi }) if there is no formula φ{displaystyle varphi }varphi such that Φ⊢φ{displaystyle Phi vdash varphi }{displaystyle Phi vdash varphi } and Φ⊢¬φ{displaystyle Phi vdash lnot varphi }{displaystyle Phi vdash lnot varphi }. Otherwise Φ{displaystyle Phi }Phi is inconsistent (written Inc⁡Φ{displaystyle operatorname {Inc} Phi }{displaystyle operatorname {Inc} Phi }).


  • Φ{displaystyle Phi }Phi is said to be simply consistent if for no formula φ{displaystyle varphi }varphi of Φ{displaystyle Phi }Phi , both φ{displaystyle varphi }varphi and the negation of φ{displaystyle varphi }varphi are theorems of Φ{displaystyle Phi }Phi .[clarification needed]


  • Φ{displaystyle Phi }Phi is said to be absolutely consistent or Post consistent if at least one formula in the language of Φ{displaystyle Phi }Phi is not a theorem of Φ{displaystyle Phi }Phi .


  • Φ{displaystyle Phi }Phi is said to be maximally consistent if for every formula φ{displaystyle varphi }varphi , if Con⁡(Φ∪φ){displaystyle operatorname {Con} (Phi cup varphi )}{displaystyle operatorname {Con} (Phi cup varphi )} implies φ∈Φ{displaystyle varphi in Phi }{displaystyle varphi in Phi }.


  • Φ{displaystyle Phi }Phi is said to contain witnesses if for every formula of the form ∃xφ{displaystyle exists x,varphi }{displaystyle exists x,varphi } there exists a term t{displaystyle t}t such that (∃xφ→φtx)∈Φ{displaystyle (exists x,varphi to varphi {t over x})in Phi }{displaystyle (exists x,varphi to varphi {t over x})in Phi }, where φtx){displaystyle varphi {t over x})}{displaystyle varphi {t over x})} denotes the substitution of each x{displaystyle x}x in φ{displaystyle varphi }varphi by a t{displaystyle t}t; see also First-order logic.[citation needed]


Basic results



  1. The following are equivalent:

    1. Inc⁡Φ{displaystyle operatorname {Inc} Phi }{displaystyle operatorname {Inc} Phi }

    2. For all φ,Φ⊢φ.{displaystyle varphi ,;Phi vdash varphi .}{displaystyle varphi ,;Phi vdash varphi .}



  2. Every satisfiable set of formulas is consistent, where a set of formulas Φ{displaystyle Phi }Phi is satisfiable if and only if there exists a model I{displaystyle {mathfrak {I}}}{mathfrak {I}} such that I⊨Φ{displaystyle {mathfrak {I}}vDash Phi }mathfrak{I} vDash Phi .

  3. For all Φ{displaystyle Phi }Phi and φ{displaystyle varphi }varphi :

    1. if not Φ⊢φ{displaystyle Phi vdash varphi }{displaystyle Phi vdash varphi }, then Con⁡(Φ∪{¬φ}){displaystyle operatorname {Con} left(Phi cup {lnot varphi }right)}{displaystyle operatorname {Con} left(Phi cup {lnot varphi }right)};

    2. if Con⁡Φ{displaystyle operatorname {Con} Phi }{displaystyle operatorname {Con} Phi } and Φ⊢φ{displaystyle Phi vdash varphi }{displaystyle Phi vdash varphi }, then Con⁡(Φ∪{φ}){displaystyle operatorname {Con} left(Phi cup {varphi }right)}{displaystyle operatorname {Con} left(Phi cup {varphi }right)};

    3. if Con⁡Φ{displaystyle operatorname {Con} Phi }{displaystyle operatorname {Con} Phi }, then Con⁡(Φ∪{φ}){displaystyle operatorname {Con} left(Phi cup {varphi }right)}{displaystyle operatorname {Con} left(Phi cup {varphi }right)} or Con⁡(Φ∪{¬φ}){displaystyle operatorname {Con} left(Phi cup {lnot varphi }right)}{displaystyle operatorname {Con} left(Phi cup {lnot varphi }right)}.



  4. Let Φ{displaystyle Phi }Phi be a maximally consistent set of formulas and contain witnesses. For all φ{displaystyle varphi }varphi and ψ{displaystyle psi }psi :

    1. if Φ⊢φ{displaystyle Phi vdash varphi }{displaystyle Phi vdash varphi }, then φ∈Φ{displaystyle varphi in Phi }{displaystyle varphi in Phi },

    2. either φ∈Φ{displaystyle varphi in Phi }{displaystyle varphi in Phi } or ¬φ∈Φ{displaystyle lnot varphi in Phi }{displaystyle lnot varphi in Phi },


    3. (φ∨ψ)∈Φ{displaystyle (varphi lor psi )in Phi }{displaystyle (varphi lor psi )in Phi } if and only if φ∈Φ{displaystyle varphi in Phi }{displaystyle varphi in Phi } or ψ∈Φ{displaystyle psi in Phi }psi in Phi,

    4. if (φ→ψ)∈Φ{displaystyle (varphi to psi )in Phi }{displaystyle (varphi to psi )in Phi } and φ∈Φ{displaystyle varphi in Phi }{displaystyle varphi in Phi }, then ψ∈Φ{displaystyle psi in Phi }psi in Phi,


    5. ∃xφ∈Φ{displaystyle exists x,varphi in Phi }{displaystyle exists x,varphi in Phi } if and only if there is a term t{displaystyle t}t such that φtx∈Φ{displaystyle varphi {t over x}in Phi }{displaystyle varphi {t over x}in Phi }.[citation needed]





Henkin's theorem


Let S{displaystyle S}S be a symbol set. Let Φ{displaystyle Phi }Phi be a maximally consistent set of S{displaystyle S}S-formulas containing witnesses.


Define an equivalence relation ∼{displaystyle sim }sim on the set of S{displaystyle S}S-terms by t0∼t1{displaystyle t_{0}sim t_{1}}t_0 sim t_1 if t0≡t1∈Φ{displaystyle ;t_{0}equiv t_{1}in Phi }; t_0 equiv t_1 in Phi, where ≡{displaystyle equiv }equiv denotes equality. Let t¯{displaystyle {overline {t}}}{displaystyle {overline {t}}} denote the equivalence class of terms containing t{displaystyle t}t; and let TΦ:={t¯∣t∈TS}{displaystyle T_{Phi }:={;{overline {t}}mid tin T^{S}}}{displaystyle T_{Phi }:={;{overline {t}}mid tin T^{S}}} where TS{displaystyle T^{S}}{displaystyle T^{S}} is the set of terms based on the symbol set S{displaystyle S}S .


Define the S{displaystyle S}S-structure TΦ{displaystyle {mathfrak {T}}_{Phi }}{displaystyle {mathfrak {T}}_{Phi }} over TΦ{displaystyle T_{Phi }}{displaystyle T_{Phi }}, also called the term-structure corresponding to Φ{displaystyle Phi }Phi , by:



  1. for each n{displaystyle n}n-ary relation symbol R∈S{displaystyle Rin S}R in S, define RTΦt0¯…tn−1¯{displaystyle R^{{mathfrak {T}}_{Phi }}{overline {t_{0}}}ldots {overline {t_{n-1}}}}{displaystyle R^{{mathfrak {T}}_{Phi }}{overline {t_{0}}}ldots {overline {t_{n-1}}}} if Rt0…tn−1∈Φ;{displaystyle ;Rt_{0}ldots t_{n-1}in Phi ;}{displaystyle ;Rt_{0}ldots t_{n-1}in Phi ;}[9]

  2. for each n{displaystyle n}n-ary function symbol f∈S{displaystyle fin S}f in S, define fTΦ(t0¯…tn−1¯):=ft0…tn−1¯;{displaystyle f^{{mathfrak {T}}_{Phi }}({overline {t_{0}}}ldots {overline {t_{n-1}}}):={overline {ft_{0}ldots t_{n-1}}};}{displaystyle 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∈S{displaystyle cin S}c in S, define cTΦ:=c¯.{displaystyle c^{{mathfrak {T}}_{Phi }}:={overline {c}}.}{displaystyle c^{{mathfrak {T}}_{Phi }}:={overline {c}}.}


Define a variable assignment βΦ{displaystyle beta _{Phi }}{displaystyle beta _{Phi }} by βΦ(x):=x¯{displaystyle beta _{Phi }(x):={bar {x}}}{displaystyle beta _{Phi }(x):={bar {x}}} for each variable x{displaystyle x}x. Let IΦ:=(TΦ,βΦ){displaystyle {mathfrak {I}}_{Phi }:=({mathfrak {T}}_{Phi },beta _{Phi })}{displaystyle {mathfrak {I}}_{Phi }:=({mathfrak {T}}_{Phi },beta _{Phi })} be the term interpretation associated with Φ{displaystyle Phi }Phi .


Then for each S{displaystyle S}S-formula φ{displaystyle varphi }varphi :



IΦ⊨φ{displaystyle {mathfrak {I}}_{Phi }vDash varphi }{displaystyle {mathfrak {I}}_{Phi }vDash varphi } if and only if φ∈Φ.{displaystyle ;varphi in Phi .}{displaystyle ;varphi in Phi .}[citation needed]




Sketch of proof


There are several things to verify. First, that ∼{displaystyle sim }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 ∼{displaystyle sim }sim is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of t0,…,tn−1{displaystyle t_{0},ldots ,t_{n-1}} t_0, ldots ,t_{n-1} class representatives. Finally, IΦ⊨φ{displaystyle {mathfrak {I}}_{Phi }vDash varphi }{displaystyle {mathfrak {I}}_{Phi }vDash varphi } can be verified by induction on formulas.



Model theory


In ZFC set theory with classical first-order logic,[10] an inconsistent theory T{displaystyle T}T is one such that there exists a closed sentence φ{displaystyle varphi }varphi such that T{displaystyle T}T contains both φ{displaystyle varphi }varphi and its negation φ′{displaystyle varphi '}varphi '. A consistent theory is one such that the following logically equivalent conditions hold




  1. {φ,φ′}⊈T{displaystyle {varphi ,varphi '}not subseteq T}{displaystyle {varphi ,varphi '}not subseteq T}[11]

  2. φ′∉T∨φ∉T{displaystyle varphi 'not in Tlor varphi not in T}{displaystyle varphi 'not in Tlor varphi not in T}



See also












  • Equiconsistency

  • Hilbert's problems

  • Hilbert's second problem

  • Jan Łukasiewicz

  • Paraconsistent logic

  • ω-consistency

  • Gentzen's consistency proof

  • Proof by contradiction



Footnotes




  1. ^ Tarski 1946 states it this way: "A deductive theory is called CONSISTENT or NON-CONTRADICTORY if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences . . . at least one cannot be proved," (p. 135) where Tarski defines contradictory as follows: "With the help of the word not one forms the NEGATION of any sentence; two sentences, of which the first is a negation of the second, are called CONTRADICTORY SENTENCES" (p. 20). This definition requires a notion of "proof". Gödel in his 1931 defines the notion this way: "The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula c of a and b is defined as an immediate consequence in terms of modus ponens or substitution; cf Gödel 1931 van Heijenoort 1967:601. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles . . . and accompanied by considerations intended to establish their validity[true conclusion for all true premises -- Reichenbach 1947:68]" cf Tarski 1946:3. Kleene 1952 defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A proof is said to be a proof of its last formula, and this formula is said to be (formally) provable or be a (formal) theorem" cf Kleene 1952:83.


  2. ^ see Paraconsistent logic


  3. ^

    Let L{displaystyle L}L be a signature, T{displaystyle T}T a theory in L∞ω{displaystyle L_{infty omega }}{displaystyle L_{infty omega }} and φ{displaystyle varphi }varphi a sentence in L∞ω{displaystyle L_{infty omega }}L_{{infty omega }}. We say that φ{displaystyle varphi }varphi is a consequence of T{displaystyle T}T, or that T{displaystyle T}T entails φ{displaystyle varphi }varphi , in symbols T⊢φ{displaystyle Tvdash varphi }T vdash varphi, if every model of T{displaystyle T}T is a model of φ{displaystyle varphi }varphi . (In particular if T{displaystyle T}T has no models then T{displaystyle T}T entails φ{displaystyle varphi }varphi .)


    Warning: we don't require that if T⊢φ{displaystyle Tvdash varphi }T vdash varphi then there is a proof of φ{displaystyle varphi }varphi from T{displaystyle T}T. In any case, with infinitary languages it's not always clear what would constitute a proof. Some writers use T⊢φ{displaystyle Tvdash varphi }{displaystyle Tvdash varphi } to mean that φ{displaystyle varphi }varphi is deducible from T{displaystyle T}T in some particular formal proof calculus, and they write T⊨φ{displaystyle Tmodels varphi }{displaystyle Tmodels varphi } for our notion of entailment (a notation which clashes with our A⊨φ{displaystyle Amodels varphi }{displaystyle Amodels varphi }). For first-order logic the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.
    We say that φ{displaystyle varphi }varphi is valid, or is a logical theorem, in symbols ⊢φ{displaystyle vdash varphi }{displaystyle vdash varphi }, if φ{displaystyle varphi }varphi is true in every L{displaystyle L}L-structure. We say that φ{displaystyle varphi }varphi is consistent if φ{displaystyle varphi }varphi is true in some L{displaystyle L}L-structure. Likewise we say that a theory T{displaystyle T}T is consistent if it has a model.



    We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T). (Please note definition of Mod(T) on p. 30 ...)



    A Shorter Model Theory by Wilfrid Hodges, p. 37





  4. ^ van Heijenoort 1967:265 states that Bernays determined the independence of the axioms of Principia Mathematica, a result not published until 1926, but he says nothing about Bernays proving their consistency.


  5. ^ Post proves both consistency and completeness of the propositional calculus of PM, cf van Heijenoort's commentary and Post's 1931 Introduction to a general theory of elementary propositions in van Heijenoort 1967:264ff. Also Tarski 1946:134ff.


  6. ^ cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967:582ff


  7. ^ cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in van Heijenoort 1967:618ff.


  8. ^ Informally, Zermelo–Fraenkel set theory is ordinarily assumed; some dialects of informal mathematics customarily assume the axiom of choice in addition.


  9. ^ This definition is indepentent of the choice of ti{displaystyle t_{i}}t_{i} due to the substitutivity properties of ≡{displaystyle equiv }equiv and the maximal consistency of Φ{displaystyle Phi }Phi .


  10. ^ the common case in many applications to other areas of mathematics as well as the ordinary mode of reasoning of informal mathematics in calculus and applications to physics, chemistry, engineering


  11. ^ according to De Morgan's laws



References




  • Stephen Kleene, 1952 10th impression 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterday, New York, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 0-7204-2103-9.


  • Hans Reichenbach, 1947, Elements of Symbolic Logic, Dover Publications, Inc. New York,
    ISBN 0-486-24004-5,


  • Alfred Tarski, 1946, Introduction to Logic and to the Methodology of Deductive Sciences, Second Edition, Dover Publications, Inc., New York,
    ISBN 0-486-28462-X.


  • Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge, MA,
    ISBN 0-674-32449-8 (pbk.)


  • The Cambridge Dictionary of Philosophy, consistency

  • H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic

  • Jevons, W.S., 1870, Elementary Lessons in Logic



External links


  • Chris Mortensen, Inconsistent Mathematics, Stanford Encyclopedia of Philosophy









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