adj.

(言行・思想など)首尾一貫した、矛盾がない(with)。無矛盾の
(人が)言行が一致した、堅実な

関: accord, accordance, agree, agreement, coincide, coincidence, concert, concerted, concordance, concur, concurrent, conform, congruence, congruent, consensus, consistency, consonant, correspond, correspondence, corresponding, fit, match, unanimous, unity

WordNet

(sometimes followed by `with'
the same throughout in structure or composition; "bituminous coal is often treated as a consistent and homogeneous product" (同)uniform
provide funds complementary to; "The company matched the employees contributions"
the score needed to win a match
lighter consisting of a thin piece of wood or cardboard tipped with combustible chemical; ignites with friction; "he always carries matches to light his pipe"; "as long youve a lucifer to light your fag" (同)lucifer, friction match
an exact duplicate; "when a match is found an entry is made in the notebook" (同)mate
a burning piece of wood or cardboard; "if you drop a match in there the whole place will explode"
a formal contest in which two or more persons or teams compete
something that resembles or harmonizes with; "that tie makes a good match with your jacket"
make correspond or harmonize; "Match my sweater" (同)fit
be compatible, similar or consistent; coincide in their characteristics; "The two stories dont agree in many details"; "The handwriting checks with the signature on the check"; "The suspects fingerprints dont match those on the gun" (同)fit, correspond, check, jibe, gibe, tally, agree
bring two objects, ideas, or people together; "This fact is coupled to the other one"; "Matchmaker, can you match my daughter with a nice young man?"; "The student was paired with a partner for collaboration on the project" (同)mate, couple, pair, twin
be equal or harmonize; "The two pieces match"
give or join in marriage
physically and mentally sound or healthy; "felt relaxed and fit after their holiday"; "keeps fit with diet and exercise"
a display of bad temper; "he had a fit"; "she threw a tantrum"; "he made a scene" (同)tantrum, scene, conniption
insert or adjust several objects or people; "Can you fit the toy into the box?"; "This man cant fit himself into our work environment"
make fit; "fit a dress"; "He fitted other pieces of paper to his cut-out"
the manner in which something fits; "I admired the fit of her coat"
be the right size or shape; fit correctly or as desired; "This piece wont fit into the puzzle" (同)go
(usually followed by `to or `for' (同)primed, set
conform to some shape or size; "How does this shirt fit?"
meeting adequate standards for a purpose; "a fit subject for discussion"; "it is fit and proper that you be there"; "water fit to drink"; "fit for duty"; "do as you see fit to"
show grammatical agreement; "Subjects and verbs must always agree in English"
be in accord; be in agreement; "We agreed on the terms of the settlement"; "I cant agree with you!"; "I hold with those who say life is sacred"; "Both philosophers concord on this point" (同)hold, concur, concord
achieve harmony of opinion, feeling, or purpose; "No two of my colleagues would agree on whom to elect chairman"
be agreeable or suitable; "White wine doesnt agree with me"
consent or assent to a condition, or agree to do something; "She agreed to all my conditions"; "He agreed to leave her alone"
compatibility of observations; "there was no agreement between theory and measurement"; "the results of two tests were in correspondence" (同)correspondence
harmony of peoples opinions or actions or characters; "the two parties were in agreement" (同)accord
the determination of grammatical inflection on the basis of word relations (同)concord
the thing arranged or agreed to; "they made arrangements to meet in Chicago" (同)arrangement
the statement (oral or written) of an exchange of promises; "they had an agreement that they would not interfere in each others business"; "there was an understanding between management and the workers" (同)understanding
the verbal act of agreeing
sympathetic compatibility
allow to have; "grant a privilege" (同)allot, grant
concurrence of opinion; "we are in accord with your proposal" (同)conformity, accordance
not capable of being made consistent or harmonious; "inconsistent accounts"
displaying a lack of consistency; "inconsistent statements cannot both be true at the same time"; "inconsistent with the roadmap"
without showing consistency; "he acted inconsistently when he bought these stocks"
not self-contradictory

PrepTutorEJDIC

『矛盾のない』 / (主義・考え方などが)不変の,(人が)節操のある
(1本の)『まっち』 / (昔の)火縄[銃]
《単数形で》(…に)『匹敵する相手』,(…の)好敵手《+『for』+『名』》 / 《単数形で》対[の一方],片われ,生き写し / 《単数形で》(…と)『よくつり合うもの』《+『for』+『名』》 / 『試合』,競技 / 《古》縁組,結婚;結婚相手 / (…で)…‘に'『匹敵する』,‘と'対等である《+『名』+『in』(『for』)+『名』》 / (…と)…‘を'対抗させる,試合させる《+『名』+『against』(『with』)+『名』》 / 〈物が〉…‘と'『調和する』,つり合う / 〈人が〉…‘に'調和するものを見つける / 《古》(…と)…‘を'結婚させる《+『名』+『to』(『with』)+『名』》 / (…と)『調和がとれている』,つり合う《+『up with』+『名』》
『適した』,適切な,ぴったりの / 《補語にのみ用いて》よい調子で;健康で / 《be fit to do》いまにも…しそうである《explode,burst,fall,dropなど危険な意 / を含む動詞と共に用いる》 / 〈意図・目的など〉‘に'『合う』,ふさわしい / (…に)…‘を'『合わせる』,はめ込む,取り付ける《+『名』+『to』(『into,in,on』)+『名』》;(…を)…‘に'合わせる《+『名』+『with』+『名』》 / 〈人〉‘に'資格をつける,‘を'適させる / 適する,合う(服・靴などの)でき具合;体に合うもの / 〈U〉適合,対応
(病気の)『発作』 / (感情などの)『一時的な爆発』《+『of』+『名』》
〈人が〉『意見が一致する』,合意する;『同意する』,賛成する / 〈物事が〉(…と)合致する,一致する《+『with』+『名』》 / (…と)いっしょにうまくやる《+『with』+『名』》 / (文章で)〈人称・性・数・格が〉一致する / …に意見が一致する,同意する
〈U〉(意見・感情などの)(…についての)『一致』,調和,同意《+『on』(『upon,about,with』)+『名』(『wh-節・句』)》 / 〈C〉(2者以上の間で結ばれた)『協定』,協約 / 〈U〉(人称・性・格の)一致(concord)
(…と)『一致する』,調和する《+『with』+『名』(『wh-』節)》 / 〈許可・賞賛・歓迎など〉'を'与える / 〈U〉(…との)『一致』,調和(harmony)《+『with』+『名』》 / 〈C〉国際協定,講和 / 〈U〉〈C〉和音,協和音;(色の)調和
食い違っている,矛盾する
一貫して;矛盾なく,きまって

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/01/28 08:39:38」(JST)

wiki en

[Wiki en表示]

For other uses, see Consistency (disambiguation).

In logic, a consistent theory is one that does not contain a contradiction.^[1] 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 that a theory is consistent if and only 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.^{[clarification needed]}^{[citation needed]} The completeness of sentential calculus was proved by Paul Bernays in 1918^{[citation needed]}^[2] and Emil Post in 1921,^[3] while the completeness of predicate calculus was proved by Kurt Gödel in 1930,^[4] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).^[5] 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 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 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–Fraenkel set theory. These set theories cannot prove their own Gödel sentences – 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

A set of formulas in first-order logic is consistent (written Con) if and only if there is no formula such that and . Otherwise is inconsistent and is written Inc.

is said to be simply consistent if and only if for no formula of , both and the negation of are 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.

Basic results

The following are equivalent:
1. Inc
2. 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 :
1. if not , then Con;
2. if Con and , then Con;
3. if Con , then Con or Con.
Let be a maximally consistent set of formulas and contain witnesses. For all and :
1. if , then ,
2. either or ,
3. if and only if or ,
4. if and , then ,
5. 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 -terms such that 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 -structure over the term-structure corresponding to by:

for -ary , if and only if ;
for -ary , ;
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. 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.

Footnotes

^ 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.
^ 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.
^ 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 propositons in van Heijenoort 1967:264ff. Also Tarski 1946:134ff.
^ cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967:582ff
^ cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in van Heijenoort 1967:618ff.

References

Stephen Kleene, 1952 10th impression 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterday, New York, ISBN 0-7204-2103-9.
Hans Reichenbach, 1947, Elements of Symbolic Logic, Dover Publications, Inc. New York, ISBN 0-486-24004-5,
T. J. Stepien and L. T. Stepien, On the consistency of Peano's Arithmetic System, Bull. Symb. Logic 16, 132 (2010), http://www.math.ucla.edu/~asl/bsl/1601-toc.htm
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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