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338:, for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that is suitably flexible,
293:, among many others, has argued that the lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as the time available to utter one), can be explained as the consequence of recursion in natural language.
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scientists find themselves when producing knowledge about the world they are always already part of. According to Audrey
Alejandro, âas social scientists, the recursivity of our condition deals with the fact that we are both subjects (as discourses are the medium through which we analyse) and objects
1707:
we are socialised into discourses and dispositions produced by the socio-political order we aim to challenge, a socio-political order that we may, therefore, reproduce unconsciously while aiming to do the contrary. The recursivity of our situation as scholars â and, more precisely, the fact that the
662:
Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with a collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the
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Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to the problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is in
280:
Even if it is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of the procedure; this requires some administration as to how far various simultaneous instances of the procedures have
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Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity of instructions. The main disadvantage is that the memory usage of recursive algorithms may grow very quickly, rendering them impractical for larger instances.
269:
To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, while the running of a procedure involves actually following the rules and performing the steps.
312:. There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another. Over the years, languages in general have proved amenable to this kind of analysis.
304:
occurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion.
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is typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one.
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of the academic discourses we produce (as we are social agents belonging to the world we analyse).â From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise of
276:
When a procedure is thus defined, this immediately creates the possibility of an endless loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the procedure can complete.
78:
being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur.
1677:
Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example is
1708:
dispositional tools we use to produce knowledge about the world are themselves produced by this world â both evinces the vital necessity of implementing reflexivity in practice and poses the main challenge in doing so.
439:
Another joke is that "To understand recursion, you must understand recursion." In the
English-language version of the Google web search engine, when a search for "recursion" is made, the site suggests "Did you mean:
1543:
and is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach is
47:. The woman in this image holds an object that contains a smaller image of her holding an identical object, which in turn contains a smaller image of herself holding an identical object, and so forth. 1904 Droste
212:
can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number." By this base case and recursive rule, one can generate the set of all natural numbers.
2228:
266:
Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be 'recursive'.
1655:
for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program.
296:
This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence:
308:
This provides a way of understanding the creativity of languageâthe unbounded number of grammatical sentencesâbecause it immediately predicts that sentences can be of arbitrary length:
1833:, made in 1320. Its central panel contains the kneeling figure of Cardinal Stefaneschi, holding up the triptych itself as an offering. This practice is more generally known as the
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are equations which define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain a non-recursive definition (e.g., a
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occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from
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1940:
1548:. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.
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More examples of recursion: Russian
Matryoshka dolls. Each doll is made of solid wood or is hollow and contains another Matryoshka doll inside it.
621:. The Peano Axioms define the natural numbers referring to a recursive successor function and addition and multiplication as recursive functions.
4845:
762:, which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step).
5003:
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Nevins, Andrew and David
Pesetsky and Cilene Rodrigues. Evidence and Argumentation: A Reply to Everett (2009). Language 85.3: 671--681 (2009)
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as the process of iterating through levels of abstraction in large business entities. A common example is the recursive nature of management
613:(or Peano postulates or DedekindâPeano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician
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273:
Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
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being stirred into flour to produce sourdough: the recipe calls for some sourdough left over from the last time the same recipe was made.
105:
In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:
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408:; the index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in
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Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving a
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folklore and was already widespread in the functional programming community before the publication of the aforementioned books.
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that restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is the
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If a proposition can be derived from true reachable propositions by means of inference rules, it is a provable proposition.
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labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating the
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450:"If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to
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404:
331:
31:
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by
Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears in
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711:â 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case
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by
Laurent SiklĂłssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and in
255:
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The generally accepted idea that recursion is an essential property of human language has been challenged by
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323:. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this. Literary
39:
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by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?"
514:
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1802:, 1320, recursively contains an image of itself (held up by the kneeling figure in the central panel).
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374:, in which the putative recursive step does not get closer to a base case, but instead leads to an
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225:
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Many mathematical axioms are based upon recursive rules. For example, the formal definition of the
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1967: â Parallel or angled mirrors, creating smaller reflections that appear to recede to infinity
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A common method of simplification is to divide a problem into subproblems of the same type. As a
740:
647:
The set of provable propositions is the smallest set of propositions satisfying these conditions.
451:
378:. It is not unusual for such books to include a joke entry in their glossary along the lines of:
233:
151:
4895:
4351:
3019:
2149:
Pinker, Steven; Jackendoff, Ray (2005). "The faculty of language: What's so special about it?".
964:
Dedekind was the first to pose the problem of unique definition of set-theoretical functions on
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Proceedings of the 40th Annual
Meeting on Association for Computational Linguistics (ACL '02)
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A plaque commemorates the
Toronto Recursive History Project of Toronto's Recursive History.
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Nederhof, Mark-Jan; Satta, Giorgio (2002), "Parsing Non-recursive
Context-free Grammars",
1964:
1961: â Philosophical view that knowledge may be justified by an infinite chain of reasons
1925:
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634:
520:
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can in any case be argued to be different in kind from mathematical or logical recursion.
3220:
2630:
2453:", pp.50--52. Bulletin of Symbolic Logic, vol. 18, no. 1 (2012). Accessed 21 August 2023.
281:
progressed. For this reason, recursive definitions are very rare in everyday situations.
1985: â Technique of placing a copy of an image within itself, or a story within a story
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2110:
2015:
1997:
774:, this is a theorem guaranteeing that recursively defined functions exist. Given a set
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371:
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324:
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2402:"Introduction to Computer Science and Programming in C; Session 8: September 25, 2008"
2203:
605:
The set of natural numbers is the smallest set satisfying the previous two properties.
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Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001).
2790:
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2324:, Stroudsburg, PA, USA: Association for Computational Linguistics, pp. 112â119,
1982:
1946:
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48:
44:
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113:(or cases) â a terminating scenario that does not use recursion to produce an answer
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2009:
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2012: â Cyclic structure that goes through several levels in a hierarchical system
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to recursively defined sets or functions, as in the preceding sections, yields
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3438:
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3414:
3148:
1958:
771:
664:
2544:
2535:
2517:"Reflexive discourse analysis: A methodology for the practice of reflexivity"
5379:
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1853:(1956) is a print which depicts a distorted city containing a gallery which
1737:
1552:
629:
Another interesting example is the set of all "provable" propositions in an
217:
98:
2180:
17:
1943: â Mathematical theory about infinitely iterated function composition
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5162:
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3310:
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683:
may be recursively defined in terms of itself. A familiar example is the
2450:
1635:
The function calls itself recursively on a smaller version of the input
120:â a set of rules that reduces all successive cases toward the base case.
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3752:
3340:
3257:
3060:
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2247:
237:
175:
1901: â Technique for defining number-theoretic functions by recursion
3650:
3328:
1824:
1793:
1652:
474:
243:
There are various more tongue-in-cheek definitions of recursion; see
101:, an ancient symbol depicting a serpent or dragon eating its own tail
519:
The canonical example of a recursively defined set is given by the
124:
For example, the following is a recursive definition of a person's
4504:
3850:
3695:
2888:. Stanford Univ Center for the Study of Language and Information.
2655:
Physical (A)Causality: Determinism, Randomness and
Uncaused Events
1787:
1767:
430:
254:
93:
63:
38:
420:
by Kernighan and Pike. It did not appear in the first edition of
2018: â Subroutine call performed as final action of a procedure
3654:
3033:
2567:"The Canadian Small BusinessâBank Interface: A Recursive Model"
2302:
Barbara Partee and Mats Rooth. 1983. In Rainer BĂ€uerle et al.,
2057:(2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers.
330:
Recursion plays a crucial role not only in syntax, but also in
470:
462:
1973: â Result of repeatedly applying a mathematical function
1928: â Vivid and convincing dream about awakening from sleep
856:
denotes the set of natural numbers including zero) such that
2864:
Recursion Theory, Gödel's Theorems, Set Theory, Model Theory
2027: â Formula that visually represents itself when graphed
3029:
2306:. Reprinted in Paul Portner and Barbara Partee, eds. 2002.
2227:
Nevins, Andrew; Pesetsky, David; Rodrigues, Cilene (2009).
1883:
to a noun to jokingly indicate the recursion of something.
1647:, analogously to the mathematical definition of factorial.
641:
If a proposition is an axiom, it is a provable proposition.
310:
Dorothy thinks that Toto suspects that Tin Man said that...
637:
which is inductively (or recursively) defined as follows:
2862:
Cori, Rene; Lascar, Daniel; Pelletier, Donald H. (2001).
1820:
is a physical artistic example of the recursive concept.
2631:"Giotto di Bondone and assistants: Stefaneschi triptych"
1551:
A classic example of recursion is the definition of the
465:, for example, stands for "PHP Hypertext Preprocessor",
2229:"Evidence and argumentation: A reply to Everett (2009)"
2020:
Pages displaying short descriptions of redirect targets
2000: â Sentence, idea or formula that refers to itself
1993:
Pages displaying short descriptions of redirect targets
1936:
Pages displaying short descriptions of redirect targets
1934: â Higher-order function Y for which Y f = f (Y f)
1915:
Pages displaying short descriptions of redirect targets
216:
Other recursively defined mathematical objects include
2485:
Bourdieu, Pierre (1992). "Double Bind et Conversion".
2282:. Springer Science & Business Media. p. 110.
493:âa confined recursion of triangles that form a fractal
477:
denotes the "SPARQL Protocol and RDF Query Language".
2967:
Stokey, Nancy; Robert Lucas; Edward Prescott (1989).
1499:
1360:
1203:
1143:
1107:
1071:
1034:
1000:
970:
901:
865:
840:
806:
800:, the theorem states that there is a unique function
587:
561:
533:
454:
than you are; then ask him or her what recursion is."
2702:"-ception â The Rice University Neologisms Database"
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3309:
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3132:
3067:
1640:
1639:and multiplies the result of the recursive call by
1636:
2840:
2814:
1694:to foreground the situation in which specifically
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1374:
1248:
1188:
1128:
1092:
1054:
1020:
978:
946:
886:
848:
826:
595:
569:
541:
2279:Perspectives on the History of Mathematical Logic
66:. The most common application of recursion is in
27:Process of repeating items in a self-similar way
1877:has colloquialized the appending of the suffix
1705:
444:." An alternative form is the following, from
3666:
3045:
2006: â 1978 musical composition by Arvo PĂ€rt
1895: â Type of algorithm in computer science
8:
2792:Gödel, Escher, Bach: an Eternal Golden Braid
2382:. University of Illinois at Urbana-Champaign
2304:Meaning, Use, and Interpretation of Language
2522:European Journal of International Relations
1941:Infinite compositions of analytic functions
1823:Recursion has been used in paintings since
4492:
4087:
3855:
3673:
3659:
3651:
3052:
3038:
3030:
1752:. It also encompasses the larger issue of
2910:Discrete Mathematics and Its Applications
2534:
2464:"Picture of the Day: Fractal Cauliflower"
2343:
2341:
2329:
2162:
1714:
1507:
1506:
1498:
1368:
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839:
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589:
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154:is another classic example of recursion:
2308:Formal Semantics: The Essential Readings
484:
390:A variation is found on page 269 in the
43:A visual form of recursion known as the
3606:List of fractals by Hausdorff dimension
2204:"What Is Recursion in English Grammar?"
2043:
1991: â Concept in computer programming
1907: â Term in theoretical linguistics
461:are other examples of recursive humor.
2969:Recursive Methods in Economic Dynamics
1732:Recursion is sometimes referred to in
723:Proofs involving recursive definitions
469:stands for "WINE Is Not an Emulator",
2679:Cooper, Jonathan (5 September 2007).
2033: â Statement of infinite regress
1772:Recursive dolls: the original set of
319:on the basis of his claims about the
82:A process that exhibits recursion is
7:
298:Dorothy thinks witches are dangerous
244:
2946:Kernighan, B.; Ritchie, D. (1988).
2354:. Jones and Bartlett. p. 494.
1055:{\displaystyle G:\mathbb {N} \to X}
1021:{\displaystyle F:\mathbb {N} \to X}
827:{\displaystyle F:\mathbb {N} \to X}
727:Applying the standard technique of
2502:Social Theory and Modern Sociology
2351:Essentials of Discrete Mathematics
2276:Drucker, Thomas (4 January 2008).
1979: â Form of mathematical proof
25:
3588:How Long Is the Coast of Britain?
2735:(1960). "Recursive Programming".
2708:from the original on July 5, 2017
2025:Tupper's self-referential formula
1514:{\displaystyle n\in \mathbb {N} }
1375:{\displaystyle k\in \mathbb {N} }
617:and by the Italian mathematician
473:stands for "GNU's not Unix", and
5392:
2487:Pour Une Anthropologie RĂ©flexive
739:widely used to derive proofs in
418:The UNIX Programming Environment
2173:10.1016/j.cognition.2004.08.004
1922: â Recursive visual effect
1913: â Poem by Edgar Allan Poe
735:â a powerful generalization of
633:that are defined in terms of a
3612:The Fractal Geometry of Nature
3002:, first chapter on set theory.
1249:{\displaystyle G(n+1)=f(G(n))}
1243:
1240:
1234:
1228:
1219:
1207:
1189:{\displaystyle F(n+1)=f(F(n))}
1183:
1180:
1174:
1168:
1159:
1147:
1117:
1111:
1081:
1075:
1046:
1012:
947:{\displaystyle F(n+1)=f(F(n))}
941:
938:
932:
926:
917:
905:
875:
869:
818:
1:
5353:History of mathematical logic
1955: â Philosophical problem
1911:A Dream Within a Dream (poem)
1857:contains the picture, and so
667:is a subdivision rule, as is
5278:Primitive recursive function
2971:. Harvard University Press.
2884:; Moss, Lawrence S. (1996).
2377:"CS 173:Discrete Structures"
2087:"Peano axioms | mathematics"
1531:Recursion (computer science)
979:{\displaystyle \mathbb {N} }
849:{\displaystyle \mathbb {N} }
596:{\displaystyle \mathbb {N} }
570:{\displaystyle \mathbb {N} }
542:{\displaystyle \mathbb {N} }
509:Example: the natural numbers
128:. One's ancestor is either:
3628:Chaos: Making a New Science
2866:. Oxford University Press.
2427:"recursion - Google Search"
1690:Authors use the concept of
609:In mathematical logic, the
51:tin, designed by Jan Misset
5450:
4342:SchröderâBernstein theorem
4069:Monadic predicate calculus
3728:Foundations of mathematics
3020:Zip Files All The Way Down
2949:The C programming Language
2929:Introduction to Algorithms
2908:Rosen, Kenneth H. (2002).
2843:Logic, Sets, and Recursion
2515:Alejandro, Audrey (2021).
2054:Logic, sets, and recursion
2051:Causey, Robert L. (2006).
1899:Course-of-values recursion
1805:
1725:
1539:technique, this is called
1528:
1319:so the equality holds for
655:
512:
501:
424:. The joke is part of the
422:The C Programming Language
405:The C Programming Language
332:natural language semantics
32:Recursion (disambiguation)
29:
5388:
5375:Philosophy of mathematics
5324:Automated theorem proving
4495:
4449:Von NeumannâBernaysâGödel
4090:
2500:Giddens, Anthony (1987).
2111:"Definition of RECURSIVE"
1949: â Programming idiom
1293:for all natural numbers
142:One's parent's ancestor (
3016:- tutorial by Alan Gauld
2900:- offers a treatment of
2847:. Jones & Bartlett.
2658:. Springer. p. 12.
2536:10.1177/1354066120969789
2451:In Praise of Replacement
2031:Turtles all the way down
1561:
1555:function, given here in
652:Finite subdivision rules
625:Example: Proof procedure
498:Recursively defined sets
354:that contains recursive
300:, in which the sentence
5025:Self-verifying theories
4846:Tarski's axiomatization
3797:Tarski's undefinability
3792:incompleteness theorems
2912:. McGraw-Hill College.
2580:Beer, Stafford (1972).
2331:10.3115/1073083.1073104
2134:Pinker, Steven (1994).
2115:www.merriam-webster.com
2091:Encyclopedia Britannica
957:for any natural number
669:barycentric subdivision
658:Finite subdivision rule
5399:Mathematics portal
5010:Proof of impossibility
4658:propositional variable
3968:Propositional calculus
3620:The Beauty of Fractals
2348:Hunter, David (2011).
1989:Reentrant (subroutine)
1977:Mathematical induction
1932:Fixed point combinator
1803:
1785:
1728:Management cybernetics
1719:
1686:In the social sciences
1663:closed-form expression
1515:
1376:
1272:mathematical induction
1250:
1190:
1130:
1129:{\displaystyle G(0)=a}
1094:
1093:{\displaystyle F(0)=a}
1056:
1022:
980:
948:
888:
887:{\displaystyle F(0)=a}
850:
828:
747:Recursive optimization
743:and computer science.
737:mathematical induction
597:
571:
543:
494:
436:
426:functional programming
263:
102:
52:
5424:Theory of computation
5268:Kolmogorov complexity
5221:Computably enumerable
5121:Model complete theory
4913:Principia Mathematica
3973:Propositional formula
3802:BanachâTarski paradox
2811:Shoenfield, Joseph R.
2765:Johnsonbaugh, Richard
2737:Numerische Mathematik
2681:"Art and Mathematics"
2652:Svozil, Karl (2018).
2407:. Columbia University
2136:The Language Instinct
1791:
1771:
1726:Further information:
1643:, until reaching the
1516:
1377:
1251:
1191:
1131:
1095:
1057:
1023:
981:
949:
889:
851:
829:
766:The recursion theorem
598:
572:
544:
515:Closure (mathematics)
488:
434:
302:witches are dangerous
258:
97:
42:
5216:ChurchâTuring thesis
5203:Computability theory
4412:continuum hypothesis
3930:Square of opposition
3788:Gödel's completeness
3566:Lewis Fry Richardson
3561:Hamid Naderi Yeganeh
3351:Burning Ship fractal
3283:Weierstrass function
2769:Discrete Mathematics
2202:Nordquist, Richard.
1837:, an example of the
1830:Stefaneschi Triptych
1799:Stefaneschi Triptych
1758:corporate governance
1659:Recurrence relations
1537:computer programming
1497:
1358:
1270:It can be proved by
1201:
1141:
1105:
1069:
1032:
998:
968:
899:
863:
838:
804:
733:structural induction
675:Functional recursion
585:
559:
531:
504:Recursive definition
394:of some editions of
226:recurrence relations
30:For other uses, see
5370:Mathematical object
5261:P versus NP problem
5226:Computable function
5020:Reverse mathematics
4946:Logical consequence
4823:primitive recursive
4818:elementary function
4591:Free/bound variable
4444:TarskiâGrothendieck
3963:Logical connectives
3893:Logical equivalence
3743:Logical consequence
3324:Space-filling curve
3301:Multifractal system
3184:Space-filling curve
3169:Sierpinski triangle
2986:Hungerford (1980).
2787:Hofstadter, Douglas
2733:Dijkstra, Edsger W.
2704:. Rice University.
1808:Mathematics and art
1546:dynamic programming
1525:In computer science
994:Take two functions
990:Proof of uniqueness
752:Dynamic programming
491:Sierpinski triangle
368:circular definition
251:Informal definition
5168:Transfer principle
5131:Semantics of logic
5116:Categorical theory
5092:Non-standard model
4606:Logical connective
3733:Information theory
3682:Mathematical logic
3551:Aleksandr Lyapunov
3531:Desmond Paul Henry
3495:Self-avoiding walk
3490:Percolation theory
3134:Iterated function
3075:Fractal dimensions
2821:. A K Peters Ltd.
2749:10.1007/BF01386232
2489:. Paris: Le Seuil.
2466:. 28 December 2012
2248:10.1353/lan.0.0140
2004:Spiegel im Spiegel
1804:
1786:
1734:management science
1713:Audrey Alejandro,
1680:Romanesco broccoli
1541:divide and conquer
1511:
1372:
1246:
1186:
1126:
1090:
1052:
1018:
976:
944:
884:
846:
824:
754:is an approach to
741:mathematical logic
593:
567:
539:
495:
459:Recursive acronyms
452:Douglas Hofstadter
437:
264:
234:Cantor ternary set
152:Fibonacci sequence
103:
90:Formal definitions
53:
5406:
5405:
5338:Abstract category
5141:Theories of truth
4951:Rule of inference
4941:Natural deduction
4922:
4921:
4467:
4466:
4172:Cartesian product
4077:
4076:
3983:Many-valued logic
3958:Boolean functions
3841:Russell's paradox
3816:diagonal argument
3713:First-order logic
3648:
3647:
3594:Coastline paradox
3571:WacĆaw SierpiĆski
3556:Benoit Mandelbrot
3480:Fractal landscape
3388:Misiurewicz point
3293:Strange attractor
3174:Apollonian gasket
3164:Sierpinski carpet
2997:978-0-387-90518-1
2978:978-0-674-75096-8
2959:978-0-13-110362-7
2952:. Prentice Hall.
2938:978-0-262-03293-3
2919:978-0-07-293033-7
2895:978-0-19-850050-6
2873:978-0-19-850050-6
2854:978-0-7637-1695-0
2837:Causey, Robert L.
2828:978-1-56881-149-9
2802:978-0-465-02656-2
2778:978-0-13-117686-7
2771:. Prentice Hall.
2582:Brain Of The Firm
2289:978-0-8176-4768-1
2138:. William Morrow.
1971:Iterated function
1754:capital structure
1750:middle management
1746:senior management
1263:is an element of
348:recursive grammar
260:Sourdough starter
16:(Redirected from
5441:
5397:
5396:
5348:History of logic
5343:Category of sets
5236:Decision problem
5015:Ordinal analysis
4956:Sequent calculus
4854:Boolean algebras
4794:
4793:
4768:
4739:logical/constant
4493:
4479:
4402:ZermeloâFraenkel
4153:Set operations:
4088:
4025:
3856:
3836:LöwenheimâSkolem
3723:Formal semantics
3675:
3668:
3661:
3652:
3511:Michael Barnsley
3378:Lyapunov fractal
3236:SierpiĆski curve
3189:Blancmange curve
3054:
3047:
3040:
3031:
3001:
2982:
2963:
2942:
2923:
2899:
2877:
2858:
2846:
2832:
2820:
2817:Recursion Theory
2806:
2782:
2760:
2718:
2717:
2715:
2713:
2698:
2692:
2691:
2689:
2687:
2676:
2670:
2669:
2649:
2643:
2642:
2640:
2638:
2627:
2621:
2620:
2615:
2613:
2602:
2596:
2595:
2577:
2571:
2570:
2569:. SAGE Journals.
2563:
2557:
2556:
2538:
2512:
2506:
2505:
2497:
2491:
2490:
2482:
2476:
2475:
2473:
2471:
2460:
2454:
2447:
2441:
2440:
2438:
2437:
2423:
2417:
2416:
2414:
2412:
2406:
2398:
2392:
2391:
2389:
2387:
2381:
2372:
2366:
2365:
2345:
2336:
2334:
2333:
2317:
2311:
2300:
2294:
2293:
2273:
2267:
2266:
2264:
2258:. Archived from
2233:
2224:
2218:
2217:
2215:
2214:
2199:
2193:
2192:
2166:
2146:
2140:
2139:
2131:
2125:
2124:
2122:
2121:
2107:
2101:
2100:
2098:
2097:
2083:
2077:
2076:
2048:
2021:
1994:
1953:Infinite regress
1937:
1916:
1905:Digital infinity
1774:Matryoshka dolls
1717:
1715:Alejandro (2021)
1642:
1638:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1520:
1518:
1517:
1512:
1510:
1492:
1467:
1448:
1426:
1383:
1381:
1379:
1378:
1373:
1371:
1352:
1325:
1318:
1296:
1292:
1266:
1262:
1255:
1253:
1252:
1247:
1195:
1193:
1192:
1187:
1135:
1133:
1132:
1127:
1099:
1097:
1096:
1091:
1061:
1059:
1058:
1053:
1045:
1027:
1025:
1024:
1019:
1011:
985:
983:
982:
977:
975:
960:
953:
951:
950:
945:
893:
891:
890:
885:
855:
853:
852:
847:
845:
833:
831:
830:
825:
817:
799:
785:
781:
777:
760:Bellman equation
685:Fibonacci number
631:axiomatic system
615:Richard Dedekind
602:
600:
599:
594:
592:
576:
574:
573:
568:
566:
548:
546:
545:
540:
538:
376:infinite regress
356:production rules
199:
183:
168:
160:
72:computer science
21:
5449:
5448:
5444:
5443:
5442:
5440:
5439:
5438:
5409:
5408:
5407:
5402:
5391:
5384:
5329:Category theory
5319:Algebraic logic
5302:
5273:Lambda calculus
5211:Church encoding
5197:
5173:Truth predicate
5029:
4995:Complete theory
4918:
4787:
4783:
4779:
4774:
4766:
4486: and
4482:
4477:
4463:
4439:New Foundations
4407:axiom of choice
4390:
4352:Gödel numbering
4292: and
4284:
4188:
4073:
4023:
4004:
3953:Boolean algebra
3939:
3903:Equiconsistency
3868:Classical logic
3845:
3826:Halting problem
3814: and
3790: and
3778: and
3777:
3772:Theorems (
3767:
3684:
3679:
3649:
3644:
3575:
3526:Felix Hausdorff
3499:
3463:Brownian motion
3448:
3419:
3342:
3335:
3305:
3287:
3278:Pythagoras tree
3135:
3128:
3124:Self-similarity
3068:Characteristics
3063:
3058:
3010:
3005:
2998:
2985:
2979:
2966:
2960:
2945:
2939:
2926:
2920:
2907:
2896:
2886:Vicious Circles
2880:
2874:
2861:
2855:
2835:
2829:
2809:
2803:
2795:. Basic Books.
2785:
2779:
2763:
2731:
2727:
2722:
2721:
2711:
2709:
2700:
2699:
2695:
2685:
2683:
2678:
2677:
2673:
2666:
2651:
2650:
2646:
2636:
2634:
2629:
2628:
2624:
2611:
2609:
2604:
2603:
2599:
2592:
2579:
2578:
2574:
2565:
2564:
2560:
2514:
2513:
2509:
2504:. Polity Press.
2499:
2498:
2494:
2484:
2483:
2479:
2469:
2467:
2462:
2461:
2457:
2448:
2444:
2435:
2433:
2425:
2424:
2420:
2410:
2408:
2404:
2400:
2399:
2395:
2385:
2383:
2379:
2375:Shaffer, Eric.
2374:
2373:
2369:
2362:
2347:
2346:
2339:
2319:
2318:
2314:
2301:
2297:
2290:
2275:
2274:
2270:
2262:
2231:
2226:
2225:
2221:
2212:
2210:
2201:
2200:
2196:
2164:10.1.1.116.7784
2148:
2147:
2143:
2133:
2132:
2128:
2119:
2117:
2109:
2108:
2104:
2095:
2093:
2085:
2084:
2080:
2065:
2050:
2049:
2045:
2040:
2019:
1992:
1965:Infinity mirror
1935:
1926:False awakening
1914:
1889:
1869:
1818:Matryoshka doll
1814:
1812:Infinity mirror
1766:
1742:line management
1740:, ranging from
1730:
1724:
1718:
1712:
1688:
1675:
1633:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1533:
1527:
1495:
1494:
1475:
1450:
1431:
1385:
1356:
1355:
1354:
1335:
1320:
1305:
1294:
1275:
1264:
1260:
1199:
1198:
1139:
1138:
1103:
1102:
1067:
1066:
1030:
1029:
996:
995:
992:
966:
965:
958:
897:
896:
861:
860:
836:
835:
802:
801:
787:
786:and a function
783:
779:
775:
768:
749:
725:
677:
660:
654:
635:proof procedure
627:
583:
582:
557:
556:
529:
528:
521:natural numbers
517:
511:
506:
500:
483:
410:Let's talk Lisp
396:Brian Kernighan
364:
362:Recursive humor
321:PirahĂŁ language
287:
253:
245:recursive humor
206:natural numbers
185:
178:
169:as base case 2,
166:
161:as base case 1,
158:
92:
35:
28:
23:
22:
15:
12:
11:
5:
5447:
5445:
5437:
5436:
5431:
5429:Self-reference
5426:
5421:
5411:
5410:
5404:
5403:
5389:
5386:
5385:
5383:
5382:
5377:
5372:
5367:
5362:
5361:
5360:
5350:
5345:
5340:
5331:
5326:
5321:
5316:
5314:Abstract logic
5310:
5308:
5304:
5303:
5301:
5300:
5295:
5293:Turing machine
5290:
5285:
5280:
5275:
5270:
5265:
5264:
5263:
5258:
5253:
5248:
5243:
5233:
5231:Computable set
5228:
5223:
5218:
5213:
5207:
5205:
5199:
5198:
5196:
5195:
5190:
5185:
5180:
5175:
5170:
5165:
5160:
5159:
5158:
5153:
5148:
5138:
5133:
5128:
5126:Satisfiability
5123:
5118:
5113:
5112:
5111:
5101:
5100:
5099:
5089:
5088:
5087:
5082:
5077:
5072:
5067:
5057:
5056:
5055:
5050:
5043:Interpretation
5039:
5037:
5031:
5030:
5028:
5027:
5022:
5017:
5012:
5007:
4997:
4992:
4991:
4990:
4989:
4988:
4978:
4973:
4963:
4958:
4953:
4948:
4943:
4938:
4932:
4930:
4924:
4923:
4920:
4919:
4917:
4916:
4908:
4907:
4906:
4905:
4900:
4899:
4898:
4893:
4888:
4868:
4867:
4866:
4864:minimal axioms
4861:
4850:
4849:
4848:
4837:
4836:
4835:
4830:
4825:
4820:
4815:
4810:
4797:
4795:
4776:
4775:
4773:
4772:
4771:
4770:
4758:
4753:
4752:
4751:
4746:
4741:
4736:
4726:
4721:
4716:
4711:
4710:
4709:
4704:
4694:
4693:
4692:
4687:
4682:
4677:
4667:
4662:
4661:
4660:
4655:
4650:
4640:
4639:
4638:
4633:
4628:
4623:
4618:
4613:
4603:
4598:
4593:
4588:
4587:
4586:
4581:
4576:
4571:
4561:
4556:
4554:Formation rule
4551:
4546:
4545:
4544:
4539:
4529:
4528:
4527:
4517:
4512:
4507:
4502:
4496:
4490:
4473:Formal systems
4469:
4468:
4465:
4464:
4462:
4461:
4456:
4451:
4446:
4441:
4436:
4431:
4426:
4421:
4416:
4415:
4414:
4409:
4398:
4396:
4392:
4391:
4389:
4388:
4387:
4386:
4376:
4371:
4370:
4369:
4362:Large cardinal
4359:
4354:
4349:
4344:
4339:
4325:
4324:
4323:
4318:
4313:
4298:
4296:
4286:
4285:
4283:
4282:
4281:
4280:
4275:
4270:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4210:
4205:
4199:
4197:
4190:
4189:
4187:
4186:
4185:
4184:
4179:
4174:
4169:
4164:
4159:
4151:
4150:
4149:
4144:
4134:
4129:
4127:Extensionality
4124:
4122:Ordinal number
4119:
4109:
4104:
4103:
4102:
4091:
4085:
4079:
4078:
4075:
4074:
4072:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4040:
4039:
4029:
4028:
4027:
4014:
4012:
4006:
4005:
4003:
4002:
4001:
4000:
3995:
3990:
3980:
3975:
3970:
3965:
3960:
3955:
3949:
3947:
3941:
3940:
3938:
3937:
3932:
3927:
3922:
3917:
3912:
3907:
3906:
3905:
3895:
3890:
3885:
3880:
3875:
3870:
3864:
3862:
3853:
3847:
3846:
3844:
3843:
3838:
3833:
3828:
3823:
3818:
3806:Cantor's
3804:
3799:
3794:
3784:
3782:
3769:
3768:
3766:
3765:
3760:
3755:
3750:
3745:
3740:
3735:
3730:
3725:
3720:
3715:
3710:
3705:
3704:
3703:
3692:
3690:
3686:
3685:
3680:
3678:
3677:
3670:
3663:
3655:
3646:
3645:
3643:
3642:
3637:
3632:
3624:
3616:
3608:
3603:
3598:
3597:
3596:
3583:
3581:
3577:
3576:
3574:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3541:Helge von Koch
3538:
3533:
3528:
3523:
3518:
3513:
3507:
3505:
3501:
3500:
3498:
3497:
3492:
3487:
3482:
3477:
3476:
3475:
3473:Brownian motor
3470:
3459:
3457:
3450:
3449:
3447:
3446:
3444:Pickover stalk
3441:
3436:
3430:
3428:
3421:
3420:
3418:
3417:
3412:
3407:
3402:
3400:Newton fractal
3397:
3392:
3391:
3390:
3383:Mandelbrot set
3380:
3375:
3374:
3373:
3368:
3366:Newton fractal
3363:
3353:
3347:
3345:
3337:
3336:
3334:
3333:
3332:
3331:
3321:
3319:Fractal canopy
3315:
3313:
3307:
3306:
3304:
3303:
3297:
3295:
3289:
3288:
3286:
3285:
3280:
3275:
3270:
3265:
3263:Vicsek fractal
3260:
3255:
3250:
3245:
3244:
3243:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3203:
3202:
3201:
3191:
3181:
3179:Fibonacci word
3176:
3171:
3166:
3161:
3156:
3154:Koch snowflake
3151:
3146:
3140:
3138:
3130:
3129:
3127:
3126:
3121:
3116:
3115:
3114:
3109:
3104:
3099:
3094:
3093:
3092:
3082:
3071:
3069:
3065:
3064:
3059:
3057:
3056:
3049:
3042:
3034:
3028:
3027:
3022:
3017:
3009:
3008:External links
3006:
3004:
3003:
2996:
2983:
2977:
2964:
2958:
2943:
2937:
2924:
2918:
2905:
2894:
2878:
2872:
2859:
2853:
2833:
2827:
2807:
2801:
2783:
2777:
2761:
2743:(1): 312â318.
2728:
2726:
2723:
2720:
2719:
2693:
2671:
2664:
2644:
2622:
2597:
2591:978-0471948391
2590:
2572:
2558:
2507:
2492:
2477:
2455:
2449:A. Kanamori, "
2442:
2431:www.google.com
2418:
2393:
2367:
2360:
2337:
2312:
2295:
2288:
2268:
2265:on 2012-01-06.
2242:(3): 671â681.
2219:
2194:
2157:(2): 201â236.
2141:
2126:
2102:
2078:
2063:
2042:
2041:
2039:
2036:
2035:
2034:
2028:
2022:
2016:Tail recursion
2013:
2007:
2001:
1998:Self-reference
1995:
1986:
1980:
1974:
1968:
1962:
1956:
1950:
1944:
1938:
1929:
1923:
1917:
1908:
1902:
1896:
1888:
1885:
1868:
1865:
1792:Front face of
1765:
1762:
1723:
1720:
1710:
1687:
1684:
1674:
1671:
1562:
1529:Main article:
1526:
1523:
1509:
1505:
1502:
1474:By induction,
1472:
1471:
1470:
1469:
1370:
1366:
1363:
1332:Inductive Step
1328:
1327:
1257:
1256:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1196:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1136:
1125:
1122:
1119:
1116:
1113:
1110:
1100:
1089:
1086:
1083:
1080:
1077:
1074:
1051:
1048:
1044:
1040:
1037:
1017:
1014:
1010:
1006:
1003:
991:
988:
974:
955:
954:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
894:
883:
880:
877:
874:
871:
868:
844:
823:
820:
816:
812:
809:
767:
764:
748:
745:
729:proof by cases
724:
721:
676:
673:
656:Main article:
653:
650:
649:
648:
645:
642:
626:
623:
619:Giuseppe Peano
607:
606:
603:
591:
565:
549:
537:
510:
507:
502:Main article:
499:
496:
482:
481:In mathematics
479:
446:Andrew Plotkin
414:Software Tools
400:Dennis Ritchie
388:
387:
372:self-reference
363:
360:
352:formal grammar
325:self-reference
317:Daniel Everett
286:
283:
252:
249:
202:
201:
171:
170:
163:
162:
148:
147:
144:recursive step
140:
132:One's parent (
122:
121:
118:recursive step
114:
91:
88:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5446:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5416:
5414:
5401:
5400:
5395:
5387:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5359:
5356:
5355:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5311:
5309:
5305:
5299:
5296:
5294:
5291:
5289:
5288:Recursive set
5286:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5238:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5208:
5206:
5204:
5200:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5157:
5154:
5152:
5149:
5147:
5144:
5143:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5110:
5107:
5106:
5105:
5102:
5098:
5097:of arithmetic
5095:
5094:
5093:
5090:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5062:
5061:
5058:
5054:
5051:
5049:
5046:
5045:
5044:
5041:
5040:
5038:
5036:
5032:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5005:
5004:from ZFC
5001:
4998:
4996:
4993:
4987:
4984:
4983:
4982:
4979:
4977:
4974:
4972:
4969:
4968:
4967:
4964:
4962:
4959:
4957:
4954:
4952:
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4933:
4931:
4929:
4925:
4915:
4914:
4910:
4909:
4904:
4903:non-Euclidean
4901:
4897:
4894:
4892:
4889:
4887:
4886:
4882:
4881:
4879:
4876:
4875:
4873:
4869:
4865:
4862:
4860:
4857:
4856:
4855:
4851:
4847:
4844:
4843:
4842:
4838:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4809:
4806:
4805:
4803:
4799:
4798:
4796:
4791:
4785:
4780:Example
4777:
4769:
4764:
4763:
4762:
4759:
4757:
4754:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4731:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4708:
4705:
4703:
4700:
4699:
4698:
4695:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4672:
4671:
4668:
4666:
4663:
4659:
4656:
4654:
4651:
4649:
4646:
4645:
4644:
4641:
4637:
4634:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4608:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4566:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4543:
4540:
4538:
4537:by definition
4535:
4534:
4533:
4530:
4526:
4523:
4522:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4497:
4494:
4491:
4489:
4485:
4480:
4474:
4470:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4424:KripkeâPlatek
4422:
4420:
4417:
4413:
4410:
4408:
4405:
4404:
4403:
4400:
4399:
4397:
4393:
4385:
4382:
4381:
4380:
4377:
4375:
4372:
4368:
4365:
4364:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4337:
4333:
4329:
4326:
4322:
4319:
4317:
4314:
4312:
4309:
4308:
4307:
4303:
4300:
4299:
4297:
4295:
4291:
4287:
4279:
4276:
4274:
4271:
4269:
4268:constructible
4266:
4265:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4200:
4198:
4196:
4191:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4154:
4152:
4148:
4145:
4143:
4140:
4139:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4114:
4110:
4108:
4105:
4101:
4098:
4097:
4096:
4093:
4092:
4089:
4086:
4084:
4080:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4038:
4035:
4034:
4033:
4030:
4026:
4021:
4020:
4019:
4016:
4015:
4013:
4011:
4007:
3999:
3996:
3994:
3991:
3989:
3986:
3985:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3950:
3948:
3946:
3945:Propositional
3942:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3904:
3901:
3900:
3899:
3896:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3873:Logical truth
3871:
3869:
3866:
3865:
3863:
3861:
3857:
3854:
3852:
3848:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3817:
3813:
3809:
3805:
3803:
3800:
3798:
3795:
3793:
3789:
3786:
3785:
3783:
3781:
3775:
3770:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3702:
3699:
3698:
3697:
3694:
3693:
3691:
3687:
3683:
3676:
3671:
3669:
3664:
3662:
3657:
3656:
3653:
3641:
3638:
3636:
3633:
3630:
3629:
3625:
3622:
3621:
3617:
3614:
3613:
3609:
3607:
3604:
3602:
3599:
3595:
3592:
3591:
3589:
3585:
3584:
3582:
3578:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3508:
3506:
3502:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3474:
3471:
3469:
3468:Brownian tree
3466:
3465:
3464:
3461:
3460:
3458:
3455:
3451:
3445:
3442:
3440:
3437:
3435:
3432:
3431:
3429:
3426:
3422:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3395:Multibrot set
3393:
3389:
3386:
3385:
3384:
3381:
3379:
3376:
3372:
3371:Douady rabbit
3369:
3367:
3364:
3362:
3359:
3358:
3357:
3354:
3352:
3349:
3348:
3346:
3344:
3338:
3330:
3327:
3326:
3325:
3322:
3320:
3317:
3316:
3314:
3312:
3308:
3302:
3299:
3298:
3296:
3294:
3290:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3242:
3241:Z-order curve
3239:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3211:Hilbert curve
3209:
3207:
3204:
3200:
3197:
3196:
3195:
3194:De Rham curve
3192:
3190:
3187:
3186:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3159:Menger sponge
3157:
3155:
3152:
3150:
3147:
3145:
3144:Barnsley fern
3142:
3141:
3139:
3137:
3131:
3125:
3122:
3120:
3117:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3091:
3088:
3087:
3086:
3083:
3081:
3078:
3077:
3076:
3073:
3072:
3070:
3066:
3062:
3055:
3050:
3048:
3043:
3041:
3036:
3035:
3032:
3026:
3023:
3021:
3018:
3015:
3012:
3011:
3007:
2999:
2993:
2989:
2984:
2980:
2974:
2970:
2965:
2961:
2955:
2951:
2950:
2944:
2940:
2934:
2930:
2925:
2921:
2915:
2911:
2906:
2903:
2897:
2891:
2887:
2883:
2879:
2875:
2869:
2865:
2860:
2856:
2850:
2845:
2844:
2838:
2834:
2830:
2824:
2819:
2818:
2812:
2808:
2804:
2798:
2794:
2793:
2788:
2784:
2780:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2746:
2742:
2738:
2734:
2730:
2729:
2724:
2707:
2703:
2697:
2694:
2682:
2675:
2672:
2667:
2665:9783319708157
2661:
2657:
2656:
2648:
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2633:. The Vatican
2632:
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2623:
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2608:
2605:Tang, Daisy.
2601:
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1983:Mise en abyme
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1947:Infinite loop
1945:
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1920:Droste effect
1918:
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1850:Print Gallery
1846:
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1839:Mise en abyme
1836:
1835:Droste effect
1832:
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778:, an element
773:
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385:
384:see Recursion
381:
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61:
57:
50:
46:
45:Droste effect
41:
37:
33:
19:
5390:
5282:
5188:Ultraproduct
5035:Model theory
5000:Independence
4936:Formal proof
4928:Proof theory
4911:
4884:
4841:real numbers
4813:second-order
4724:Substitution
4601:Metalanguage
4542:conservative
4515:Axiom schema
4459:Constructive
4429:MorseâKelley
4395:Set theories
4374:Aleph number
4367:inaccessible
4273:Grothendieck
4157:intersection
4044:Higher-order
4032:Second-order
3978:Truth tables
3935:Venn diagram
3718:Formal proof
3640:Chaos theory
3635:Kaleidoscope
3626:
3618:
3610:
3536:Gaston Julia
3516:Georg Cantor
3341:Escape-time
3273:Gosper curve
3221:LĂ©vy C curve
3206:Dragon curve
3118:
3085:Box-counting
2990:. Springer.
2987:
2968:
2948:
2928:
2909:
2885:
2882:Barwise, Jon
2863:
2842:
2816:
2791:
2768:
2740:
2736:
2725:Bibliography
2712:December 23,
2710:. Retrieved
2696:
2684:. Retrieved
2674:
2654:
2647:
2637:16 September
2635:. Retrieved
2625:
2617:
2612:24 September
2610:. Retrieved
2600:
2581:
2575:
2561:
2526:
2520:
2510:
2501:
2495:
2486:
2480:
2468:. Retrieved
2458:
2445:
2434:. Retrieved
2430:
2421:
2409:. Retrieved
2396:
2384:. Retrieved
2370:
2350:
2321:
2315:
2310:. Blackwell.
2307:
2303:
2298:
2278:
2271:
2260:the original
2239:
2235:
2222:
2211:. Retrieved
2207:
2197:
2154:
2150:
2144:
2135:
2129:
2118:. Retrieved
2114:
2105:
2094:. Retrieved
2090:
2081:
2053:
2046:
2010:Strange loop
1878:
1872:
1870:
1860:ad infinitum
1858:
1848:
1845:M. C. Escher
1843:
1828:
1822:
1815:
1797:
1778:Zvyozdochkin
1731:
1706:
1695:
1691:
1689:
1676:
1667:
1657:
1649:
1634:
1550:
1534:
1488:
1484:
1480:
1476:
1473:
1463:
1459:
1455:
1451:
1444:
1440:
1436:
1432:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1348:
1344:
1340:
1336:
1331:
1321:
1314:
1310:
1306:
1301:
1288:
1284:
1280:
1276:
1269:
1258:
993:
963:
956:
796:
792:
788:
769:
756:optimization
750:
726:
716:
715:(0) = 0 and
712:
708:
704:
700:
696:
692:
688:
678:
661:
628:
611:Peano axioms
608:
578:
552:
518:
457:
449:
441:
438:
421:
417:
413:
409:
403:
389:
383:
365:
345:
339:
335:
329:
314:
309:
307:
301:
297:
295:
291:Noam Chomsky
288:
279:
275:
272:
268:
265:
242:
215:
210:Peano axioms
203:
195:
191:
187:
179:
149:
143:
137:
133:
125:
123:
117:
110:
104:
83:
81:
55:
54:
36:
5298:Type theory
5246:undecidable
5178:Truth value
5065:equivalence
4744:non-logical
4357:Enumeration
4347:Isomorphism
4294:cardinality
4278:Von Neumann
4243:Ultrafilter
4208:Uncountable
4142:equivalence
4059:Quantifiers
4049:Fixed-point
4018:First-order
3898:Consistency
3883:Proposition
3860:Traditional
3831:Lindström's
3821:Compactness
3763:Type theory
3708:Cardinality
3631:(1987 book)
3623:(1986 book)
3615:(1982 book)
3601:Fractal art
3521:Bill Gosper
3485:LĂ©vy flight
3231:Peano curve
3226:Moore curve
3112:Topological
3097:Correlation
2902:corecursion
2607:"Recursion"
1893:Corecursion
1855:recursively
1841:technique.
1738:hierarchies
1722:In business
1692:recursivity
1062:such that:
382:Recursion,
334:. The word
285:In language
194:â 1) + Fib(
68:mathematics
60:linguistics
18:Recursively
5413:Categories
5109:elementary
4802:arithmetic
4670:Quantifier
4648:functional
4520:Expression
4238:Transitive
4182:identities
4167:complement
4100:hereditary
4083:Set theory
3439:Orbit trap
3434:Buddhabrot
3427:techniques
3415:Mandelbulb
3216:Koch curve
3149:Cantor set
2931:. Mit Pr.
2529:(1): 171.
2436:2019-10-24
2213:2019-10-24
2120:2019-10-24
2096:2019-10-24
2038:References
1959:Infinitism
1867:In culture
1806:See also:
1748: via
1673:In biology
1334:: Suppose
772:set theory
687:sequence:
665:Cantor set
581:+ 1 is in
513:See also:
218:factorials
167:Fib(1) = 1
159:Fib(0) = 0
74:, where a
5419:Recursion
5380:Supertask
5283:Recursion
5241:decidable
5075:saturated
5053:of models
4976:deductive
4971:axiomatic
4891:Hilbert's
4878:Euclidean
4859:canonical
4782:axiomatic
4714:Signature
4643:Predicate
4532:Extension
4454:Ackermann
4379:Operation
4258:Universal
4248:Recursive
4223:Singleton
4218:Inhabited
4203:Countable
4193:Types of
4177:power set
4147:partition
4064:Predicate
4010:Predicate
3925:Syllogism
3915:Soundness
3888:Inference
3878:Tautology
3780:paradoxes
3546:Paul LĂ©vy
3425:Rendering
3410:Mandelbox
3356:Julia set
3268:Hexaflake
3199:Minkowski
3119:Recursion
3102:Hausdorff
3014:Recursion
2757:127891023
2553:229461433
2545:1354-0661
2208:ThoughtCo
2159:CiteSeerX
2151:Cognition
1874:Inception
1871:The film
1701:reflexive
1645:base case
1603:factorial
1567:factorial
1553:factorial
1504:∈
1365:∈
1353:for some
1302:Base Case
1047:→
1013:→
819:→
719:(1) = 1.
442:recursion
289:Linguist
222:functions
134:base case
111:base case
109:A simple
99:Ouroboros
84:recursive
56:Recursion
5434:Feedback
5365:Logicism
5358:timeline
5334:Concrete
5193:Validity
5163:T-schema
5156:Kripke's
5151:Tarski's
5146:semantic
5136:Strength
5085:submodel
5080:spectrum
5048:function
4896:Tarski's
4885:Elements
4872:geometry
4828:Robinson
4749:variable
4734:function
4707:spectrum
4697:Sentence
4653:variable
4596:Language
4549:Relation
4510:Automata
4500:Alphabet
4484:language
4338:-jection
4316:codomain
4302:Function
4263:Universe
4233:Infinite
4137:Relation
3920:Validity
3910:Argument
3808:theorem,
3456:fractals
3343:fractals
3311:L-system
3253:T-square
3061:Fractals
2839:(2001).
2813:(2000).
2789:(1999).
2767:(2004).
2706:Archived
2470:19 April
2256:16915455
2236:Language
2181:15694646
2073:62093042
1887:See also
1880:-ception
1782:Malyutin
1711:â
1703:efforts:
1493:for all
1449:implies
681:function
527:0 is in
402:'s book
238:fractals
190:) = Fib(
176:integers
174:For all
126:ancestor
76:function
5307:Related
5104:Diagram
5002: (
4981:Hilbert
4966:Systems
4961:Theorem
4839:of the
4784:systems
4564:Formula
4559:Grammar
4475: (
4419:General
4132:Forcing
4117:Element
4037:Monadic
3812:paradox
3753:Theorem
3689:General
3405:Tricorn
3258:n-flake
3107:Packing
3090:Higuchi
3080:Assouad
2988:Algebra
2189:1599505
1653:parsers
1637:(n - 1)
1458:+ 1) =
1393:+ 1) =
834:(where
703:â 1) +
577:, then
236:), and
232:(e.g.,
224:(e.g.,
208:by the
5070:finite
4833:Skolem
4786:
4761:Theory
4729:Symbol
4719:String
4702:atomic
4579:ground
4574:closed
4569:atomic
4525:ground
4488:syntax
4384:binary
4311:domain
4228:Finite
3993:finite
3851:Logics
3810:
3758:Theory
3504:People
3454:Random
3361:Filled
3329:H tree
3248:String
3136:system
2994:
2975:
2956:
2935:
2916:
2892:
2870:
2851:
2825:
2799:
2775:
2755:
2686:5 July
2662:
2588:
2551:
2543:
2411:7 July
2386:7 July
2358:
2286:
2254:
2187:
2179:
2161:
2071:
2061:
1825:Giotto
1794:Giotto
1784:, 1892
1764:In art
1696:social
1627:return
1594:return
1559:code:
1557:Python
1430:Hence
1309:(0) =
1259:where
555:is in
475:SPARQL
182:> 1
5060:Model
4808:Peano
4665:Proof
4505:Arity
4434:Naive
4321:image
4253:Fuzzy
4213:Empty
4162:union
4107:Class
3748:Model
3738:Lemma
3696:Axiom
3580:Other
2753:S2CID
2549:S2CID
2405:(PDF)
2380:(PDF)
2263:(PDF)
2252:S2CID
2232:(PDF)
2185:S2CID
1417:)) =
1405:)) =
1384:Then
1274:that
392:index
350:is a
64:logic
49:cocoa
5183:Type
4986:list
4790:list
4767:list
4756:Term
4690:rank
4584:open
4478:list
4290:Maps
4195:sets
4054:Free
4024:list
3774:list
3701:list
2992:ISBN
2973:ISBN
2954:ISBN
2933:ISBN
2914:ISBN
2890:ISBN
2868:ISBN
2849:ISBN
2823:ISBN
2797:ISBN
2773:ISBN
2714:2016
2688:2020
2660:ISBN
2639:2015
2614:2015
2586:ISBN
2541:ISSN
2472:2020
2413:2023
2388:2023
2356:ISBN
2284:ISBN
2177:PMID
2069:OCLC
2059:ISBN
1816:The
1810:and
1780:and
1621:else
1585:>
1483:) =
1466:+ 1)
1439:) =
1425:+ 1)
1343:) =
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1028:and
695:) =
489:The
467:WINE
398:and
230:sets
198:â 2)
186:Fib(
150:The
70:and
4870:of
4852:of
4800:of
4332:Sur
4306:Map
4113:Ur-
4095:Set
2745:doi
2531:doi
2326:doi
2244:doi
2169:doi
1847:'s
1827:'s
1796:'s
1776:by
1756:in
1744:to
1665:).
1564:def
1324:= 0
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782:of
770:In
551:if
471:GNU
463:PHP
370:or
340:and
336:and
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62:to
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5256:NP
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1175:(
1172:F
1169:(
1166:f
1163:=
1160:)
1157:1
1154:+
1151:n
1148:(
1145:F
1124:a
1121:=
1118:)
1115:0
1112:(
1109:G
1088:a
1085:=
1082:)
1079:0
1076:(
1073:F
1050:X
1043:N
1039::
1036:G
1016:X
1009:N
1005::
1002:F
973:N
959:n
942:)
939:)
936:n
933:(
930:F
927:(
924:f
921:=
918:)
915:1
912:+
909:n
906:(
903:F
882:a
879:=
876:)
873:0
870:(
867:F
843:N
822:X
815:N
811::
808:F
797:X
793:X
789:f
784:X
780:a
776:X
717:F
713:F
709:n
707:(
705:F
701:n
699:(
697:F
693:n
691:(
689:F
590:N
579:n
564:N
553:n
536:N
386:.
200:.
196:n
192:n
188:n
180:n
34:.
20:)
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