First-order logicFrom Wikipedia, the free encyclopediaContents1 Consistency 11.1 Consistency and completeness in arithmetic and set theory . . . . . . . . . . . . . . . . . . . . . . 11.2 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Henkins theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.5 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Equiconsistency 52.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 First-order logic 73.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.3 Validity, satisability, and logical consequence. . . . . . . . . . . . . . . . . . . . . . . . 143.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 153.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iii CONTENTS3.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5.1 First-order logic without equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8.6 Fixpoint logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.9 Automated theorem proving and formal methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 233.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 If and only if 284.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30CONTENTS iii4.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Theory (mathematical logic) 315.1 Theories expressed in formal language generally . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.1 Subtheories and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.2 Deductive theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.3 Consistency and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.4 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1.5 Theories associated with a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 First-order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.1 Derivation in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.2 Syntactic consequence in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.3 Interpretation of a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.4 First order theories with identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.5 Topics related to rst order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Well-formed formula 356.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 -consistent theory 407.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2.1 Consistent, -inconsistent theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2.2 Arithmetically sound, -inconsistent theories . . . . . . . . . . . . . . . . . . . . . . . . 417.2.3 Arithmetically unsound, -consistent theories . . . . . . . . . . . . . . . . . . . . . . . . 417.3 -logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.4 Relation to other consistency principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42iv CONTENTS7.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 447.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Chapter 1ConsistencyFor 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 contra-diction can be dened in either semantic or syntactic terms. The semantic denition states that a theory is consistentif and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This isthe sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisable isused instead. The syntactic denition states that a theory is consistent if and only if there is no formula P such thatboth P and its negation are provable from the axioms of the theory under its associated deductive system.If these semantic and syntactic denitions are equivalent for any theory formulated using a particular deductive logic,the logic is called complete. The completeness of the sentential calculus was proved by Paul Bernays in 1918[3]and Emil Post in 1921,[4] while the completeness of predicate calculus was proved by Kurt Gdel in 1930,[5] andconsistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann(1924), von Neumann (1927) and Herbrand (1931).[6] 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 math-ematical proof theory was driven by the desire to provide nitary consistency proofs for all of mathematics as partof Hilberts program. Hilberts program was strongly impacted by incompleteness theorems, which showed thatsuciently 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 anyneed to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlyingcalculus 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.1.1 Consistency and completeness in arithmetic and set theoryIn theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of thetheory and its completeness. A theory is complete if, for every formula in its language, at least one of or isa logical consequence of the theory.Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete.Gdels incompleteness theorems show that any suciently strong eective theory of arithmetic cannot be bothcomplete and consistent.Gdels theorem applies to the theories of Peano arithmetic (PA) and Primitive recursivearithmetic (PRA), but not to Presburger arithmetic.Moreover, Gdels second incompleteness theorem shows that the consistency of suciently strong eective theoriesof arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particularsentence, called the Gdel sentence of the theory, which is a formalized statement of the claim that the theory isindeed consistent. Thus the consistency of a suciently strong, eective, consistent theory of arithmetic can never beproven in that system itself. The same result is true for eective theories that can describe a strong enough fragmentof arithmetic including set theories such as ZermeloFraenkel set theory. These set theories cannot prove their ownGdel sentences provided that they are consistent, which is generally believed.12 CHAPTER 1. CONSISTENCYBecause consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory(and in other suciently expressive axiomatic systems). If T is a theory and A is an additional axiom, T + A is saidto 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.1.2 First-order logic1.2.1 Notation (Turnstile symbol) in the following context of Mathematical logic, means provable from. That is, a b reads:b is provable from a (in some specied formal system) -- see List of logic symbols) . In other cases, the turnstilesymbol may stand to mean infers; derived from. See: List of mathematical symbols.1.2.2 DenitionA set of formulas in rst-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 tocontainwitnesses if and only if for every formula of the form x there exists a termt such that(x tx) . See First-order logic.1.2.3 Basic results1. The following are equivalent:(a) Inc (b) For all , .2. Every satisable set of formulas is consistent, where a set of formulas is satisable if and only if there existsa model I such that I .3. For all and :(a) if not , then Con ( {}) ;(b) if Con and , then Con ( {}) ;(c) if Con , then Con ( {}) or Con ( {}) .4. Let be a maximally consistent set of formulas and contain witnesses. For all and :(a) if , then ,(b) either or ,(c) ( ) if and only if or ,(d) if ( ) and , then ,(e) x if and only if there is a term t such that tx .1.3. SEE ALSO 31.2.4 Henkins theoremLet be a maximally consistent set of S -formulas containing witnesses.Dene a binary relation on the set of S -terms such that t0 t1 if and only if t0 t1 ; and let t denotethe equivalence class of terms containing t; and let T:= {t |t TS} where TSis the set of terms based on thesymbol set S.Dene the S -structure T over Tthe term-structure corresponding to by:1. for n -ary R S , RTt0. . . tn1 if and only if Rt0. . . tn1 ;2. for n -ary f S , fT(t0. . . tn1) := ft0. . . tn1 ;3. for c S , cT:= c .Let I:= (T, ) be the term interpretation associated with , where (x) := x .For all , I if and only if .1.2.5 Sketch of proofThere are several things to verify. First, that is an equivalence relation. Then, it needs to be veried that (1), (2),and (3) are well dened. 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 t0, . . . , tn1 class representatives. Finally, I can be veried byinduction on formulas.1.3 See alsoEquiconsistencyHilberts problemsHilberts second problemJan ukasiewiczParaconsistent logic-consistencyGentzens consistency proof1.4 Footnotes[1] Tarski 1946 states it this way: A deductive theory is called CONSISTENT or NON-CONTRADICTORY if no twoasserted 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 denes contradictory as follows: With the help of the word notone forms the NEGATION of any sentence; two sentences, of which the rst is a negation of the second, are calledCONTRADICTORY SENTENCES (p. 20). This denition requires a notion of proof. Gdel in his 1931 denes thenotion this way: The class of provable formulas is dened to be the smallest class of formulas that contains the axiomsand is closed under the relation immediate consequence, i.e. formula c of a and b is dened as an immediate consequencein terms of modus ponens or substitution; cf Gdel 1931 van Heijenoort 1967:601.Tarski denes proof informally asstatements follow one another in a denite order according to certain principles . . . and accompanied by considerationsintended to establish their validity[true conclusion for all true premises -- Reichenbach 1947:68]" cf Tarski 1946:3. Kleene1952 denes the notion with respect to either an induction or as to paraphrase) a nite sequence of formulas such that eachformula in the sequence is either an axiom or an immediate consequence of the preceding formulas; A proof is said to bea proof of its last formula, and this formula is said to be (formally) provable or be a (formal) theorem cf Kleene 1952:83.[2] Paraconsistent logic tolerates contradictions, but toleration of contradiction does not entail consistency.4 CHAPTER 1. CONSISTENCY[3] van Heijenoort 1967:265 states that Bernays determined the independence of the axioms of Principia Mathematica, a resultnot published until 1926, but he says nothing about Bernays proving their consistency.[4] Post proves both consistency and completeness of the propositional calculus of PM, cf van Heijenoorts commentaryand Posts 1931 Introduction to a general theory of elementary propositons in van Heijenoort 1967:264. Also Tarski1946:134.[5] cf van Heijenoorts commentary and Gdels 1930 The completeness of the axioms of the functional calculus of logic in vanHeijenoort 1967:582[6] cf van Heijenoorts commentary and Herbrands 1930 On the consistency of arithmetic in van Heijenoort 1967:618.1.5 ReferencesStephen Kleene, 1952 10th impression 1991, IntroductiontoMetamathematics, North-Holland PublishingCompany, Amsterday, New York, ISBN 0-7204-2103-9.Hans Reichenbach, 1947, Elements of Symbolic Logic, Dover Publications, Inc. NewYork, ISBN0-486-24004-5,Alfred Tarski, 1946, Introduction to Logic and to the Methodology of Deductive Sciences, Second Edition, DoverPublications, Inc., New York, ISBN 0-486-28462-X.Jean van Heijenoort, 1967, From Frege to Gdel: A Source Book in Mathematical Logic, Harvard UniversityPress, Cambridge, MA, ISBN 0-674-32449-8 (pbk.)The Cambridge Dictionary of Philosophy, consistencyH.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical LogicJevons, W.S., 1870, Elementary Lessons in Logic1.6 External linksChris Mortensen, Inconsistent Mathematics, Stanford Encyclopedia of PhilosophyChapter 2EquiconsistencyIn mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of theother theory, and vice versa. In this case, they are, roughly speaking, as consistent as each other.In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believedto be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistentif wecan do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T areequiconsistent.2.1 ConsistencyIn mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enoughto model dierent mathematical objects, it is natural to wonder about their own consistency.Hilbert proposed a programat the beginning of the 20th century whose ultimate goal was to show, using mathematicalmethods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, theprogramquickly became the establishment of the consistency of arithmetic by methods formalizable within arithmeticitself.Gdel's incompleteness theorems show that Hilberts program cannot be realized: If a consistent recursively enumer-able theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strongenough to model a weak fragment of arithmetic (Robinson arithmetic suces), then the theory cannot prove its ownconsistency. There are some technical caveats as to what requirements the formal statement representing the meta-mathematical statement The theory is consistent needs to satisfy, but the outcome is that if a (suciently strong)theory can prove its own consistency then either there is no computable way of identifying whether a statement is evenan axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, includingfalse statements such as its own consistency).Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories.Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Twotheories are equiconsistent if each one is consistent relative to the other.2.2 Consistency strengthIf T is consistent relative to S, but S is not known to be consistent relative to T, then we say that S has greaterconsistencystrength than T. When discussing these issues of consistency strength the metatheory in which thediscussion takes places needs to be carefully addressed. For theories at the level of second-order arithmetic, thereverse mathematics program has much to say. Consistency strength issues are a usual part of set theory, since thisis a recursive theory that can certainly model most of mathematics.The usual set of axioms of set theory is calledZFC. When a set theoretic statement A is said to be equiconsistent to another B, what is being claimed is that in themetatheory (Peano Arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent.Usually, primitive recursive arithmetic can be adopted as the metatheory in question, but even if the metatheory is56 CHAPTER 2. EQUICONSISTENCYZFC (for Ernst Zermelo and Abraham Fraenkel with Zermelos axiom of choice) or an extension of it, the notion ismeaningful. Thus, the method of forcing allows one to show that the theories ZFC, ZFC+CH and ZFC+CH are allequiconsistent.When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, orZF+AD, set theory with the axiom of determinacy), the notions described above are adapted accordingly. Thus, ZFis equiconsistent with ZFC, as shown by Gdel.The consistency strength of numerous combinatorial statements can be calibrated by large cardinals.For example,the negation of Kurepas hypothesis is equiconsistent with an inaccessible cardinal, the non-existence of special 2 -Aronszajn trees is equiconsistent with a Mahlo cardinal, and the non-existence of 2 -Aronszajn trees is equiconsistentwith a weakly compact cardinal.[1]2.3 See alsoLarge cardinal property2.4 References[1] Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, p. 225, ISBN 978-1-84890-050-9, Zbl 1262.03001Akihiro Kanamori (2003). The Higher Innite. Springer. ISBN 3-540-00384-3Chapter 3First-order logicFirst-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science.It is alsoknown as rst-order predicate calculus, the lower predicate calculus, quantication theory, and predicate logic.First-order logic uses quantied variables over (non-logical) objects. This distinguishes it from propositional logicwhich does not use quantiers.A theory about some topic is usually rst-order logic together with a specied domain of discourse over which thequantied variables range, nitely many functions which map fromthat domain into it, nitely many predicates denedon that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes theory isunderstood in a more formal sense, which is just a set of sentences in rst-order logic.The adjective rst-order distinguishes rst-order logic from higher-order logic in which there are predicates havingpredicates or functions as arguments, or in which one or both of predicate quantiers or function quantiers arepermitted.[1] In rst-order theories, predicates are often associated with sets. In interpreted higher-order theories,predicates may be interpreted as sets of sets.There are many deductive systems for rst-order logic that are sound (all provable statements are true in all models)and complete (all statements which are true in all models are provable). Although the logical consequence relation isonly semidecidable, much progress has been made in automated theoremproving in rst-order logic. First-order logicalso satises several metalogical theorems that make it amenable to analysis in proof theory, such as the LwenheimSkolem theorem and the compactness theorem.First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundationsof mathematics. Mathematical theories, such as number theory and set theory, have been formalized into rst-orderaxiom schemas such as Peano arithmetic and ZermeloFraenkel set theory (ZF) respectively.No rst-order theory, however, has the strength to describe uniquely a structure with an innite domain, such as thenatural numbers or the real line. A uniquely describing, i.e. categorical, axiom system for such a structure can beobtained in stronger logics such as second-order logic.For a history of rst-order logic and how it came to dominate formal logic, see Jos Ferreirs (2001).3.1 IntroductionWhile propositional logic deals with simple declarative propositions, rst-order logic additionally covers predicatesand quantication.A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Considerthe two sentences Socrates is a philosopher and Plato is a philosopher. In propositional logic, these sentencesare viewed as being unrelated and are denoted, for example, by p and q. However, the predicate is a philosopheroccurs in both sentences which have a common structure of "a is a philosopher. The variable a is instantiated asSocrates in the rst sentence and is instantiated as Plato in the second sentence. The use of predicates, such asis a philosopher in this example, distinguishes rst-order logic from propositional logic.Predicates can be compared. Consider, for example, the rst-order formula if a is a philosopher, then a is a scholar.This formula is a conditional statement with "a is a philosopher as hypothesis and "a is a scholar as conclusion.78 CHAPTER 3. FIRST-ORDER LOGICThe truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates is aphilosopher and is a scholar.Variables can be quantied over. The variable a in the previous formula can be quantied over, for instance, in therst-order sentence For every a, if a is a philosopher, then a is a scholar.The universal quantier for every inthis sentence expresses the idea that the claim if a is a philosopher, then a is a scholar holds for all choices of a.The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to thesentence There exists a such that a is a philosopher and a is not a scholar. The existential quantier there existsexpresses the idea that the claim "a is a philosopher and a is not a scholar holds for some choice of a.The predicates is a philosopher and is a scholar each take a single variable. Predicates can take several variables.In the rst-order sentence Socrates is the teacher of Plato, the predicate is the teacher of takes two variables.To interpret a rst-order formula, one species what each predicate means and the entities that can instantiate thepredicated variables. These entities form the domain of discourse or universe, which is usually required to be anonempty set. Given that the interpretation with the domain of discourse as consisting of all human beings and thepredicate is a philosopher understood as have written the Republic, the sentence There exists a such that a is aphilosopher is seen as being true, as witnessed by Plato.3.2 SyntaxThere are two key parts of rst-order logic. The syntax determines which collections of symbols are legal expressionsin rst-order logic, while the semantics determine the meanings behind these expressions.3.2.1 AlphabetUnlike natural languages, such as English, the language of rst-order logic is completely formal, so that it can bemechanically determined whether a given expression is legal. There are two key types of legal expressions: terms,which intuitively represent objects, and formulas, which intuitively express predicates that can be true or false. Theterms and formulas of rst-order logic are strings of symbols which together form the alphabet of the language. Aswith all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are oftenregarded simply as letters and punctuation symbols.It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, andnon-logical symbols, whose meaning varies by interpretation. For example, the logical symbol always representsand"; it is never interpreted as or. On the other hand, a non-logical predicate symbol such as Phil(x) could beinterpreted to mean "x is a philosopher, "x is a man named Philip, or any other unary predicate, depending on theinterpretation at hand.Logical symbolsThere are several logical symbols in the alphabet, which vary by author but usually include:The quantier symbols and The logical connectives: for conjunction, for disjunction, for implication, for biconditional, fornegation.Occasionally other logical connective symbols are included.Some authors use Cpq, instead of ,and Epq, instead of , especially in contexts where is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may replace ; a tilde (~), Np, or Fpq, may replace ; ||, or Apq may replace; and &, Kpq, or the middle dot, , may replace , especially if these symbols are not available for technicalreasons. (Note: the aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation.)Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context.An innite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, ... . Subscriptsare often used to distinguish variables: x0, x1, x2, ... .An equality symbol (sometimes, identity symbol) =; see the section on equality below.3.2. SYNTAX 9It should be noted that not all of these symbols are required only one of the quantiers, negation and conjunc-tion, variables, brackets and equality suce. There are numerous minor variations that may dene additional logicalsymbols:Sometimes the truth constants T, Vpq, or , for true and F, Opq, or , for false are included. Without anysuch logical operators of valence 0, these two constants can only be expressed using quantiers.Sometimes additional logical connectives are included, such as the Sheer stroke, Dpq (NAND), and exclusiveor, Jpq.Non-logical symbolsThe non-logical symbols represent predicates (relations), functions and constants on the domain of discourse. It usedto be standard practice to use a xed, innite set of non-logical symbols for all purposes. A more recent practice isto use dierent non-logical symbols according to the application one has in mind. Therefore it has become necessaryto name the set of all non-logical symbols used in a particular application. This choice is made via a signature.[2]The traditional approach is to have only one, innite, set of non-logical symbols (one signature) for all applications.Consequently, under the traditional approach there is only one language of rst-order logic.[3] This approach is stillcommon, especially in philosophically oriented books.1. For every integer n 0 there is a collection of n-ary, or n-place, predicate symbols. Because they representrelations between n elements, they are also called relation symbols. For each arity n we have an innite supplyof them:Pn0, Pn1, Pn2, Pn3, ...2. For every integer n 0 there are innitely many n-ary function symbols:fn0, fn1, fn2, fn3, ...In contemporary mathematical logic, the signature varies by application. Typical signatures in mathematics are {1,} or just {} for groups, or {0, 1, +, ,