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03: Mathematical logic and foundations![[Skip all navigation and headers]](http://www.math.niu.edu/~rusin/known-math/index/03-XX.html/../images/bl_ball.gif) ![[The <b>Mathematical</b> Atlas]](http://www.math.niu.edu/~rusin/known-math/index/03-XX.html/../images/title.gif) [Search][Subject Index][MathMap][Tour][Help!]![[MathMap Icon]](http://www.math.niu.edu/~rusin/known-math/index/03-XX.html/../images/mini.gif) ABOUT:[Introduction][History][Related areas][Subfields]POINTERS:[Texts][Software][Web links][Selected topics here]03: Mathematical logic and foundationsIntroductionMathematical Logic is the study of the processes used in mathematicaldeduction.The subject has origins in philosophy, and indeed it is only bynonmathematical argument that one can show the usual rules forinference and deduction (law of excluded middle; cut rule; etc.) are valid.It is also a legacy from philosophy that we can distinguish semanticreasoning ("what is true?") from syntactic reasoning ("what can be shown?").The first leads to Model Theory, the second, to Proof Theory.Students encounter elementary (sentential) logic early in theirmathematical training. This includes techniques using truth tables,symbolic logic with only "and", "or", and "not" in the language, andvarious equivalences among methods of proof (e.g. proof bycontradiction is a proof of the contrapositive). This materialincludes somewhat deeper results such as the existence of disjunctivenormal forms for statements. Also fairly straightforward iselementary first-order logic, which adds quantifiers ("for all" and"there exists") to the language. The corresponding normal form isprenex normal form. In second-order logic, the quantifiers areallowed to apply to relations and functions -- to subsets as well aselements of a set. (For example, the well-ordering axiom of theintegers is a second-order statement).In Model Theory, one asks for a description of the structures whichsatisfy some set of axioms (e.g. the axioms for a group or topologicalspace). In first-order languages, the results are striking. Forexample, the Löwenheim-Skolem-Tarski Theorem asserts that if there areany models, then there are models of every infinite cardinality,unless there is a (finite) upper bound on the cardinalities of themodels (assuming the language is countable). The compactnesstheorem asserts that a model exists for infinite sets of axioms, aslong as every finite set of them has a model. (This is true for axiomsin sentential logic, and true but deeper in first-order logic; itfails for second-order logic). For example, since all finite maps are4-colorable, the same is true of infinite maps.Proof theory is the study of certain kinds of symbol manipulation.Begin with a language -- a set of symbols and a set (the "syntax") ofstrings of those symbols; elements of this set are "formulas" in thelanguage. A collection T of these formulas is called a "theory" (ormore precisely the "axioms" for the theory); for example the theory ofgroups is expressed by a few axioms in a language which includesymbols "=", "*", "x1", "x2", ... as well as symbols "\arrow", "\wedge", and"\forall". We are interested in the "theorems" of T, which is thesmallest set S of formulas which includes T and is closed undercertain operations (the "rules of inference"), such as modus ponens(if both "A \arrow B" and "A" are in S, then "B" must also be in S). So how can we characterize the set of theorems for the theory? Thetheorems are defined in a purely procedural way, yet they should berelated to those statements which are (semantically) "true", that is,statements which are valid in every model of those axioms. With asuitable (and reasonably natural) set of rules of inference, the twonotions coincide for any theory in first-order logic: the SoundnessTheorem assures that what is provable is true, and the CompletenessTheorem assures that what is true is provable. It follows that theset of true first-order statements is effectively enumerable, anddecidable: one can deduce in a finite number of steps whether or notsuch a statement follows from the axioms. So, for example, one couldmake a countable list of all statements which are true for all groups.In some cases, even more is true: a theory is complete if all itsmodels are elementarily equivalent (the same sentences are true ineach). In that case, any statement in that language is decidable:it's either true (in all models) or it's false (in all models). Amongexamples of this are the theory of algebraically closed fields ofcharacteristic zero (Los-Vaught) and the theory of the real field R(Tarski). Any theory T is contained in a complete theory T' using thesame language (Lindenbaum); a key issue is just how "bad" T' might be.A famous example is the theory of arithmetic: Using symbols 0,+,*,^,<,and S (successor), let T be decidable set of axioms (e.g. a finiteset!) which are valid in the set of natural numbers. Gödel showed Tcannot be complete, that is, there are statements in the languagewhich are also valid in the set of natural numbers but which are nottheorems of T; equivalently, any complete theory T' containing T(e.g. the set of all statements which are valid in the naturalnumbers) cannot be decidable -- loosely, T' must be so complicatedthat we cannot even decide whether a given statement is in thatcollection of axioms!The topic of being able to recognize membership in a set is theprovince of Recursion Theory. Here one asks what is calculable in afinite number of steps. To the extent that one "calculates" a proof ofa theorem, this is the question of decidability in proof theory. Butthe term is more commonly used specifically to mean the calculation ofvalues of natural number functions; this can be shown to embed largerquestions such as the satisfiability of a predicate. Here one usesChurch's thesis, the convention that "calculable" means what isformally defined (inductively) as recursive. Recursive functions canbe described as those which may be computed in finite steps by asimple machine (a Turing machine); a recursive predicate is one whosetruth value (0 or 1, say) can be computed in a finite number of steps.(A predicate has the weaker property of being "recursively enumerable"if an algorithm exists which will in a finite number of steps returna value of 1 if the predicate hold; since the amount of steps isnot bounded in advance, lack of a conclusion from the algorithm at anytime may or may not mean the predicate holds.) Hilbert's tenth problemwas to decide if the solvability of Diophantine equations wasdecidable; Matijasevic (1971) showed it is not, that is, one cannotdescribe a Turing machine which computes whether each Diophantineequation is solvable or not.Algebraic Logic studies logical systems via associated algebraic structures.In particular, this is a convenient setting for the study of many-valuedlogics (more truth values than just "true" or "false"). Just as ordinarylogic may be studies with Boolean algebras one may formalize the calculiwith many-valued logics using other algebraic systems. These include thealgebras of Post, Lukasiewicz, Heyting, etc. This topic leads to a studyof abstract algebra systems (lattices, filters, and so on).In the preceding discussion it has been tacitly assumed that the ideaof a "set" is unambiguous. Indeed this is not true, as was noticed acentury ago. Much of mathematical logic was developed in response to thequestions surrounding the axiomatization of set theory. From thisdeveloped the constructions and investigations of very large infinite sets.Descriptive set theory considers various classifications of sets. Cardinalarithmetic considers the "size" of various sets; ordinal arithmeticrefines this to the "ordering" of sets. Fuzzy set theory replacesthe yes/no statement of set membership with a qualitative predicate.HistoryElementary logic has been studied since ancient times, in part throughthe analysis of paradoxes, and as a part of rhetoric. Late in the nineteenthcentury Frege and others attempted to formalize mathematics and the lawsof deduction. At the turn of the 20th century a debate arose regarding thelegitimacy of non-constructive proofs; Hilbert suggested a program demonstrating the possibility of securing mathematics onto a formalfoundation. (Whitehead and Russell (1910-1913) actually tried it.) Thenecessary groundwork in logic, outlined above, was laid during the 1920s and 1930s, which is when some of the most paradoxical results were obtained.The implicit dependence on set theory and the inability to determine adecidable set of first-order axioms for set theory have caused considerableconsternation among mathematicians, particularly those confronted withdifficulties associated with the axiom of choice. Notable among postwardevelopments is Robinson's application of model theory to developNonstandard analysis and use it as a framework for ordinary calculus.Many undecidability results appeared in the 1960s and 1970s withapplications in traditional branches of algebra. With the development ofcomputer science during later decades, many topics in recursion theory andproof theory were developed from the perspective of the theory of algorithms.See also "Perspectives on the history of mathematical logic",edited by Thomas Drucker. Birkhäuser Boston, Inc., Boston, MA,1991. 195 pp. ISBN 0-8176-3444-4 MR94b:03007Applications and related fieldsThe construction of nonstandard models for certain sets of axioms leadsto "nonstandard analysis" of those axioms. In particular, a nonstandardmodel of the real line (including "infinitesimals") leads to a branch ofReal Analysis known as "Nonstandard Analysis"(Thus yielding nonstandard versions of Measure Theory, Functional Analysis,and so on.) Likewise, nonstandard models of arithmetic open a branch ofNumber TheoryThe structures introduced in Algebraic Logic are also studied (qua algebraicobjects) in General Algebraic Structures.Recursion theory is closely related to questions of computability andcomplexity in Computer Science.The related issues of the existence of effective procedures show up inalgebraic contexts, e.g. the negative solution of the Word Problem inGroup Theory, or the definability of Diophantinesets in Number Theory.Through multivalued logics one obtains a language of uncertaintyapplicable to Quantum TheoryWe remark that there is a relationship between models and theoriesakin to the relationship between varieties and ideals in algebraic geometry;though this is too vague a parallel to be exploited.![[Schematic of subareas and <b>related</b> areas]](http://www.math.niu.edu/~rusin/known-math/index/03-XX.html/../images/03b.gif) Subfields03A05: Philosophical and critical03B: General logic03C: Model theory03D: Computability and Recursion theory 03E: Set theory03F: Proof theory and constructive mathematics03G: Algebraic logic03H: Nonstandard models, see also 03C62Note that material in Mathematical Logic was classified as section 02before 1980. A separate section 04: Set Theory was used from then until 1999but (reflecting ongoing usage) has been returned to this section.Browse all (old) classifications for this area at the AMS.Textbooks, reference works, and tutorialsSome survey articles:Crossley, J. N.; Ash, C. J.; Brickhill, C. J.; Stillwell, J. C.;Williams, N. H.: "What is mathematical logic?"Oxford Paperbacks University Series, No. 60.Oxford University Press, London-New York, 1972. 82 pp. MR54#2411Mangione, Corrado: "What is contemporary logic talking about?" Italian studies in the philosophy of science, pp. 89--111, Boston Stud. Philos. Sci., 47,Reidel, Dordrecht-Boston, Mass., 1981. MR82c:03006Smorynski, C.: "What's new in logic?" Math. Intelligencer 7 (1985), no. 3, 53--54. MR86f:03078Well-known textbooks include "A mathematical introduction to logic", Herbert B. Enderton, Academic Press, New York-London, 1972 -- possibly an undergraduate text"Mathematical logic", Joseph R. Shoenfield, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967More specialized is "Many-valued logics", Leonard Bolc and Piotr Borowik, Springer-Verlag, Berlin, 1992. ISBN 3-540-55926-4Online textbook in Mathematical Logic [Stefan Bilaniuk]A somewhat dated but comprehensive overview is the five-volume "Handbook of mathematical logic", edited by Jon Barwise: Studies in Logic and the Foundations of Mathematics, Vol. 90. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. ISBN 0-7204-2285-XA comprehensive bibliography is the "Omega-bibliography ofmathematical logic", Edited by Gert H. Müller, Wolfgang Lenski etal, Springer-Verlag, Berlin-New York, 1987, in six volumes (Classicallogic, Nonclassical logics, Model theory, Recursion theory, Settheory, Proof theory and constructive mathematics.)Perhaps the best-known journal in this area is the Journal of Symbolic Logic.There is a USENET newsgroup sci.logicFriedman and Simpson have established a Foundations of Mathematics mailing listSoftware and tablesOtter: An Automated Deduction SystemCinderella, featured in the paper: "Automatic theorem proving of Geometric Theorems", H. Crapo and J. Richter-Gerbert.Logic Software from CSLI including Tarski's WorldPackages for Mathematica, versions2.2and 3.0.Other web sites with this focusHere are the AMS and Goettingen resource pages for area 03.There are some useful links at the end of the IU-Logic homepage.There are some elementary exercises in first-order and predicate logicavailable for students on a separate math teaching page.Selected topics at this siteWhat are Formalism and Constructivism? [Robert Israel]Constructive mathematics and the role of the Law of Excluded MiddleCollection of logic paradoxes (liar, etc.)Quaternio terminorum and other types of logical fallaciesPointers to web sites for classical logical fallaciesEvery triangle is equilateral :-)Testing tautologies in propositional logic with MacsymaTesting for tautologies "efficiently". (Not really possible in general).Minimal lengths of proofs (Friedman, Buss, et al)What does Gödel's Incompleteness Theorem say? [Theodore Hwa]Gödel's theorem in a new light (lots of impressive big-number arithmetic).Gödel's theorem masquerading as popular literature (Smullyan)Really Gödel's theorem is unremarkable :-)Gödel numbers as a form of information complexity (Chaitan)Post's Problem: are there problems too hard to be solved by a Turing machine but not as hard as the Halting Problem?The collatz (3x+1 / Hailstone) problem is "just" a Turing machine halting problem and so may be insoluble. [For more information about the Collatz problem see this summary.]The Halting Problem for Turing machinesPointer to Busy Beaver problem.Busy Beaver referencesBig numbers used in proofs (Friedman -- strings in alphabets)What are primitive recursive functions?Independence vs undecidabilityUndecidable Diophantine Equations (Jones)Number theory is provably hard: either EEAE is undecidable or finding all integer points on an algebraic curve is nonrecursive.Church-Turing thesis on computabilityParis-Harrington variation on Ramsey's theoremEhrenfeucht-Fraisse games to establish equivalence of theoriesComparison of definability through Lambda Calculus versus axiom systemsTextbooks, overview of the Lambda CalculusA description of the Hilbert Hotel (one room per natural number)Löb's Theorem: "This statement is provable" is provableCould (say) Fermat's Last Theorem be proved from the axioms of Peano Arithmetic?Tarski's Elimination of Quantifiers can be used to answer questions like, "Is this polynomial always positive?"Does a set of real polynomials have a real solution? Tarski's Elimination of Quantifiers.Pointers to sites on Non-Standard AnalysisModal logic: what one knows vs. believes (treated formally)References on non-standard logics.Complexity of the 3-valued logics of Kleene, PostTennenbaum's theorem (non-standard arithmetic is nonrecursive)Arithmetic with multiplication and successor define additionOrdinal analysis, infinite proofs, Gentzen proof of consistency of elementary arithmeticWhat are Lindenbaum Algebras (Boolean algebras in Model Theory)Can planarity be expressed as a sentence in first-order logic? (no)Geometric theorem-proversWhere can I find a predicate calculus editor/theorem checker?References on applications of the AUTOMATH system (automatic proof checker)Pointers for connections to Logic journals.Recollections of the logic "game" WFF'N'PROOFWhat Model Theory is not!You can reach this page through http://www.math-atlas.org/welcome.htmlLast modified 2000/01/24 by Dave Rusin. Mail: rusin@math.niu.edu |
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