![Consistency](https://www.english.nina.az/wikipedia/image/aHR0cHM6Ly91cGxvYWQud2lraW1lZGlhLm9yZy93aWtpcGVkaWEvY29tbW9ucy90aHVtYi9jL2NkL1NvY3JhdGVzLnBuZy8xNjAwcHgtU29jcmF0ZXMucG5n.png )
In classical, deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences (informally "axioms") and the set of closed sentences provable from under some (specified, possibly implicitly) formal deductive system. The set of axioms is consistent when there is no formula such that and . A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial.: 7 Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.
In a sound 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 deductive logic, the logic is called complete.[citation needed] The completeness of the propositional calculus was proved by Paul Bernays in 1918[citation needed] and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). 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 the incompleteness 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. The cut-elimination (or equivalently the normalization of the underlying calculus if 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 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 (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 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
In the following context of mathematical logic, the turnstile symbol means "provable from". That is,
reads: b is provable from a (in some specified formal system).
Definition
- A set of formulas
in first-order logic is consistent (written
) if there is no formula
such that
and
. Otherwise
is inconsistent (written
).
is said to be simply consistent if for no formula
of
, both
and the negation of
are theorems of
.[clarification needed]
is said to be absolutely consistent or Post consistent if at least one formula in the language of
is not a theorem of
.
is said to be maximally consistent if
is consistent and for every formula
,
implies
.
is said to contain witnesses if for every formula of the form
there exists a term
such that
, where
denotes the substitution of each
in
by a
; see also First-order logic.[citation needed]
Basic results
- The following are equivalent:
- For all
- Every satisfiable set of formulas is consistent, where a set of formulas
is satisfiable if and only if there exists a model
such that
.
- For all
and
:
- if not
, then
;
- if
and
, then
;
- if
, then
or
.
- if not
- Let
be a maximally consistent set of formulas and suppose it contains witnesses. For all
and
:
- if
, then
,
- either
or
,
if and only if
or
,
- if
and
, then
,
if and only if there is a term
such that
.[citation needed]
- if
Henkin's theorem
Let be a set of symbols. Let
be a maximally consistent set of
-formulas containing witnesses.
Define an equivalence relation on the set of
-terms by
if
, where
denotes equality. Let
denote the equivalence class of terms containing
; and let
where
is the set of terms based on the set of symbols
.
Define the -structure
over
, also called the term-structure corresponding to
, by:
- for each
-ary relation symbol
, define
if
- for each
-ary function symbol
, define
- for each constant symbol
, define
Define a variable assignment by
for each variable
. Let
be the term interpretation associated with
.
Then for each -formula
:
Sketch of proof
There are several things to verify. First, that 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
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.
Model theory
In ZFC set theory with classical first-order logic, an inconsistent theory is one such that there exists a closed sentence
such that
contains both
and its negation
. A consistent theory is one such that the following logically equivalent conditions hold
See also
![image](https://www.english.nina.az/wikipedia/image/aHR0cHM6Ly93d3cuZW5nbGlzaC5uaW5hLmF6L3dpa2lwZWRpYS9pbWFnZS9hSFIwY0hNNkx5OTFjR3h2WVdRdWQybHJhVzFsWkdsaExtOXlaeTkzYVd0cGNHVmthV0V2WTI5dGJXOXVjeTkwYUhWdFlpOW1MMlpoTDFkcGEybHhkVzkwWlMxc2IyZHZMbk4yWnk4ek5IQjRMVmRwYTJseGRXOTBaUzFzYjJkdkxuTjJaeTV3Ym1jPS5wbmc=.png)
- Cognitive dissonance
- Equiconsistency
- Hilbert's problems
- Hilbert's second problem
- Jan Łukasiewicz
- Paraconsistent logic
- ω-consistency
- Gentzen's consistency proof
- Proof by contradiction
Notes
- 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 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, p. 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, p. 68]" cf Tarski 1946, p. 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, p. 83.
- Carnielli, Walter; Coniglio, Marcelo Esteban (2016). Paraconsistent logic: consistency, contradiction and negation. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer. doi:10.1007/978-3-319-33205-5. ISBN 978-3-319-33203-1. MR 3822731. Zbl 1355.03001.
- Hodges, Wilfrid (1997). A Shorter Model Theory. New York: Cambridge University Press. p. 37.
Let
(Please note the definition of Mod(T) on p. 30 ...)be a signature,
a theory in
and
a sentence in
. We say that
is a consequence of
, or that
entails
, in symbols
, if every model of
is a model of
. (In particular if
has no models then
entails
.)
Warning: we don't require that ifthen there is a proof of
from
. In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use
to mean that
is deducible from
in some particular formal proof calculus, and they write
for our notion of entailment (a notation which clashes with our
). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.
We say thatis valid, or is a logical theorem, in symbols
, if
is true in every
-structure. We say that
is consistent if
is true in some
-structure. Likewise, we say that a theory
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). - van Heijenoort 1967, p. 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.
- 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, pp. 264ff. Also Tarski 1946, pp. 134ff.
- cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967, pp. 582ff.
- cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in van Heijenoort 1967, pp. 618ff.
- A consistency proof often assumes the consistency of another theory. In most cases, this other theory is Zermelo–Fraenkel set theory with or without the axiom of choice (this is equivalent since these two theories have been proved equiconsistent; that is, if one is consistent, the same is true for the other).
- This definition is independent of the choice of
due to the substitutivity properties of
and the maximal consistency of
.
- 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
- according to De Morgan's laws
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. ISBN 0-7204-2103-9. 10th impression 1991.
- Reichenbach, Hans (1947). Elements of Symbolic Logic. New York: Dover. ISBN 0-486-24004-5.
- Tarski, Alfred (1946). Introduction to Logic and to the Methodology of Deductive Sciences (Second ed.). New York: Dover. ISBN 0-486-28462-X.
- van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press. ISBN 0-674-32449-8. (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
![image](https://www.english.nina.az/wikipedia/image/aHR0cHM6Ly93d3cuZW5nbGlzaC5uaW5hLmF6L3dpa2lwZWRpYS9pbWFnZS9hSFIwY0hNNkx5OTFjR3h2WVdRdWQybHJhVzFsWkdsaExtOXlaeTkzYVd0cGNHVmthV0V2WTI5dGJXOXVjeTkwYUhWdFlpODVMems1TDFkcGEzUnBiMjVoY25rdGJHOW5ieTFsYmkxMk1pNXpkbWN2TkRCd2VDMVhhV3QwYVc5dVlYSjVMV3h2WjI4dFpXNHRkakl1YzNabkxuQnVadz09LnBuZw==.png)
- Mortensen, Chris (2017). "Inconsistent Mathematics". Stanford Encyclopedia of Philosophy.
In classical deductive logic a consistent theory is one that does not lead to a logical contradiction A theory T displaystyle T is consistent if there is no formula f displaystyle varphi such that both f displaystyle varphi and its negation f displaystyle lnot varphi are elements of the set of consequences of T displaystyle T Let A displaystyle A be a set of closed sentences informally axioms and A displaystyle langle A rangle the set of closed sentences provable from A displaystyle A under some specified possibly implicitly formal deductive system The set of axioms A displaystyle A is consistent when there is no formula f displaystyle varphi such that f A displaystyle varphi in langle A rangle and f A displaystyle lnot varphi in langle A rangle A trivial theory i e one which proves every sentence in the language of the theory is clearly inconsistent Conversely in an explosive formal system e g classical or intuitionistic propositional or first order logics every inconsistent theory is trivial 7 Consistency of a theory is a syntactic notion whose semantic counterpart is satisfiability A theory is satisfiable if it has a model i e there exists an interpretation under which all axioms in the theory are true This is what consistent meant in traditional Aristotelian logic although in contemporary mathematical logic the term satisfiable is used instead In a sound 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 deductive logic the logic is called complete citation needed The completeness of the propositional calculus was proved by Paul Bernays in 1918 citation needed and Emil Post in 1921 while the completeness of first order predicate calculus was proved by Kurt Godel in 1930 and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann 1924 von Neumann 1927 and Herbrand 1931 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 the incompleteness 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 The cut elimination or equivalently the normalization of the underlying calculus if 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 theoryIn 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 f in its language at least one of f or f 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 Godel s incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent Godel s theorem applies to the theories of Peano arithmetic PA and primitive recursive arithmetic PRA but not to Presburger arithmetic Moreover Godel 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 Godel 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 ZF These set theories cannot prove their own Godel 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 logicNotation In the following context of mathematical logic the turnstile symbol displaystyle vdash means provable from That is a b displaystyle a vdash b reads b is provable from a in some specified formal system Definition A set of formulas F displaystyle Phi in first order logic is consistent written Con F displaystyle operatorname Con Phi if there is no formula f displaystyle varphi such that F f displaystyle Phi vdash varphi and F f displaystyle Phi vdash lnot varphi Otherwise F displaystyle Phi is inconsistent written Inc F displaystyle operatorname Inc Phi F displaystyle Phi is said to be simply consistent if for no formula f displaystyle varphi of F displaystyle Phi both f displaystyle varphi and the negation of f displaystyle varphi are theorems of F displaystyle Phi clarification needed F displaystyle Phi is said to be absolutely consistent or Post consistent if at least one formula in the language of F displaystyle Phi is not a theorem of F displaystyle Phi F displaystyle Phi is said to be maximally consistent if F displaystyle Phi is consistent and for every formula f displaystyle varphi Con F f displaystyle operatorname Con Phi cup varphi implies f F displaystyle varphi in Phi F displaystyle Phi is said to contain witnesses if for every formula of the form xf displaystyle exists x varphi there exists a term t displaystyle t such that xf ftx F displaystyle exists x varphi to varphi t over x in Phi where ftx displaystyle varphi t over x denotes the substitution of each x displaystyle x in f displaystyle varphi by a t displaystyle t see also First order logic citation needed Basic results The following are equivalent Inc F displaystyle operatorname Inc Phi For all f F f displaystyle varphi Phi vdash varphi Every satisfiable set of formulas is consistent where a set of formulas F displaystyle Phi is satisfiable if and only if there exists a model I displaystyle mathfrak I such that I F displaystyle mathfrak I vDash Phi For all F displaystyle Phi and f displaystyle varphi if not F f displaystyle Phi vdash varphi then Con F f displaystyle operatorname Con left Phi cup lnot varphi right if Con F displaystyle operatorname Con Phi and F f displaystyle Phi vdash varphi then Con F f displaystyle operatorname Con left Phi cup varphi right if Con F displaystyle operatorname Con Phi then Con F f displaystyle operatorname Con left Phi cup varphi right or Con F f displaystyle operatorname Con left Phi cup lnot varphi right Let F displaystyle Phi be a maximally consistent set of formulas and suppose it contains witnesses For all f displaystyle varphi and ps displaystyle psi if F f displaystyle Phi vdash varphi then f F displaystyle varphi in Phi either f F displaystyle varphi in Phi or f F displaystyle lnot varphi in Phi f ps F displaystyle varphi lor psi in Phi if and only if f F displaystyle varphi in Phi or ps F displaystyle psi in Phi if f ps F displaystyle varphi to psi in Phi and f F displaystyle varphi in Phi then ps F displaystyle psi in Phi xf F displaystyle exists x varphi in Phi if and only if there is a term t displaystyle t such that ftx F displaystyle varphi t over x in Phi citation needed Henkin s theorem Let S displaystyle S be a set of symbols Let F displaystyle Phi be a maximally consistent set of S displaystyle S formulas containing witnesses Define an equivalence relation displaystyle sim on the set of S displaystyle S terms by t0 t1 displaystyle t 0 sim t 1 if t0 t1 F displaystyle t 0 equiv t 1 in Phi where displaystyle equiv denotes equality Let t displaystyle overline t denote the equivalence class of terms containing t displaystyle t and let TF t t TS displaystyle T Phi overline t mid t in T S where TS displaystyle T S is the set of terms based on the set of symbols S displaystyle S Define the S displaystyle S structure TF displaystyle mathfrak T Phi over TF displaystyle T Phi also called the term structure corresponding to F displaystyle Phi by for each n displaystyle n ary relation symbol R S displaystyle R in S define RTFt0 tn 1 displaystyle R mathfrak T Phi overline t 0 ldots overline t n 1 if Rt0 tn 1 F displaystyle Rt 0 ldots t n 1 in Phi for each n displaystyle n ary function symbol f S displaystyle f in S define fTF 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 for each constant symbol c S displaystyle c in S define cTF c displaystyle c mathfrak T Phi overline c Define a variable assignment bF displaystyle beta Phi by bF x x displaystyle beta Phi x bar x for each variable x displaystyle x Let IF TF bF displaystyle mathfrak I Phi mathfrak T Phi beta Phi be the term interpretation associated with F displaystyle Phi Then for each S displaystyle S formula f displaystyle varphi IF f displaystyle mathfrak I Phi vDash varphi if and only if f F displaystyle varphi in Phi citation needed Sketch of proof There are several things to verify First that displaystyle 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 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 class representatives Finally IF f displaystyle mathfrak I Phi vDash varphi can be verified by induction on formulas Model theoryIn ZFC set theory with classical first order logic an inconsistent theory T displaystyle T is one such that there exists a closed sentence f displaystyle varphi such that T displaystyle T contains both f displaystyle varphi and its negation f displaystyle varphi A consistent theory is one such that the following logically equivalent conditions hold f f T displaystyle varphi varphi not subseteq T f T f T displaystyle varphi not in T lor varphi not in T See alsoPhilosophy portalWikiquote has quotations related to Consistency Cognitive dissonance Equiconsistency Hilbert s problems Hilbert s second problem Jan Lukasiewicz Paraconsistent logic w consistency Gentzen s consistency proof Proof by contradictionNotesTarski 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 Godel 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 Godel 1931 van Heijenoort 1967 p 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 p 68 cf Tarski 1946 p 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 proofofits last formula and this formula is said to be formally provableor be a formal theorem cf Kleene 1952 p 83 Carnielli Walter Coniglio Marcelo Esteban 2016 Paraconsistent logic consistency contradiction and negation Logic Epistemology and the Unity of Science Vol 40 Cham Springer doi 10 1007 978 3 319 33205 5 ISBN 978 3 319 33203 1 MR 3822731 Zbl 1355 03001 Hodges Wilfrid 1997 A Shorter Model Theory New York Cambridge University Press p 37 Let L displaystyle L be a signature T displaystyle T a theory in L w displaystyle L infty omega and f displaystyle varphi a sentence in L w displaystyle L infty omega We say that f displaystyle varphi is a consequence of T displaystyle T or that T displaystyle T entails f displaystyle varphi in symbols T f displaystyle T vdash varphi if every model of T displaystyle T is a model of f displaystyle varphi In particular if T displaystyle T has no models then T displaystyle T entails f displaystyle varphi Warning we don t require that if T f displaystyle T vdash varphi then there is a proof of f displaystyle varphi from T displaystyle T In any case with infinitary languages it s not always clear what would constitute proof Some writers use T f displaystyle T vdash varphi to mean that f displaystyle varphi is deducible from T displaystyle T in some particular formal proof calculus and they write T f displaystyle T models varphi for our notion of entailment a notation which clashes with our A f displaystyle A models varphi For first order logic the two kinds of entailment coincide by the completeness theorem for the proof calculus in question We say that f displaystyle varphi is valid or is a logical theorem in symbols f displaystyle vdash varphi if f displaystyle varphi is true in every L displaystyle L structure We say that f displaystyle varphi is consistent if f displaystyle varphi is true in some L displaystyle L structure Likewise we say that a theory T displaystyle 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 the definition of Mod T on p 30 van Heijenoort 1967 p 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 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 pp 264ff Also Tarski 1946 pp 134ff cf van Heijenoort s commentary and Godel s 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967 pp 582ff cf van Heijenoort s commentary and Herbrand s 1930 On the consistency of arithmetic in van Heijenoort 1967 pp 618ff A consistency proof often assumes the consistency of another theory In most cases this other theory is Zermelo Fraenkel set theory with or without the axiom of choice this is equivalent since these two theories have been proved equiconsistent that is if one is consistent the same is true for the other This definition is independent of the choice of ti displaystyle t i due to the substitutivity properties of displaystyle equiv and the maximal consistency of F displaystyle Phi 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 according to De Morgan s lawsReferencesGodel Kurt 1 December 1931 Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I Monatshefte fur Mathematik und Physik 38 1 173 198 doi 10 1007 BF01700692 Kleene Stephen 1952 Introduction to Metamathematics New York North Holland ISBN 0 7204 2103 9 10th impression 1991 Reichenbach Hans 1947 Elements of Symbolic Logic New York Dover ISBN 0 486 24004 5 Tarski Alfred 1946 Introduction to Logic and to the Methodology of Deductive Sciences Second ed New York Dover ISBN 0 486 28462 X van Heijenoort Jean 1967 From Frege to Godel A Source Book in Mathematical Logic Cambridge MA Harvard University Press ISBN 0 674 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 linksLook up consistency in Wiktionary the free dictionary Mortensen Chris 2017 Inconsistent Mathematics Stanford Encyclopedia of Philosophy