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| | Hello and welcome. My name is Irwin and I totally dig that name. What I love performing is playing baseball but I haven't made a dime with it. California is our beginning location. Hiring is her working day occupation now but she's usually needed her own company.<br><br>Also visit my web blog; [http://luansantana.trechosertanejo.com.br/members/gidgedavies/activity/67317/ http://luansantana.trechosertanejo.com.br/] |
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| <!-- I added the odd spaces below so that they will all be rendered with TeX. In all three browsers on all three platforms I tried, PNG phi looked different enough from HTML phi that I think many people would be confused. (dreish) -->
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| In [[classical logic|classical]] [[deductive logic|deductive]] [[logic]], a '''consistent''' [[theory (mathematical logic)|theory]] is one that does not contain a [[contradiction]].<ref>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.</ref><ref>[[Paraconsistent logic]] ''tolerates'' contradictions, but toleration of contradiction does not entail consistency.</ref> 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 theory#First-order logic|model]], i.e. there exists an [[interpretation (logic)|interpretation]] under which all [[Well-formed formula|formulas]] in the theory are true. This is the sense used in traditional [[Term logic|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 (mathematical logic)|formula]] ''P'' such that both ''P'' and its negation are provable from the axioms of the theory under its associated deductive system.
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| If these semantic and syntactic definitions are equivalent for a particular deductive logic, the logic is '''[[Completeness#Logical completeness|complete]]'''.{{clarify|date=May 2012|reason=which notion of cmpleteness is this?}}{{citation needed|date=May 2012}} The completeness of the [[sentential calculus]] was proved by [[Paul Bernays]] in 1918{{Citation needed|date=October 2009}}<ref>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''.</ref> and [[Emil Post]] in 1921,<ref>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.</ref> while the completeness of [[predicate calculus]] was proved by [[Kurt Gödel]] in 1930,<ref>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</ref> and consistency proofs for arithmetics restricted with respect to the [[Mathematical induction|induction axiom schema]] were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).<ref>cf van Heijenoort's commentary and Herbrand's 1930 ''On the consistency of arithmetic'' in van Heijenoort 1967:618ff.</ref> Stronger logics, such as [[second-order logic]], are not complete.
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| 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 [[Gödel's incompleteness theorems|incompleteness theorems]], which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).
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| 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 property|normalization]] of the [[Curry-Howard|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.
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| ==Consistency and completeness in arithmetic and set theory==
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| 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.
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| [[Presburger arithmetic]] is an axiom system for the natural numbers under addition. It is both consistent and complete.
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| [[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]].
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| 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.
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| 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 (mathematical logic)|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
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| if ''T'' is consistent then ''T'' + ''A'' is consistent. If both ''A'' and ¬''A'' are consistent with ''T'', then ''A'' is said to be [[Independence (mathematical logic)|independent]] of ''T''.
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| == First-order logic ==
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| ===Notation===
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| ⊢ (Turnstyle symbol) in the following context of [[Mathematical logic]], means "provable from". That is, a ⊢ b reads: b is provable from a (in some specified formal system) -- see [[List of logic symbols]]) . In other cases, the turnstyle symbol may stand to mean infers; derived from. See: [[List of mathematical symbols]].
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| ===Definition===
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| A set of [[formulas]] <math>\Phi</math> in first-order logic is '''consistent''' (written Con<math>\Phi</math>) [[if and only if]] there is no formula <math>\phi</math> such that <math>\Phi \vdash \phi</math> and <math>\Phi \vdash \lnot\phi</math>. Otherwise <math>\Phi</math> is '''inconsistent''' and is written Inc<math>\Phi</math>.
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| <math>\Phi</math> is said to be '''simply consistent''' [[if and only if]] for no formula <math>\phi</math> of <math>\Phi</math>, both <math>\phi</math> and the [[negation]] of <math>\phi</math> are theorems of <math>\Phi</math>.
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| <math>\Phi</math> is said to be '''absolutely consistent''' or '''Post consistent''' if and only if at least one formula of <math>\Phi</math> is not a theorem of <math>\Phi</math>.
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| <math>\Phi</math> is said to be '''maximally consistent''' if and only if for every formula <math>\phi</math>, if Con (<math>\Phi \cup \phi</math>) then <math>\phi \in \Phi</math>.
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| <math>\Phi</math> is said to '''contain witnesses''' if and only if for every formula of the form <math>\exists x \phi</math> there exists a term <math>t</math> such that <math>(\exists x \phi \to \phi {t \over x}) \in \Phi</math>. See [[First-order logic]].
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| ===Basic results===
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| # The following are equivalent:
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| ## Inc<math>\Phi</math>
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| ## For all <math>\phi,\; \Phi \vdash \phi.</math>
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| # Every satisfiable set of formulas is consistent, where a set of formulas <math>\Phi</math> is satisfiable if and only if there exists a model <math>\mathfrak{I}</math> such that <math>\mathfrak{I} \vDash \Phi </math>.
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| # For all <math>\Phi</math> and <math>\phi</math>:
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| ## if not <math> \Phi \vdash \phi</math>, then Con<math>\left( \Phi \cup \{\lnot\phi\}\right)</math>;
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| ## if Con <math>\Phi</math> and <math>\Phi \vdash \phi</math>, then Con<math> \left(\Phi \cup \{\phi\}\right)</math>;
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| ## if Con <math>\Phi</math>, then Con<math>\left( \Phi \cup \{\phi\}\right)</math> or Con<math>\left( \Phi \cup \{\lnot \phi\}\right)</math>.
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| # Let <math>\Phi</math> be a maximally consistent set of formulas and contain [[Witness (mathematics)|witnesses]]. For all <math>\phi</math> and <math> \psi </math>:
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| ## if <math> \Phi \vdash \phi</math>, then <math>\phi \in \Phi</math>,
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| ## either <math>\phi \in \Phi</math> or <math>\lnot \phi \in \Phi</math>,
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| ## <math>(\phi \or \psi) \in \Phi</math> if and only if <math>\phi \in \Phi</math> or <math>\psi \in \Phi</math>,
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| ## if <math>(\phi\to\psi) \in \Phi</math> and <math>\phi \in \Phi </math>, then <math>\psi \in \Phi</math>,
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| ## <math>\exists x \phi \in \Phi</math> if and only if there is a term <math>t</math> such that <math>\phi{t \over x}\in\Phi</math>.
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| ===Henkin's theorem===
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| Let <math>\Phi</math> be a maximally consistent set of <math>S</math>-formulas containing [[Witness_(mathematics)#Henkin_witnesses|witnesses]].
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| Define a binary relation <math>\sim</math> on the set of <math>S</math>-terms such that <math>t_0 \sim t_1</math> if and only if <math>\; t_0 \equiv t_1 \in \Phi</math>; and let <math>\overline t \!</math> denote the equivalence class of terms containing <math>t \!</math>; and let <math>T_{\Phi} := \{ \; \overline t \; |\; t \in T^S \} </math> where <math>T^S \!</math> is the set of terms based on the symbol set <math>S \!</math>.
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| Define the <math>S</math>-structure <math>\mathfrak T_{\Phi} </math> over <math> T_{\Phi} \!</math> the '''term-structure''' corresponding to <math>\Phi</math> by:
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| # for <math>n</math>-ary <math>R \in S</math>, <math>R^{\mathfrak T_{\Phi}} \overline {t_0} \ldots \overline {t_{n-1}}</math> if and only if <math>\; R t_0 \ldots t_{n-1} \in \Phi</math>;
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| # for <math>n</math>-ary <math>f \in S</math>, <math>f^{\mathfrak T_{\Phi}} (\overline {t_0} \ldots \overline {t_{n-1}}) := \overline {f t_0 \ldots t_{n-1}}</math>;
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| # for <math>c \in S</math>, <math>c^{\mathfrak T_{\Phi}}:= \overline c</math>.
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| Let <math>\mathfrak I_{\Phi} := (\mathfrak T_{\Phi},\beta_{\Phi})</math> be the '''term interpretation''' associated with <math>\Phi</math>, where <math>\beta _{\Phi} (x) := \bar x</math>.
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| <center>
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| For all <math>\phi</math>, <math>\; \mathfrak I_{\Phi} \vDash \phi</math> if and only if <math> \; \phi \in \Phi</math>.
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| </center>
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| ===Sketch of proof===
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| There are several things to verify. First, that <math>\sim</math> 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 <math>\sim</math> is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of <math> t_0, \ldots ,t_{n-1} </math> class representatives. Finally, <math> \mathfrak I_{\Phi} \vDash \Phi </math> can be verified by induction on formulas.
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| ==See also==
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| {{Portal|Logic}}
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| {{Wikiquote}}
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| *[[Equiconsistency]]
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| *[[Hilbert's problems]]
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| *[[Hilbert's second problem]]
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| *[[Jan Łukasiewicz]]
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| *[[Paraconsistent logic]]
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| *[[ω-consistency]]
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| ==Footnotes==
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| <references/>
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| ==References==
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| * [[Stephen Kleene]], 1952 10th impression 1991, ''Introduction to Metamathematics'', North-Holland Publishing Company, Amsterday, New York, ISBN 0-7204-2103-9.
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| * [[Hans Reichenbach]], 1947, ''Elements of Symbolic Logic'', Dover Publications, Inc. New York, ISBN 0-486-24004-5,
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| * [[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.
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| * [[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.)
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| * [[The Cambridge Dictionary of Philosophy]], ''consistency''
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| * H.D. Ebbinghaus, J. Flum, W. Thomas, '''Mathematical Logic'''
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| * Jevons, W.S., 1870, ''Elementary Lessons in Logic''
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| ==External links==
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| * Chris Mortensen, [http://plato.stanford.edu/entries/mathematics-inconsistent/ Inconsistent Mathematics], [[Stanford Encyclopedia of Philosophy]]
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| {{Logic}}
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| [[Category:Proof theory]]
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| [[Category:Hilbert's problems]]
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| [[Category:Metalogic]]
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