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Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a
724:. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the
637:(or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger
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About the Author
Compiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff who represents the cutting-edge of skill, knowledge, and innovation in science publishing.
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approach is the most common. A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language. The reductive or
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technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language."
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or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a
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Hunter, Geoffrey, Metalogic: An
Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
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are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
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is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
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as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a
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1121:) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language.
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The TREE-META Compiler-Compiler System: A Meta
Compiler System for the Univac 1108 and General Electric 645
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1380:: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48â64.
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The point of view that generating formal proofs is all there is to mathematics is often called
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represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.
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The two main types of deductive systems are proof systems and formal semantics.
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that are expressed in the system. Usually the quality we are concerned with is
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of the language that gets involved with the deductive nature of the system.
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An example of a deductive system would be the rules of inference and
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is a language that is defined by a formal system. Like languages in
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proposed to use formal systems as the foundation of knowledge in
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of the lines that precede it. There should be no element of any
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855:(c. 325â250 BCE) . In more recent times, contributors include
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1006:"Formal system | Logic, Symbols & Axioms | Britannica"
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and gives the symbols their usual meaning. There are also
645:) consistent with the usage in modern mathematics such as
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of the language. The aim is to ensure that each line of a
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This section needs better organization and more citations.
736:, which, in order to avoid confusion, are usually called
969: â Replacing subterm in a formula with another term
462:. The two main categories of formal grammar are that of
1249:, Princeton University Press (April 1, 1961) 156 pages
1247:
Theory of Formal
Systems: Annals of Mathematics Studies
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Pages displaying short descriptions of redirect targets
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Pages displaying short descriptions of redirect targets
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for deciding whether a given WFF is a theorem or not.
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of a formal language is considered via the notion of a
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Hilbert's
Program, Stanford Encyclopedia of Philosophy
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80:
The term formalism is sometimes a rough synonym for
963: â Translation of a text into a logical system
27:Mathematical model for deduction or proof systems
4072:
1040:. Metaphysics Research Lab, Stanford University.
189:, both of which are part of the formal language.
1214:Kaye, Richard (1991). "1. The Standard Model".
610:In order to sustain its deductive integrity, a
435:, formal languages generally have two aspects:
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987: â Set of sentences in a formal language
847:Early logic systems includes Indian logic of
390:
1282:Gödel, Escher, Bach: An Eternal Golden Braid
1182:"Definition:Deductive Apparatus - ProofWiki"
817:. The standard model of arithmetic sets the
1359:Formal Systems and Machines: An Isomorphism
948: â Mathematical program specifications
685:(or WFF for short) that might either be an
614:must be definable without reference to any
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3164:
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1392:
1077:"Syntax & Semantics of Formal Systems"
397:
383:
732:just defined should not be confused with
527:Learn how and when to remove this message
84:, but it also refers to a given style of
1329:Syntax & Semantics of Formal Systems
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595:as opposed to falsehood. However, other
103:
1218:. Oxford: Clarendon Press. p. 10.
14:
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1413:
1095:"Definition:Formal System - ProofWiki"
1075:Rapaport, William J. (25 March 2010).
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839:Formalism (philosophy of mathematics)
769:is a deductive system (most commonly
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3715:Systems theory in political science
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1323:Formal systems, logic and semantics
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136:A formal system has the following:
24:
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893:, that was eventually tempered by
891:foundational crisis of mathematics
813:An example of a logical system is
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3891:List of mathematical logic topics
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827:non-standard models of arithmetic
264:Semantics (programming languages)
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781:, a logical system may be given
734:theorems about the formal system
583:Such deductive systems preserve
482:
4106:1st-millennium BC introductions
1207:
895:Gödel's incompleteness theorems
785:which describe whether a given
681:Formal proofs are sequences of
666:
4060:List of category theory topics
3705:Systems theory in anthropology
1347:Definition:Deductive Apparatus
1198:
1134:
1111:
1087:
1068:
1044:
1032:Zach, Richard (31 July 2003).
998:
889:as a proposed solution to the
871:was developed in 19th century
193:A formal system is said to be
13:
1:
3710:Systems theory in archaeology
3086:History of mathematical logic
1351:Encyclopedia of Mathematics,
991:
112:that may be constructed from
3011:Primitive recursive function
1126:(6th ed.). McGraw-Hill.
120:may be broadly divided into
7:
4055:Glossary of category theory
3929:ZermeloâFraenkel set theory
3881:Mathematical constructivism
985:Theory (mathematical logic)
904:
773:) together with additional
502:. The specific problem is:
349:Programming language theory
344:Natural language processing
99:
10:
4122:
4050:Mathematical structuralism
3987:Intuitionistic type theory
3823:Foundations of Mathematics
2075:SchröderâBernstein theorem
1802:Monadic predicate calculus
1461:Foundations of mathematics
1266:Reprinted by Dover, 2002.
1216:Models of Peano arithmetic
836:
832:
750:
670:
607:may be preserved instead.
538:
408:
173:, deductive apparatus, or
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3954:List of set theory topics
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3409:
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3294:
3285:
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3206:Coupled humanâenvironment
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3121:
3108:Philosophy of mathematics
3057:Automated theorem proving
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1823:
1814:
1741:
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1504:
1421:
1314:EncyclopĂŠdia Britannica,
957: â Branch of science
654:axioms regarding equality
369:Automated theorem proving
354:Computational linguistics
1378:What is a Formal System?
1341:Definition:Formal System
978: â Concept in logic
793:of the logical system.
4081:4th century BC in India
3934:Constructive set theory
3539:Charles A. S. Hall
2758:Self-verifying theories
2579:Tarski's axiomatization
1530:Tarski's undefinability
1525:incompleteness theorems
616:intended interpretation
147:, which are strings of
108:This diagram shows the
4035:Higher category theory
3939:Descriptive set theory
3844:Mathematical induction
3459:Ludwig von Bertalanffy
3132:Mathematics portal
2743:Proof of impossibility
2391:propositional variable
1701:Propositional calculus
941:List of formal systems
919:Systems science portal
843:Formal logical systems
757:Interpretation (logic)
329:Propositional calculus
133:
3997:Univalent foundations
3982:Dependent type theory
3972:Axiom of reducibility
3736:Principia Cybernetica
3449:Anthony Stafford Beer
3319:Sociotechnical system
3001:Kolmogorov complexity
2954:Computably enumerable
2854:Model complete theory
2646:Principia Mathematica
1706:Propositional formula
1535:BanachâTarski paradox
1327:William J. Rapaport,
1117:Reductive grammar: (
1081:University of Buffalo
976:Substitution instance
807:Semantically complete
796:A logical system is:
339:Mathematical notation
107:
3992:Homotopy type theory
3919:Axiomatic set theory
3599:Mihajlo D. Mesarovic
3574:Edward Norton Lorenz
3529:Jay Wright Forrester
3334:World-systems theory
3309:Earth system science
2949:ChurchâTuring thesis
2936:Computability theory
2145:continuum hypothesis
1663:Square of opposition
1521:Gödel's completeness
1311:at Wikimedia Commons
823:nonnegative integers
683:well-formed formulas
573:that can be used to
509:improve this section
498:to meet Knowledge's
145:well-formed formulas
143:, which is a set of
126:well-formed formulas
3474:Kenneth E. Boulding
3103:Mathematical object
2994:P versus NP problem
2959:Computable function
2753:Reverse mathematics
2679:Logical consequence
2556:primitive recursive
2551:elementary function
2324:Free/bound variable
2177:TarskiâGrothendieck
1696:Logical connectives
1626:Logical equivalence
1476:Logical consequence
1321:Daniel Richardson,
1260:Stephen Cole Kleene
1243:Raymond M. Smullyan
1034:"Hilbert's Program"
819:domain of discourse
635:logical consequence
624:logical consequence
612:deductive apparatus
559:deductive apparatus
549:Deductive reasoning
545:Logical consequence
464:generative grammars
364:Formal verification
279:Well-formed formula
3977:Simple type theory
3924:Zermelo set theory
3871:Mathematical proof
3831:Mathematical logic
3604:James Grier Miller
3559:Faina M. Kirillova
3519:Heinz von Foerster
3509:Edsger W. Dijkstra
3469:Alexander Bogdanov
3454:Richard E. Bellman
3429:William Ross Ashby
2901:Transfer principle
2864:Semantics of logic
2849:Categorical theory
2825:Non-standard model
2339:Logical connective
1466:Information theory
1415:Mathematical logic
1364:2011-05-24 at the
1277:Douglas Hofstadter
1264:Mathematical Logic
1146:analytical grammar
1010:www.britannica.com
869:Mathematical logic
861:Augustus De Morgan
775:non-logical axioms
753:Semantics of logic
699:decision procedure
643:Euclidean geometry
571:rules of inference
561:, consists of the
289:Regular expression
203:semidecidable sets
179:rules of inference
134:
118:strings of symbols
116:. The symbols and
110:syntactic entities
52:rules of inference
36:abstract structure
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3949:Russell's paradox
3864:Natural deduction
3788:
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3669:Manuela M. Veloso
3584:Humberto Maturana
3524:Stephanie Forrest
3494:C. West Churchman
3419:Russell L. Ackoff
3405:
3404:
3277:Positive feedback
3272:Negative feedback
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3071:Abstract category
2874:Theories of truth
2684:Rule of inference
2674:Natural deduction
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1905:Cartesian product
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1716:Many-valued logic
1691:Boolean functions
1574:Russell's paradox
1549:diagonal argument
1446:First-order logic
1318:definition, 2007.
1307:Media related to
1290:978-0-465-02656-2
961:Logic translation
933:Philosophy portal
887:Hilbertâs program
771:first order logic
747:of logical system
658:first order logic
587:qualities in the
537:
536:
529:
500:quality standards
491:This section may
468:analytic grammars
454:Usually only the
407:
406:
299:Ground expression
259:Semantics (logic)
132:and non-theorems.
16:(Redirected from
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4101:Formal languages
4030:Category of sets
4002:Girard's paradox
3914:Naive set theory
3854:Axiomatic system
3821:Major topics in
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3396:Systems thinking
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3081:History of logic
3076:Category of sets
2969:Decision problem
2748:Ordinal analysis
2689:Sequent calculus
2587:Boolean algebras
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2472:logical/constant
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2135:ZermeloâFraenkel
1886:Set operations:
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951:
935:
930:
929:
928:
921:
916:
915:
914:
885:movement called
815:Peano arithmetic
745:Formal semantics
557:, also called a
555:deductive system
532:
525:
521:
518:
512:
486:
485:
478:
474:Deductive system
399:
392:
385:
228:Formal languages
213:
212:
205:, respectively.
171:Deductive system
161:production rules
114:formal languages
94:braâket notation
44:axiomatic system
21:
4121:
4120:
4116:
4115:
4114:
4112:
4111:
4110:
4071:
4070:
4069:
4064:
4012:Category theory
4006:
3958:
3895:
3825:
3819:
3789:
3784:
3740:
3719:
3693:
3634:Anatol Rapoport
3619:Talcott Parsons
3594:Donella Meadows
3569:Allenna Leonard
3489:Mary Cartwright
3484:Kathleen Carley
3444:Gregory Bateson
3439:BĂ©la H. BĂĄnĂĄthy
3401:
3338:
3287:
3281:
3267:Limiting factor
3262:Leverage points
3245:
3183:
3175:
3173:Systems science
3170:
3140:
3135:
3124:
3117:
3062:Category theory
3052:Algebraic logic
3035:
3006:Lambda calculus
2944:Church encoding
2930:
2906:Truth predicate
2762:
2728:Complete theory
2651:
2520:
2516:
2512:
2507:
2499:
2219: and
2215:
2210:
2196:
2172:New Foundations
2140:axiom of choice
2123:
2085:Gödel numbering
2025: and
2017:
1921:
1806:
1756:
1737:
1686:Boolean algebra
1672:
1636:Equiconsistency
1601:Classical logic
1578:
1559:Halting problem
1547: and
1523: and
1511: and
1510:
1505:Theorems (
1500:
1417:
1412:
1366:Wayback Machine
1299:
1239:
1237:Further reading
1234:
1233:
1226:
1212:
1208:
1203:
1199:
1190:
1188:
1180:
1179:
1172:
1162:
1160:
1157:
1149:
1139:
1135:
1122:
1116:
1112:
1103:
1101:
1093:
1092:
1088:
1073:
1069:
1060:
1058:
1052:"formal system"
1050:
1049:
1045:
1030:
1023:
1014:
1012:
1004:
1003:
999:
994:
979:
970:
949:
931:
926:
924:
917:
912:
910:
907:
845:
837:Main articles:
835:
783:interpretations
777:. According to
763:
751:Main articles:
749:
726:object language
716:metamathematics
679:
671:Main articles:
669:
641:or field (e.g.
580:of the system.
551:
539:Main articles:
533:
522:
516:
513:
506:
487:
483:
476:
429:formal language
425:
411:Formal language
409:Main articles:
403:
374:
373:
359:Syntax analysis
334:Predicate logic
319:
318:
309:
308:
284:Automata theory
239:
238:
211:
209:Formal language
165:formation rules
159:(consisting of
141:Formal language
102:
88:, for example,
64:inference rules
28:
23:
22:
15:
12:
11:
5:
4119:
4109:
4108:
4103:
4098:
4096:Formal systems
4093:
4091:Syntax (logic)
4088:
4083:
4066:
4065:
4063:
4062:
4057:
4052:
4047:
4045:â-topos theory
4042:
4037:
4032:
4027:
4022:
4016:
4014:
4008:
4007:
4005:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3968:
3966:
3960:
3959:
3957:
3956:
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3916:
3911:
3905:
3903:
3897:
3896:
3894:
3893:
3888:
3883:
3878:
3873:
3868:
3867:
3866:
3861:
3859:Hilbert system
3856:
3846:
3841:
3835:
3833:
3827:
3826:
3818:
3817:
3810:
3803:
3795:
3786:
3785:
3783:
3782:
3770:
3758:
3745:
3742:
3741:
3739:
3738:
3733:
3727:
3725:
3721:
3720:
3718:
3717:
3712:
3707:
3701:
3699:
3695:
3694:
3692:
3691:
3689:Anthony Wilden
3686:
3684:Jennifer Wilby
3681:
3679:Norbert Wiener
3676:
3671:
3666:
3661:
3656:
3651:
3649:Claude Shannon
3646:
3641:
3636:
3631:
3626:
3624:Ilya Prigogine
3621:
3616:
3614:Howard T. Odum
3611:
3609:Radhika Nagpal
3606:
3601:
3596:
3591:
3586:
3581:
3579:Niklas Luhmann
3576:
3571:
3566:
3561:
3556:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3516:
3511:
3506:
3504:George Dantzig
3501:
3499:Manfred Clynes
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3464:Margaret Boden
3461:
3456:
3451:
3446:
3441:
3436:
3431:
3426:
3424:Victor Aladjev
3421:
3415:
3413:
3407:
3406:
3403:
3402:
3400:
3399:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3343:
3340:
3339:
3337:
3336:
3331:
3326:
3321:
3316:
3314:Living systems
3311:
3306:
3301:
3299:Control theory
3295:
3289:
3283:
3282:
3280:
3279:
3274:
3269:
3264:
3259:
3253:
3251:
3247:
3246:
3244:
3243:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3203:
3198:
3193:
3187:
3185:
3177:
3176:
3169:
3168:
3161:
3154:
3146:
3137:
3136:
3122:
3119:
3118:
3116:
3115:
3110:
3105:
3100:
3095:
3094:
3093:
3083:
3078:
3073:
3064:
3059:
3054:
3049:
3047:Abstract logic
3043:
3041:
3037:
3036:
3034:
3033:
3028:
3026:Turing machine
3023:
3018:
3013:
3008:
3003:
2998:
2997:
2996:
2991:
2986:
2981:
2976:
2966:
2964:Computable set
2961:
2956:
2951:
2946:
2940:
2938:
2932:
2931:
2929:
2928:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2892:
2891:
2886:
2881:
2871:
2866:
2861:
2859:Satisfiability
2856:
2851:
2846:
2845:
2844:
2834:
2833:
2832:
2822:
2821:
2820:
2815:
2810:
2805:
2800:
2790:
2789:
2788:
2783:
2776:Interpretation
2772:
2770:
2764:
2763:
2761:
2760:
2755:
2750:
2745:
2740:
2730:
2725:
2724:
2723:
2722:
2721:
2711:
2706:
2696:
2691:
2686:
2681:
2676:
2671:
2665:
2663:
2657:
2656:
2653:
2652:
2650:
2649:
2641:
2640:
2639:
2638:
2633:
2632:
2631:
2626:
2621:
2601:
2600:
2599:
2597:minimal axioms
2594:
2583:
2582:
2581:
2570:
2569:
2568:
2563:
2558:
2553:
2548:
2543:
2530:
2528:
2509:
2508:
2506:
2505:
2504:
2503:
2491:
2486:
2485:
2484:
2479:
2474:
2469:
2459:
2454:
2449:
2444:
2443:
2442:
2437:
2427:
2426:
2425:
2420:
2415:
2410:
2400:
2395:
2394:
2393:
2388:
2383:
2373:
2372:
2371:
2366:
2361:
2356:
2351:
2346:
2336:
2331:
2326:
2321:
2320:
2319:
2314:
2309:
2304:
2294:
2289:
2287:Formation rule
2284:
2279:
2278:
2277:
2272:
2262:
2261:
2260:
2250:
2245:
2240:
2235:
2229:
2223:
2206:Formal systems
2202:
2201:
2198:
2197:
2195:
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2148:
2147:
2142:
2131:
2129:
2125:
2124:
2122:
2121:
2120:
2119:
2109:
2104:
2103:
2102:
2095:Large cardinal
2092:
2087:
2082:
2077:
2072:
2058:
2057:
2056:
2051:
2046:
2031:
2029:
2019:
2018:
2016:
2015:
2014:
2013:
2008:
2003:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1932:
1930:
1923:
1922:
1920:
1919:
1918:
1917:
1912:
1907:
1902:
1897:
1892:
1884:
1883:
1882:
1877:
1867:
1862:
1860:Extensionality
1857:
1855:Ordinal number
1852:
1842:
1837:
1836:
1835:
1824:
1818:
1812:
1811:
1808:
1807:
1805:
1804:
1799:
1794:
1789:
1784:
1779:
1774:
1773:
1772:
1762:
1761:
1760:
1747:
1745:
1739:
1738:
1736:
1735:
1734:
1733:
1728:
1723:
1713:
1708:
1703:
1698:
1693:
1688:
1682:
1680:
1674:
1673:
1671:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1639:
1638:
1628:
1623:
1618:
1613:
1608:
1603:
1597:
1595:
1586:
1580:
1579:
1577:
1576:
1571:
1566:
1561:
1556:
1551:
1539:Cantor's
1537:
1532:
1527:
1517:
1515:
1502:
1501:
1499:
1498:
1493:
1488:
1483:
1478:
1473:
1468:
1463:
1458:
1453:
1448:
1443:
1438:
1437:
1436:
1425:
1423:
1419:
1418:
1411:
1410:
1403:
1396:
1388:
1382:
1381:
1375:
1373:Formal Systems
1369:
1355:
1349:
1343:
1337:
1331:
1325:
1319:
1312:
1309:Formal systems
1298:
1297:External links
1295:
1294:
1293:
1274:
1257:
1238:
1235:
1232:
1231:
1224:
1206:
1197:
1170:
1133:
1110:
1086:
1067:
1056:planetmath.org
1043:
1021:
996:
995:
993:
990:
989:
988:
982:
973:
964:
958:
955:Formal science
952:
943:
937:
936:
922:
906:
903:
834:
831:
811:
810:
804:
767:logical system
748:
742:
668:
665:
628:interpretation
567:axiom schemata
535:
534:
490:
488:
481:
475:
472:
460:formal grammar
452:
451:
444:
419:Syntax (logic)
415:Formal grammar
405:
404:
402:
401:
394:
387:
379:
376:
375:
372:
371:
366:
361:
356:
351:
346:
341:
336:
331:
326:
324:Formal methods
320:
316:
315:
314:
311:
310:
307:
306:
304:Atomic formula
301:
296:
291:
286:
281:
276:
274:Formation rule
271:
269:Formal grammar
266:
261:
256:
251:
246:
240:
236:
235:
234:
231:
230:
224:
223:
210:
207:
199:decidable sets
191:
190:
168:
157:formal grammar
155:, formed by a
101:
98:
26:
18:Logical system
9:
6:
4:
3:
2:
4118:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4078:
4076:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4017:
4015:
4013:
4009:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3969:
3967:
3965:
3961:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3906:
3904:
3902:
3898:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3865:
3862:
3860:
3857:
3855:
3852:
3851:
3850:
3849:Formal system
3847:
3845:
3842:
3840:
3837:
3836:
3834:
3832:
3828:
3824:
3816:
3811:
3809:
3804:
3802:
3797:
3796:
3793:
3781:
3780:
3771:
3769:
3768:
3763:
3759:
3757:
3756:
3747:
3746:
3743:
3737:
3734:
3732:
3729:
3728:
3726:
3724:Organizations
3722:
3716:
3713:
3711:
3708:
3706:
3703:
3702:
3700:
3696:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3674:Kevin Warwick
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3589:Margaret Mead
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3549:Lydia Kavraki
3547:
3545:
3542:
3540:
3537:
3535:
3534:Barbara Grosz
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3434:Ruzena Bajcsy
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3416:
3414:
3412:
3408:
3397:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3344:
3341:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3296:
3293:
3290:
3284:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3257:Doubling time
3255:
3254:
3252:
3248:
3242:
3239:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3188:
3186:
3182:
3178:
3174:
3167:
3162:
3160:
3155:
3153:
3148:
3147:
3144:
3134:
3133:
3128:
3120:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3092:
3089:
3088:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3044:
3042:
3038:
3032:
3029:
3027:
3024:
3022:
3021:Recursive set
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2971:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2941:
2939:
2937:
2933:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2890:
2887:
2885:
2882:
2880:
2877:
2876:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2843:
2840:
2839:
2838:
2835:
2831:
2830:of arithmetic
2828:
2827:
2826:
2823:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2795:
2794:
2791:
2787:
2784:
2782:
2779:
2778:
2777:
2774:
2773:
2771:
2769:
2765:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2738:
2737:from ZFC
2734:
2731:
2729:
2726:
2720:
2717:
2716:
2715:
2712:
2710:
2707:
2705:
2702:
2701:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2666:
2664:
2662:
2658:
2648:
2647:
2643:
2642:
2637:
2636:non-Euclidean
2634:
2630:
2627:
2625:
2622:
2620:
2619:
2615:
2614:
2612:
2609:
2608:
2606:
2602:
2598:
2595:
2593:
2590:
2589:
2588:
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2001:constructible
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1678:Propositional
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1606:Logical truth
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1353:Formal system
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1335:Formal System
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1225:9780198532132
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1186:proofwiki.org
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946:Formal method
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892:
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884:
881:instigated a
880:
879:David Hilbert
876:
874:
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866:
865:Gottlob Frege
862:
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712:David Hilbert
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82:formal system
78:
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72:
71:David Hilbert
67:
65:
61:
57:
53:
49:
45:
41:
40:formalization
37:
33:
32:formal system
19:
4025:Topos theory
3876:Model theory
3848:
3839:Peano axioms
3777:
3765:
3753:
3698:Applications
3654:Katia Sycara
3554:James J. Kay
3544:Mike Jackson
3479:Murray Bowen
3377:Pharmacology
3372:Neuroscience
3123:
2921:Ultraproduct
2768:Model theory
2733:Independence
2698:
2669:Formal proof
2661:Proof theory
2644:
2617:
2574:real numbers
2546:second-order
2457:Substitution
2334:Metalanguage
2275:conservative
2248:Axiom schema
2205:
2192:Constructive
2162:MorseâKelley
2128:Set theories
2107:Aleph number
2100:inaccessible
2006:Grothendieck
1890:intersection
1777:Higher-order
1765:Second-order
1711:Truth tables
1668:Venn diagram
1451:Formal proof
1333:PlanetMath,
1292:. 777 pages.
1280:
1263:
1246:
1215:
1209:
1200:
1189:. Retrieved
1185:
1161:. Retrieved
1152:
1136:
1118:
1113:
1102:. Retrieved
1098:
1089:
1080:
1070:
1059:. Retrieved
1055:
1046:
1037:
1013:. Retrieved
1009:
1000:
877:
857:George Boole
853:Gongsun Long
846:
812:
795:
779:model theory
766:
764:
761:Model theory
738:metatheorems
733:
729:
725:
721:metalanguage
719:
705:
703:
695:
680:
677:Formal proof
673:Proof system
667:Proof system
662:
651:
647:model theory
632:
622:is merely a
611:
609:
582:
558:
554:
552:
523:
517:October 2023
514:
507:Please help
503:
492:
453:
426:
423:Logical form
317:Applications
243:
237:Key concepts
218:
192:
177:, which has
175:proof system
135:
81:
79:
68:
62:by a set of
31:
29:
3964:Type theory
3944:Determinacy
3886:Modal logic
3644:Peter Senge
3639:John Seddon
3629:Qian Xuesen
3564:George Klir
3367:Engineering
3304:Cybernetics
3286:Theoretical
3236:Recommender
3226:Multi-agent
3221:Information
3031:Type theory
2979:undecidable
2911:Truth value
2798:equivalence
2477:non-logical
2090:Enumeration
2080:Isomorphism
2027:cardinality
2011:Von Neumann
1976:Ultrafilter
1941:Uncountable
1875:equivalence
1792:Quantifiers
1782:Fixed-point
1751:First-order
1631:Consistency
1616:Proposition
1593:Traditional
1564:Lindström's
1554:Compactness
1496:Type theory
1441:Cardinality
511:if you can.
433:linguistics
185:and infers
75:mathematics
4075:Categories
4040:â-groupoid
3901:Set theory
3659:Eric Trist
3514:Fred Emery
3411:Scientists
3387:Psychology
3382:Philosophy
3211:Ecological
3196:Biological
2842:elementary
2535:arithmetic
2403:Quantifier
2381:functional
2253:Expression
1971:Transitive
1915:identities
1900:complement
1833:hereditary
1816:Set theory
1371:Ray Taol,
1345:PrâfWiki,
1339:PrâfWiki,
1191:2023-10-10
1104:2023-10-16
1061:2023-10-10
1015:2023-10-10
992:References
821:to be the
620:derivation
599:, such as
597:modalities
294:Production
181:that take
90:Paul Dirac
4086:Metalogic
3324:Systemics
3113:Supertask
3016:Recursion
2974:decidable
2808:saturated
2786:of models
2709:deductive
2704:axiomatic
2624:Hilbert's
2611:Euclidean
2592:canonical
2515:axiomatic
2447:Signature
2376:Predicate
2265:Extension
2187:Ackermann
2112:Operation
1991:Universal
1981:Recursive
1956:Singleton
1951:Inhabited
1936:Countable
1926:Types of
1910:power set
1880:partition
1797:Predicate
1743:Predicate
1658:Syllogism
1648:Soundness
1621:Inference
1611:Tautology
1513:paradoxes
1163:5 January
883:formalist
787:structure
707:formalism
585:deductive
541:Inference
448:semantics
195:recursive
69:In 1921,
46:used for
4020:Category
3755:Category
3357:Dynamics
3347:Analysis
3250:Concepts
3216:Economic
3098:Logicism
3091:timeline
3067:Concrete
2926:Validity
2896:T-schema
2889:Kripke's
2884:Tarski's
2879:semantic
2869:Strength
2818:submodel
2813:spectrum
2781:function
2629:Tarski's
2618:Elements
2605:geometry
2561:Robinson
2482:variable
2467:function
2440:spectrum
2430:Sentence
2386:variable
2329:Language
2282:Relation
2243:Automata
2233:Alphabet
2217:language
2071:-jection
2049:codomain
2035:Function
1996:Universe
1966:Infinite
1870:Relation
1653:Validity
1643:Argument
1541:theorem,
1362:Archived
1279:, 1979.
1262:, 1967.
1245:, 1961.
905:See also
714:founded
656:used in
589:formulas
578:theorems
493:require
249:Alphabet
219:a series
216:Part of
187:theorems
153:alphabet
151:from an
130:theorems
122:nonsense
100:Concepts
86:notation
56:theorems
50:, using
48:deducing
3779:Commons
3362:Ecology
3352:Biology
3231:Nervous
3201:Complex
3040:Related
2837:Diagram
2735: (
2714:Hilbert
2699:Systems
2694:Theorem
2572:of the
2517:systems
2297:Formula
2292:Grammar
2208: (
2152:General
1865:Forcing
1850:Element
1770:Monadic
1545:paradox
1486:Theorem
1422:General
1368:, 1997.
1142:grammar
897:. The
833:History
730:theorem
691:theorem
495:cleanup
149:symbols
3767:Portal
3392:Theory
3288:fields
3241:Social
3181:System
2803:finite
2566:Skolem
2519:
2494:Theory
2462:Symbol
2452:String
2435:atomic
2312:ground
2307:closed
2302:atomic
2258:ground
2221:syntax
2117:binary
2044:domain
1961:Finite
1726:finite
1584:Logics
1543:
1491:Theory
1288:
1270:
1253:
1222:
873:Europe
863:, and
849:PÄáčini
759:, and
639:theory
605:belief
575:derive
569:) and
563:axioms
547:, and
456:syntax
441:syntax
421:, and
254:Syntax
183:axioms
60:axioms
42:of an
34:is an
3184:types
2793:Model
2541:Peano
2398:Proof
2238:Arity
2167:Naive
2054:image
1986:Fuzzy
1946:Empty
1895:union
1840:Class
1481:Model
1471:Lemma
1429:Axiom
1158:(PDF)
801:Sound
791:model
687:axiom
593:truth
58:from
3731:List
2916:Type
2719:list
2523:list
2500:list
2489:Term
2423:rank
2317:open
2211:list
2023:Maps
1928:sets
1787:Free
1757:list
1507:list
1434:list
1286:ISBN
1268:ISBN
1251:ISBN
1220:ISBN
1165:2015
841:and
675:and
633:The
565:(or
446:the
439:the
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38:and
3909:Set
3191:Art
2603:of
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1828:Set
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