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327:, 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,
282:, 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.
1687:
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
1696:
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
651:
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
1639:
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
269:
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
1657:
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.
258:
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.
301:. 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.
293:
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.
331:
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.
1688:
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
265:
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.
67:
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.
1666:
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
3576:
1697:
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.
428:
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:
1532:
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
36:. 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
201:
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.
2217:
255:
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'.
1644:
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.
285:
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:
297:
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:
1822:, 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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1123:
1087:
881:
47:
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
3762:
1537:. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.
2607:
More examples of recursion: Russian
Matryoshka dolls. Each doll is made of solid wood or is hollow and contains another Matryoshka doll inside it.
610:. The Peano Axioms define the natural numbers referring to a recursive successor function and addition and multiplication as recursive functions.
751:, 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).
4437:
3014:
Nevins, Andrew and David
Pesetsky and Cilene Rodrigues. Evidence and Argumentation: A Reply to Everett (2009). Language 85.3: 671--681 (2009)
2694:
1725:
as the process of iterating through levels of abstraction in large business entities. A common example is the recursive nature of management
602:(or Peano postulates or DedekindâPeano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician
262:
Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
4520:
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2510:
1929:
251:
being stirred into flour to produce sourdough: the recipe calls for some sourdough left over from the last time the same recipe was made.
94:
In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:
397:; the index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in
3002:
355:
Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving a
417:
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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2051:
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439:"If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to
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405:
by
Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears in
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700:â 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case
5412:
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by
Laurent SiklĂłssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and in
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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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4011:
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312:. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this. Literary
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by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?"
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393:
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20:
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1791:, 1320, recursively contains an image of itself (held up by the kneeling figure in the central panel).
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2019:
363:, in which the putative recursive step does not get closer to a base case, but instead leads to an
193:
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the
2690:
2555:
1956: 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
657:
646:
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The set of provable propositions is the smallest set of propositions satisfying these conditions.
367:. It is not unusual for such books to include a joke entry in their glossary along the lines of:
3008:
2138:
Pinker, Steven; Jackendoff, Ray (2005). "The faculty of language: What's so special about it?".
953:
Dedekind was the first to pose the problem of unique definition of set-theoretical functions on
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1977:
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Proceedings of the 40th Annual
Meeting on Association for Computational Linguistics (ACL '02)
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492:
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A plaque commemorates the
Toronto Recursive History Project of Toronto's Recursive History.
2309:
Nederhof, Mark-Jan; Satta, Giorgio (2002), "Parsing Non-recursive
Context-free Grammars",
1950: â Philosophical view that knowledge may be justified by an infinite chain of reasons
316:
can in any case be argued to be different in kind from mathematical or logical recursion.
244:
8:
5358:
5249:
5234:
5214:
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2442:", pp.50--52. Bulletin of Symbolic Logic, vol. 18, no. 1 (2012). Accessed 21 August 2023.
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270:
progressed. For this reason, recursive definitions are very rare in everyday situations.
214:
3224:
1974: â Technique of placing a copy of an image within itself, or a story within a story
28:
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5387:
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1992:
1722:
1668:
763:, this is a theorem guaranteeing that recursively defined functions exist. Given a set
729:
440:
222:
140:
4340:
2804:
2391:"Introduction to Computer Science and Programming in C; Session 8: September 25, 2008"
2192:
594:
The set of natural numbers is the smallest set satisfying the previous two properties.
309:
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5382:
5322:
5129:
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4929:
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2916:
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001).
2902:
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2811:
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2313:, Stroudsburg, PA, USA: Association for Computational Linguistics, pp. 112â119,
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2057:
2047:
1959:
1742:
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1734:
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336:
248:
218:
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102:(or cases) â a terminating scenario that does not use recursion to produce an answer
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5181:
5003:
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4624:
4422:
4372:
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3499:
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3177:
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2519:
2314:
2232:
2177:
2157:
1941:
1893:
748:
673:
619:
603:
420:
364:
60:
2075:
2001: â Cyclic structure that goes through several levels in a hierarchical system
1868:
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5307:
5261:
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5199:
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5063:
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4790:
4717:
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1953:
1914:
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623:
509:
455:
384:
3209:
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3354:
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2004:
1986:
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to recursively defined sets or functions, as in the preceding sections, yields
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434:
388:
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3182:
3147:
3132:
2533:
2524:
2506:"Reflexive discourse analysis: A methodology for the practice of reflexivity"
1971:
1935:
1908:
1827:
1823:
37:
33:
2850:
2415:
2319:
2061:
1842:(1956) is a print which depicts a distorted city containing a gallery which
618:
Another interesting example is the set of all "provable" propositions in an
5176:
5023:
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4916:
4796:
4744:
4653:
4589:
4572:
4503:
4362:
4221:
3923:
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3628:
3623:
3524:
3504:
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3194:
2169:
1998:
1848:
1833:
599:
279:
198:
1932: â Mathematical theory about infinitely iterated function composition
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1881:
672:
may be recursively defined in terms of itself. A familiar example is the
56:
48:
2955:
2439:
1624:
The function calls itself recursively on a smaller version of the input
109:â a set of rules that reduces all successive cases toward the base case.
4216:
4071:
4042:
3848:
3442:
3427:
3422:
3403:
3137:
2737:
2236:
1947:
760:
653:
1890: â Technique for defining number-theoretic functions by recursion
474:
83:
5368:
4324:
4241:
4201:
4165:
4101:
3913:
3903:
3876:
3639:
3398:
3344:
3256:
1862:
1843:
1726:
1541:
232:
There are various more tongue-in-cheek definitions of recursion; see
206:
90:, an ancient symbol depicting a serpent or dragon eating its own tail
87:
5353:
5151:
4599:
4304:
3898:
3299:
508:
The canonical example of a recursively defined set is given by the
113:
For example, the following is a recursive definition of a person's
2877:. Stanford Univ Center for the Study of Language and Information.
2644:
Physical (A)Causality: Determinism, Randomness and
Uncaused Events
4949:
3741:
3246:
3049:
409:
by Kernighan and Pike. It did not appear in the first edition of
226:
164:
2007: â Subroutine call performed as final action of a procedure
3317:
2556:"The Canadian Small BusinessâBank Interface: A Recursive Model"
2291:
Barbara Partee and Mats Rooth. 1983. In Rainer BĂ€uerle et al.,
2046:(2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers.
1813:
1782:
1641:
463:
319:
Recursion plays a crucial role not only in syntax, but also in
4493:
3839:
3684:
2869:
1962: â Result of repeatedly applying a mathematical function
1917: â Vivid and convincing dream about awakening from sleep
845:
denotes the set of natural numbers including zero) such that
52:
2853:
Recursion Theory, Gödel's Theorems, Set Theory, Model Theory
2016: â Formula that visually represents itself when graphed
3018:
2295:. Reprinted in Paul Portner and Barbara Partee, eds. 2002.
2216:
Nevins, Andrew; Pesetsky, David; Rodrigues, Cilene (2009).
1872:
to a noun to jokingly indicate the recursion of something.
1636:, analogously to the mathematical definition of factorial.
630:
If a proposition is an axiom, it is a provable proposition.
299:
Dorothy thinks that Toto suspects that Tin Man said that...
2915:
626:
which is inductively (or recursively) defined as follows:
2851:
Cori, Rene; Lascar, Daniel; Pelletier, Donald H. (2001).
1809:
is a physical artistic example of the recursive concept.
711:
459:
451:
2934:
2620:"Giotto di Bondone and assistants: Stefaneschi triptych"
1540:
A classic example of recursion is the definition of the
454:, for example, stands for "PHP Hypertext Preprocessor",
2218:"Evidence and argumentation: A reply to Everett (2009)"
2215:
2009:
Pages displaying short descriptions of redirect targets
1989: â Sentence, idea or formula that refers to itself
1982:
Pages displaying short descriptions of redirect targets
1925:
Pages displaying short descriptions of redirect targets
1923: â Higher-order function Y for which Y f = f (Y f)
1904:
Pages displaying short descriptions of redirect targets
205:
Other recursively defined mathematical objects include
2474:
Bourdieu, Pierre (1992). "Double Bind et Conversion".
2271:. Springer Science & Business Media. p. 110.
482:âa confined recursion of triangles that form a fractal
466:
denotes the "SPARQL Protocol and RDF Query Language".
2956:
Stokey, Nancy; Robert Lucas; Edward Prescott (1989).
2798:
2752:
1488:
1349:
1192:
1132:
1096:
1060:
1023:
989:
959:
890:
854:
829:
795:
789:, the theorem states that there is a unique function
576:
550:
522:
443:
than you are; then ask him or her what recursion is."
2691:"-ception â The Rice University Neologisms Database"
2774:
2383:
1628:and multiplies the result of the recursive call by
2829:
2803:
1683:to foreground the situation in which specifically
1502:
1363:
1237:
1177:
1117:
1081:
1043:
1009:
967:
935:
875:
837:
815:
584:
558:
530:
2896:
2824:
2268:Perspectives on the History of Mathematical Logic
55:. The most common application of recursion is in
5399:
2357:
2137:
16:Process of repeating items in a self-similar way
1866:has colloquialized the appending of the suffix
497:
433:." An alternative form is the following, from
3655:
3034:
2497:
2308:
1995: â 1978 musical composition by Arvo PĂ€rt
1884: â Type of algorithm in computer science
2781:Gödel, Escher, Bach: an Eternal Golden Braid
2371:. University of Illinois at Urbana-Champaign
2293:Meaning, Use, and Interpretation of Language
2511:European Journal of International Relations
1930:Infinite compositions of analytic functions
1812:Recursion has been used in paintings since
640:
613:
486:
3847:
3662:
3648:
3041:
3027:
2974:
2258:
1741:. It also encompasses the larger issue of
2899:Discrete Mathematics and Its Applications
2523:
2503:
2453:"Picture of the Day: Fractal Cauliflower"
2332:
2330:
2318:
2151:
1703:
1674:
1496:
1357:
1031:
997:
961:
831:
803:
735:
578:
552:
524:
143:is another classic example of recursion:
2720:
2473:
2297:Formal Semantics: The Essential Readings
1776:
1756:
754:
473:
419:
379:A variation is found on page 269 in the
243:
82:
32:A visual form of recursion known as the
27:
3595:List of fractals by Hausdorff dimension
2488:
2264:
2193:"What Is Recursion in English Grammar?"
1980: â Concept in computer programming
1896: â Term in theoretical linguistics
663:
450:are other examples of recursive humor.
5400:
3669:
2958:Recursive Methods in Economic Dynamics
2667:
2640:
2336:
2327:
2122:
2039:
1721:Recursion is sometimes referred to in
1513:
978:
712:Proofs involving recursive definitions
458:stands for "WINE Is Not an Emulator",
239:
3643:
3022:
2668:Cooper, Jonathan (5 September 2007).
2190:
2022: â Statement of infinite regress
1761:Recursive dolls: the original set of
308:on the basis of his claims about the
78:
71:A process that exhibits recursion is
2568:
2302:
287:Dorothy thinks witches are dangerous
233:
2935:Kernighan, B.; Ritchie, D. (1988).
2363:
2343:. Jones and Bartlett. p. 494.
1044:{\displaystyle G:\mathbb {N} \to X}
1010:{\displaystyle F:\mathbb {N} \to X}
816:{\displaystyle F:\mathbb {N} \to X}
716:Applying the standard technique of
13:
2491:Social Theory and Modern Sociology
2340:Essentials of Discrete Mathematics
2265:Drucker, Thomas (4 January 2008).
1968: â Form of mathematical proof
350:
14:
5434:
3577:How Long Is the Coast of Britain?
2996:
2724:(1960). "Recursive Programming".
2697:from the original on July 5, 2017
2014:Tupper's self-referential formula
1503:{\displaystyle n\in \mathbb {N} }
1364:{\displaystyle k\in \mathbb {N} }
606:and by the Italian mathematician
469:
462:stands for "GNU's not Unix", and
5381:
2593:
2476:Pour Une Anthropologie RĂ©flexive
728:widely used to derive proofs in
407:The UNIX Programming Environment
2713:
2683:
2661:
2634:
2612:
2587:
2562:
2548:
2482:
2467:
2445:
2432:
2408:
2162:10.1016/j.cognition.2004.08.004
1911: â Recursive visual effect
1902: â Poem by Edgar Allan Poe
1629:
1625:
724:â a powerful generalization of
622:that are defined in terms of a
3601:The Fractal Geometry of Nature
2991:, first chapter on set theory.
2285:
2209:
2184:
2131:
2116:
2092:
2068:
2033:
1710:
1238:{\displaystyle G(n+1)=f(G(n))}
1232:
1229:
1223:
1217:
1208:
1196:
1178:{\displaystyle F(n+1)=f(F(n))}
1172:
1169:
1163:
1157:
1148:
1136:
1106:
1100:
1070:
1064:
1035:
1001:
936:{\displaystyle F(n+1)=f(F(n))}
930:
927:
921:
915:
906:
894:
864:
858:
807:
273:
1:
5342:History of mathematical logic
2026:
1944: â Philosophical problem
1900:A Dream Within a Dream (poem)
1855:
1846:contains the picture, and so
1661:
656:is a subdivision rule, as is
5267:Primitive recursive function
3048:
2960:. Harvard University Press.
2873:; Moss, Lawrence S. (1996).
2366:"CS 173:Discrete Structures"
2076:"Peano axioms | mathematics"
1520:Recursion (computer science)
968:{\displaystyle \mathbb {N} }
838:{\displaystyle \mathbb {N} }
585:{\displaystyle \mathbb {N} }
559:{\displaystyle \mathbb {N} }
531:{\displaystyle \mathbb {N} }
498:Example: the natural numbers
117:. One's ancestor is either:
7:
3617:Chaos: Making a New Science
2855:. Oxford University Press.
2416:"recursion - Google Search"
1875:
1679:Authors use the concept of
598:In mathematical logic, the
40:tin, designed by Jan Misset
10:
5439:
4331:SchröderâBernstein theorem
4058:Monadic predicate calculus
3717:Foundations of mathematics
3009:Zip Files All The Way Down
2938:The C programming Language
2918:Introduction to Algorithms
2897:Rosen, Kenneth H. (2002).
2832:Logic, Sets, and Recursion
2504:Alejandro, Audrey (2021).
2043:Logic, sets, and recursion
2040:Causey, Robert L. (2006).
1888:Course-of-values recursion
1794:
1714:
1528:technique, this is called
1517:
1308:so the equality holds for
644:
501:
490:
413:. The joke is part of the
411:The C Programming Language
394:The C Programming Language
321:natural language semantics
21:Recursion (disambiguation)
18:
5377:
5364:Philosophy of mathematics
5313:Automated theorem proving
5295:
5190:
5022:
4915:
4767:
4484:
4460:
4438:Von NeumannâBernaysâGödel
4383:
4277:
4181:
4079:
4070:
3997:
3932:
3838:
3760:
3677:
3568:
3492:
3441:
3412:
3328:
3298:
3280:
3121:
3056:
2489:Giddens, Anthony (1987).
2100:"Definition of RECURSIVE"
1938: â Programming idiom
1752:
1282:for all natural numbers
131:One's parent's ancestor (
3005:- tutorial by Alan Gauld
2889:- offers a treatment of
2836:. Jones & Bartlett.
2647:. Springer. p. 12.
2525:10.1177/1354066120969789
2440:In Praise of Replacement
2020:Turtles all the way down
1550:
1544:function, given here in
641:Finite subdivision rules
614:Example: Proof procedure
487:Recursively defined sets
343:that contains recursive
289:, in which the sentence
5014:Self-verifying theories
4835:Tarski's axiomatization
3786:Tarski's undefinability
3781:incompleteness theorems
2901:. McGraw-Hill College.
2569:Beer, Stafford (1972).
2320:10.3115/1073083.1073104
2123:Pinker, Steven (1994).
2104:www.merriam-webster.com
2080:Encyclopedia Britannica
946:for any natural number
658:barycentric subdivision
647:Finite subdivision rule
5388:Mathematics portal
4999:Proof of impossibility
4647:propositional variable
3957:Propositional calculus
3609:The Beauty of Fractals
2337:Hunter, David (2011).
1978:Reentrant (subroutine)
1966:Mathematical induction
1921:Fixed point combinator
1792:
1774:
1717:Management cybernetics
1708:
1675:In the social sciences
1652:closed-form expression
1504:
1365:
1261:mathematical induction
1239:
1179:
1119:
1118:{\displaystyle G(0)=a}
1083:
1082:{\displaystyle F(0)=a}
1045:
1011:
969:
937:
877:
876:{\displaystyle F(0)=a}
839:
817:
736:Recursive optimization
732:and computer science.
726:mathematical induction
586:
560:
532:
483:
425:
415:functional programming
252:
91:
41:
5413:Theory of computation
5257:Kolmogorov complexity
5210:Computably enumerable
5110:Model complete theory
4902:Principia Mathematica
3962:Propositional formula
3791:BanachâTarski paradox
2800:Shoenfield, Joseph R.
2754:Johnsonbaugh, Richard
2726:Numerische Mathematik
2670:"Art and Mathematics"
2641:Svozil, Karl (2018).
2396:. Columbia University
2125:The Language Instinct
1780:
1760:
1715:Further information:
1694:
1632:, until reaching the
1505:
1366:
1240:
1180:
1120:
1084:
1046:
1012:
970:
938:
878:
840:
818:
755:The recursion theorem
587:
561:
533:
504:Closure (mathematics)
477:
423:
291:witches are dangerous
247:
86:
31:
5205:ChurchâTuring thesis
5192:Computability theory
4401:continuum hypothesis
3919:Square of opposition
3777:Gödel's completeness
3555:Lewis Fry Richardson
3550:Hamid Naderi Yeganeh
3340:Burning Ship fractal
3272:Weierstrass function
2758:Discrete Mathematics
2191:Nordquist, Richard.
1826:, an example of the
1819:Stefaneschi Triptych
1788:Stefaneschi Triptych
1747:corporate governance
1648:Recurrence relations
1526:computer programming
1486:
1347:
1259:It can be proved by
1190:
1130:
1094:
1058:
1021:
987:
957:
888:
852:
827:
793:
722:structural induction
664:Functional recursion
574:
548:
520:
493:Recursive definition
383:of some editions of
215:recurrence relations
19:For other uses, see
5359:Mathematical object
5250:P versus NP problem
5215:Computable function
5009:Reverse mathematics
4935:Logical consequence
4812:primitive recursive
4807:elementary function
4580:Free/bound variable
4433:TarskiâGrothendieck
3952:Logical connectives
3882:Logical equivalence
3732:Logical consequence
3313:Space-filling curve
3290:Multifractal system
3173:Space-filling curve
3158:Sierpinski triangle
2975:Hungerford (1980).
2776:Hofstadter, Douglas
2722:Dijkstra, Edsger W.
2693:. Rice University.
1797:Mathematics and art
1535:dynamic programming
1514:In computer science
983:Take two functions
979:Proof of uniqueness
741:Dynamic programming
480:Sierpinski triangle
357:circular definition
240:Informal definition
5157:Transfer principle
5120:Semantics of logic
5105:Categorical theory
5081:Non-standard model
4595:Logical connective
3722:Information theory
3671:Mathematical logic
3540:Aleksandr Lyapunov
3520:Desmond Paul Henry
3484:Self-avoiding walk
3479:Percolation theory
3123:Iterated function
3064:Fractal dimensions
2810:. A K Peters Ltd.
2738:10.1007/BF01386232
2478:. Paris: Le Seuil.
2455:. 28 December 2012
2237:10.1353/lan.0.0140
1993:Spiegel im Spiegel
1793:
1775:
1723:management science
1702:Audrey Alejandro,
1669:Romanesco broccoli
1530:divide and conquer
1500:
1361:
1235:
1175:
1115:
1079:
1041:
1007:
965:
933:
873:
835:
813:
743:is an approach to
730:mathematical logic
582:
556:
528:
484:
448:Recursive acronyms
441:Douglas Hofstadter
426:
253:
223:Cantor ternary set
141:Fibonacci sequence
92:
79:Formal definitions
42:
5395:
5394:
5327:Abstract category
5130:Theories of truth
4940:Rule of inference
4930:Natural deduction
4911:
4910:
4456:
4455:
4161:Cartesian product
4066:
4065:
3972:Many-valued logic
3947:Boolean functions
3830:Russell's paradox
3805:diagonal argument
3702:First-order logic
3637:
3636:
3583:Coastline paradox
3560:WacĆaw SierpiĆski
3545:Benoit Mandelbrot
3469:Fractal landscape
3377:Misiurewicz point
3282:Strange attractor
3163:Apollonian gasket
3153:Sierpinski carpet
2986:978-0-387-90518-1
2967:978-0-674-75096-8
2948:978-0-13-110362-7
2941:. Prentice Hall.
2927:978-0-262-03293-3
2908:978-0-07-293033-7
2884:978-0-19-850050-6
2862:978-0-19-850050-6
2843:978-0-7637-1695-0
2826:Causey, Robert L.
2817:978-1-56881-149-9
2791:978-0-465-02656-2
2767:978-0-13-117686-7
2760:. Prentice Hall.
2571:Brain Of The Firm
2278:978-0-8176-4768-1
2127:. William Morrow.
1960:Iterated function
1743:capital structure
1739:middle management
1735:senior management
1252:is an element of
337:recursive grammar
249:Sourdough starter
5430:
5386:
5385:
5337:History of logic
5332:Category of sets
5225:Decision problem
5004:Ordinal analysis
4945:Sequent calculus
4843:Boolean algebras
4783:
4782:
4757:
4728:logical/constant
4482:
4481:
4468:
4391:ZermeloâFraenkel
4142:Set operations:
4077:
4076:
4014:
3845:
3844:
3825:LöwenheimâSkolem
3712:Formal semantics
3664:
3657:
3650:
3641:
3640:
3500:Michael Barnsley
3367:Lyapunov fractal
3225:SierpiĆski curve
3178:Blancmange curve
3043:
3036:
3029:
3020:
3019:
2990:
2971:
2952:
2931:
2912:
2888:
2866:
2847:
2835:
2821:
2809:
2806:Recursion Theory
2795:
2771:
2749:
2707:
2706:
2704:
2702:
2687:
2681:
2680:
2678:
2676:
2665:
2659:
2658:
2638:
2632:
2631:
2629:
2627:
2616:
2610:
2609:
2604:
2602:
2591:
2585:
2584:
2566:
2560:
2559:
2558:. SAGE Journals.
2552:
2546:
2545:
2527:
2501:
2495:
2494:
2486:
2480:
2479:
2471:
2465:
2464:
2462:
2460:
2449:
2443:
2436:
2430:
2429:
2427:
2426:
2412:
2406:
2405:
2403:
2401:
2395:
2387:
2381:
2380:
2378:
2376:
2370:
2361:
2355:
2354:
2334:
2325:
2323:
2322:
2306:
2300:
2289:
2283:
2282:
2262:
2256:
2255:
2253:
2247:. Archived from
2222:
2213:
2207:
2206:
2204:
2203:
2188:
2182:
2181:
2155:
2135:
2129:
2128:
2120:
2114:
2113:
2111:
2110:
2096:
2090:
2089:
2087:
2086:
2072:
2066:
2065:
2037:
2010:
1983:
1942:Infinite regress
1926:
1905:
1894:Digital infinity
1763:Matryoshka dolls
1706:
1704:Alejandro (2021)
1631:
1627:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1509:
1507:
1506:
1501:
1499:
1481:
1456:
1437:
1415:
1372:
1370:
1368:
1367:
1362:
1360:
1341:
1314:
1307:
1285:
1281:
1255:
1251:
1244:
1242:
1241:
1236:
1184:
1182:
1181:
1176:
1124:
1122:
1121:
1116:
1088:
1086:
1085:
1080:
1050:
1048:
1047:
1042:
1034:
1016:
1014:
1013:
1008:
1000:
974:
972:
971:
966:
964:
949:
942:
940:
939:
934:
882:
880:
879:
874:
844:
842:
841:
836:
834:
822:
820:
819:
814:
806:
788:
774:
770:
766:
749:Bellman equation
674:Fibonacci number
620:axiomatic system
604:Richard Dedekind
591:
589:
588:
583:
581:
565:
563:
562:
557:
555:
537:
535:
534:
529:
527:
365:infinite regress
345:production rules
188:
172:
157:
149:
61:computer science
5438:
5437:
5433:
5432:
5431:
5429:
5428:
5427:
5398:
5397:
5396:
5391:
5380:
5373:
5318:Category theory
5308:Algebraic logic
5291:
5262:Lambda calculus
5200:Church encoding
5186:
5162:Truth predicate
5018:
4984:Complete theory
4907:
4776:
4772:
4768:
4763:
4755:
4475: and
4471:
4466:
4452:
4428:New Foundations
4396:axiom of choice
4379:
4341:Gödel numbering
4281: and
4273:
4177:
4062:
4012:
3993:
3942:Boolean algebra
3928:
3892:Equiconsistency
3857:Classical logic
3834:
3815:Halting problem
3803: and
3779: and
3767: and
3766:
3761:Theorems (
3756:
3673:
3668:
3638:
3633:
3564:
3515:Felix Hausdorff
3488:
3452:Brownian motion
3437:
3408:
3331:
3324:
3294:
3276:
3267:Pythagoras tree
3124:
3117:
3113:Self-similarity
3057:Characteristics
3052:
3047:
2999:
2994:
2987:
2968:
2949:
2928:
2909:
2885:
2875:Vicious Circles
2863:
2844:
2818:
2792:
2784:. Basic Books.
2768:
2716:
2711:
2710:
2700:
2698:
2689:
2688:
2684:
2674:
2672:
2666:
2662:
2655:
2639:
2635:
2625:
2623:
2618:
2617:
2613:
2600:
2598:
2592:
2588:
2581:
2567:
2563:
2554:
2553:
2549:
2502:
2498:
2493:. Polity Press.
2487:
2483:
2472:
2468:
2458:
2456:
2451:
2450:
2446:
2437:
2433:
2424:
2422:
2414:
2413:
2409:
2399:
2397:
2393:
2389:
2388:
2384:
2374:
2372:
2368:
2364:Shaffer, Eric.
2362:
2358:
2351:
2335:
2328:
2307:
2303:
2290:
2286:
2279:
2263:
2259:
2251:
2220:
2214:
2210:
2201:
2199:
2189:
2185:
2153:10.1.1.116.7784
2136:
2132:
2121:
2117:
2108:
2106:
2098:
2097:
2093:
2084:
2082:
2074:
2073:
2069:
2054:
2038:
2034:
2029:
2008:
1981:
1954:Infinity mirror
1924:
1915:False awakening
1903:
1878:
1858:
1807:Matryoshka doll
1803:
1801:Infinity mirror
1755:
1731:line management
1729:, ranging from
1719:
1713:
1707:
1701:
1677:
1664:
1622:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1522:
1516:
1495:
1487:
1484:
1483:
1464:
1439:
1420:
1374:
1356:
1348:
1345:
1344:
1343:
1324:
1309:
1294:
1283:
1264:
1253:
1249:
1191:
1188:
1187:
1131:
1128:
1127:
1095:
1092:
1091:
1059:
1056:
1055:
1030:
1022:
1019:
1018:
996:
988:
985:
984:
981:
960:
958:
955:
954:
947:
889:
886:
885:
853:
850:
849:
830:
828:
825:
824:
802:
794:
791:
790:
776:
775:and a function
772:
768:
764:
757:
738:
714:
666:
649:
643:
624:proof procedure
616:
577:
575:
572:
571:
551:
549:
546:
545:
523:
521:
518:
517:
510:natural numbers
506:
500:
495:
489:
472:
399:Let's talk Lisp
385:Brian Kernighan
353:
351:Recursive humor
310:PirahĂŁ language
276:
242:
234:recursive humor
195:natural numbers
174:
167:
158:as base case 2,
155:
150:as base case 1,
147:
81:
24:
17:
12:
11:
5:
5436:
5426:
5425:
5420:
5418:Self-reference
5415:
5410:
5393:
5392:
5378:
5375:
5374:
5372:
5371:
5366:
5361:
5356:
5351:
5350:
5349:
5339:
5334:
5329:
5320:
5315:
5310:
5305:
5303:Abstract logic
5299:
5297:
5293:
5292:
5290:
5289:
5284:
5282:Turing machine
5279:
5274:
5269:
5264:
5259:
5254:
5253:
5252:
5247:
5242:
5237:
5232:
5222:
5220:Computable set
5217:
5212:
5207:
5202:
5196:
5194:
5188:
5187:
5185:
5184:
5179:
5174:
5169:
5164:
5159:
5154:
5149:
5148:
5147:
5142:
5137:
5127:
5122:
5117:
5115:Satisfiability
5112:
5107:
5102:
5101:
5100:
5090:
5089:
5088:
5078:
5077:
5076:
5071:
5066:
5061:
5056:
5046:
5045:
5044:
5039:
5032:Interpretation
5028:
5026:
5020:
5019:
5017:
5016:
5011:
5006:
5001:
4996:
4986:
4981:
4980:
4979:
4978:
4977:
4967:
4962:
4952:
4947:
4942:
4937:
4932:
4927:
4921:
4919:
4913:
4912:
4909:
4908:
4906:
4905:
4897:
4896:
4895:
4894:
4889:
4888:
4887:
4882:
4877:
4857:
4856:
4855:
4853:minimal axioms
4850:
4839:
4838:
4837:
4826:
4825:
4824:
4819:
4814:
4809:
4804:
4799:
4786:
4784:
4765:
4764:
4762:
4761:
4760:
4759:
4747:
4742:
4741:
4740:
4735:
4730:
4725:
4715:
4710:
4705:
4700:
4699:
4698:
4693:
4683:
4682:
4681:
4676:
4671:
4666:
4656:
4651:
4650:
4649:
4644:
4639:
4629:
4628:
4627:
4622:
4617:
4612:
4607:
4602:
4592:
4587:
4582:
4577:
4576:
4575:
4570:
4565:
4560:
4550:
4545:
4543:Formation rule
4540:
4535:
4534:
4533:
4528:
4518:
4517:
4516:
4506:
4501:
4496:
4491:
4485:
4479:
4462:Formal systems
4458:
4457:
4454:
4453:
4451:
4450:
4445:
4440:
4435:
4430:
4425:
4420:
4415:
4410:
4405:
4404:
4403:
4398:
4387:
4385:
4381:
4380:
4378:
4377:
4376:
4375:
4365:
4360:
4359:
4358:
4351:Large cardinal
4348:
4343:
4338:
4333:
4328:
4314:
4313:
4312:
4307:
4302:
4287:
4285:
4275:
4274:
4272:
4271:
4270:
4269:
4264:
4259:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4188:
4186:
4179:
4178:
4176:
4175:
4174:
4173:
4168:
4163:
4158:
4153:
4148:
4140:
4139:
4138:
4133:
4123:
4118:
4116:Extensionality
4113:
4111:Ordinal number
4108:
4098:
4093:
4092:
4091:
4080:
4074:
4068:
4067:
4064:
4063:
4061:
4060:
4055:
4050:
4045:
4040:
4035:
4030:
4029:
4028:
4018:
4017:
4016:
4003:
4001:
3995:
3994:
3992:
3991:
3990:
3989:
3984:
3979:
3969:
3964:
3959:
3954:
3949:
3944:
3938:
3936:
3930:
3929:
3927:
3926:
3921:
3916:
3911:
3906:
3901:
3896:
3895:
3894:
3884:
3879:
3874:
3869:
3864:
3859:
3853:
3851:
3842:
3836:
3835:
3833:
3832:
3827:
3822:
3817:
3812:
3807:
3795:Cantor's
3793:
3788:
3783:
3773:
3771:
3758:
3757:
3755:
3754:
3749:
3744:
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3693:
3692:
3681:
3679:
3675:
3674:
3667:
3666:
3659:
3652:
3644:
3635:
3634:
3632:
3631:
3626:
3621:
3613:
3605:
3597:
3592:
3587:
3586:
3585:
3572:
3570:
3566:
3565:
3563:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3530:Helge von Koch
3527:
3522:
3517:
3512:
3507:
3502:
3496:
3494:
3490:
3489:
3487:
3486:
3481:
3476:
3471:
3466:
3465:
3464:
3462:Brownian motor
3459:
3448:
3446:
3439:
3438:
3436:
3435:
3433:Pickover stalk
3430:
3425:
3419:
3417:
3410:
3409:
3407:
3406:
3401:
3396:
3391:
3389:Newton fractal
3386:
3381:
3380:
3379:
3372:Mandelbrot set
3369:
3364:
3363:
3362:
3357:
3355:Newton fractal
3352:
3342:
3336:
3334:
3326:
3325:
3323:
3322:
3321:
3320:
3310:
3308:Fractal canopy
3304:
3302:
3296:
3295:
3293:
3292:
3286:
3284:
3278:
3277:
3275:
3274:
3269:
3264:
3259:
3254:
3252:Vicsek fractal
3249:
3244:
3239:
3234:
3233:
3232:
3227:
3222:
3217:
3212:
3207:
3202:
3197:
3192:
3191:
3190:
3180:
3170:
3168:Fibonacci word
3165:
3160:
3155:
3150:
3145:
3143:Koch snowflake
3140:
3135:
3129:
3127:
3119:
3118:
3116:
3115:
3110:
3105:
3104:
3103:
3098:
3093:
3088:
3083:
3082:
3081:
3071:
3060:
3058:
3054:
3053:
3046:
3045:
3038:
3031:
3023:
3017:
3016:
3011:
3006:
2998:
2997:External links
2995:
2993:
2992:
2985:
2972:
2966:
2953:
2947:
2932:
2926:
2913:
2907:
2894:
2883:
2867:
2861:
2848:
2842:
2822:
2816:
2796:
2790:
2772:
2766:
2750:
2732:(1): 312â318.
2717:
2715:
2712:
2709:
2708:
2682:
2660:
2653:
2633:
2611:
2586:
2580:978-0471948391
2579:
2561:
2547:
2496:
2481:
2466:
2444:
2438:A. Kanamori, "
2431:
2420:www.google.com
2407:
2382:
2356:
2349:
2326:
2301:
2284:
2277:
2257:
2254:on 2012-01-06.
2231:(3): 671â681.
2208:
2183:
2146:(2): 201â236.
2130:
2115:
2091:
2067:
2052:
2031:
2030:
2028:
2025:
2024:
2023:
2017:
2011:
2005:Tail recursion
2002:
1996:
1990:
1987:Self-reference
1984:
1975:
1969:
1963:
1957:
1951:
1945:
1939:
1933:
1927:
1918:
1912:
1906:
1897:
1891:
1885:
1877:
1874:
1857:
1854:
1781:Front face of
1754:
1751:
1712:
1709:
1699:
1676:
1673:
1663:
1660:
1551:
1518:Main article:
1515:
1512:
1498:
1494:
1491:
1463:By induction,
1461:
1460:
1459:
1458:
1359:
1355:
1352:
1321:Inductive Step
1317:
1316:
1246:
1245:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1185:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1125:
1114:
1111:
1108:
1105:
1102:
1099:
1089:
1078:
1075:
1072:
1069:
1066:
1063:
1040:
1037:
1033:
1029:
1026:
1006:
1003:
999:
995:
992:
980:
977:
963:
944:
943:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
883:
872:
869:
866:
863:
860:
857:
833:
812:
809:
805:
801:
798:
756:
753:
737:
734:
718:proof by cases
713:
710:
665:
662:
645:Main article:
642:
639:
638:
637:
634:
631:
615:
612:
608:Giuseppe Peano
596:
595:
592:
580:
554:
538:
526:
499:
496:
491:Main article:
488:
485:
471:
470:In mathematics
468:
435:Andrew Plotkin
403:Software Tools
389:Dennis Ritchie
377:
376:
361:self-reference
352:
349:
341:formal grammar
314:self-reference
306:Daniel Everett
275:
272:
241:
238:
191:
190:
160:
159:
152:
151:
137:
136:
133:recursive step
129:
121:One's parent (
111:
110:
107:recursive step
103:
80:
77:
15:
9:
6:
4:
3:
2:
5435:
5424:
5421:
5419:
5416:
5414:
5411:
5409:
5406:
5405:
5403:
5390:
5389:
5384:
5376:
5370:
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5348:
5345:
5344:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5300:
5298:
5294:
5288:
5285:
5283:
5280:
5278:
5277:Recursive set
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5197:
5195:
5193:
5189:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5146:
5143:
5141:
5138:
5136:
5133:
5132:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5099:
5096:
5095:
5094:
5091:
5087:
5086:of arithmetic
5084:
5083:
5082:
5079:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5051:
5050:
5047:
5043:
5040:
5038:
5035:
5034:
5033:
5030:
5029:
5027:
5025:
5021:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4994:
4993:from ZFC
4990:
4987:
4985:
4982:
4976:
4973:
4972:
4971:
4968:
4966:
4963:
4961:
4958:
4957:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4922:
4920:
4918:
4914:
4904:
4903:
4899:
4898:
4893:
4892:non-Euclidean
4890:
4886:
4883:
4881:
4878:
4876:
4875:
4871:
4870:
4868:
4865:
4864:
4862:
4858:
4854:
4851:
4849:
4846:
4845:
4844:
4840:
4836:
4833:
4832:
4831:
4827:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4794:
4792:
4788:
4787:
4785:
4780:
4774:
4769:Example
4766:
4758:
4753:
4752:
4751:
4748:
4746:
4743:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4720:
4719:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4697:
4694:
4692:
4689:
4688:
4687:
4684:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4661:
4660:
4657:
4655:
4652:
4648:
4645:
4643:
4640:
4638:
4635:
4634:
4633:
4630:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4597:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4555:
4554:
4551:
4549:
4546:
4544:
4541:
4539:
4536:
4532:
4529:
4527:
4526:by definition
4524:
4523:
4522:
4519:
4515:
4512:
4511:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4486:
4483:
4480:
4478:
4474:
4469:
4463:
4459:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4413:KripkeâPlatek
4411:
4409:
4406:
4402:
4399:
4397:
4394:
4393:
4392:
4389:
4388:
4386:
4382:
4374:
4371:
4370:
4369:
4366:
4364:
4361:
4357:
4354:
4353:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4326:
4322:
4318:
4315:
4311:
4308:
4306:
4303:
4301:
4298:
4297:
4296:
4292:
4289:
4288:
4286:
4284:
4280:
4276:
4268:
4265:
4263:
4260:
4258:
4257:constructible
4255:
4254:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4189:
4187:
4185:
4180:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4143:
4141:
4137:
4134:
4132:
4129:
4128:
4127:
4124:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4103:
4099:
4097:
4094:
4090:
4087:
4086:
4085:
4082:
4081:
4078:
4075:
4073:
4069:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4027:
4024:
4023:
4022:
4019:
4015:
4010:
4009:
4008:
4005:
4004:
4002:
4000:
3996:
3988:
3985:
3983:
3980:
3978:
3975:
3974:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3939:
3937:
3935:
3934:Propositional
3931:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3893:
3890:
3889:
3888:
3885:
3883:
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3862:Logical truth
3860:
3858:
3855:
3854:
3852:
3850:
3846:
3843:
3841:
3837:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3802:
3798:
3794:
3792:
3789:
3787:
3784:
3782:
3778:
3775:
3774:
3772:
3770:
3764:
3759:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3691:
3688:
3687:
3686:
3683:
3682:
3680:
3676:
3672:
3665:
3660:
3658:
3653:
3651:
3646:
3645:
3642:
3630:
3627:
3625:
3622:
3619:
3618:
3614:
3611:
3610:
3606:
3603:
3602:
3598:
3596:
3593:
3591:
3588:
3584:
3581:
3580:
3578:
3574:
3573:
3571:
3567:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3497:
3495:
3491:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3463:
3460:
3458:
3457:Brownian tree
3455:
3454:
3453:
3450:
3449:
3447:
3444:
3440:
3434:
3431:
3429:
3426:
3424:
3421:
3420:
3418:
3415:
3411:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3384:Multibrot set
3382:
3378:
3375:
3374:
3373:
3370:
3368:
3365:
3361:
3360:Douady rabbit
3358:
3356:
3353:
3351:
3348:
3347:
3346:
3343:
3341:
3338:
3337:
3335:
3333:
3327:
3319:
3316:
3315:
3314:
3311:
3309:
3306:
3305:
3303:
3301:
3297:
3291:
3288:
3287:
3285:
3283:
3279:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3231:
3230:Z-order curve
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3200:Hilbert curve
3198:
3196:
3193:
3189:
3186:
3185:
3184:
3183:De Rham curve
3181:
3179:
3176:
3175:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3148:Menger sponge
3146:
3144:
3141:
3139:
3136:
3134:
3133:Barnsley fern
3131:
3130:
3128:
3126:
3120:
3114:
3111:
3109:
3106:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3080:
3077:
3076:
3075:
3072:
3070:
3067:
3066:
3065:
3062:
3061:
3059:
3055:
3051:
3044:
3039:
3037:
3032:
3030:
3025:
3024:
3021:
3015:
3012:
3010:
3007:
3004:
3001:
3000:
2988:
2982:
2978:
2973:
2969:
2963:
2959:
2954:
2950:
2944:
2940:
2939:
2933:
2929:
2923:
2919:
2914:
2910:
2904:
2900:
2895:
2892:
2886:
2880:
2876:
2872:
2868:
2864:
2858:
2854:
2849:
2845:
2839:
2834:
2833:
2827:
2823:
2819:
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2654:9783319708157
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2622:. The Vatican
2621:
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2594:Tang, Daisy.
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2018:
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1972:Mise en abyme
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1940:
1937:
1936:Infinite loop
1934:
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1909:Droste effect
1907:
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1839:Print Gallery
1835:
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1828:Mise en abyme
1825:
1824:Droste effect
1821:
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870:
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810:
799:
796:
787:
783:
779:
767:, an element
762:
752:
750:
746:
742:
733:
731:
727:
723:
719:
709:
707:
703:
699:
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374:
373:see Recursion
370:
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358:
348:
346:
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333:
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317:
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97:
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89:
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76:
74:
69:
66:
62:
58:
54:
50:
46:
39:
35:
34:Droste effect
30:
26:
22:
5379:
5271:
5177:Ultraproduct
5024:Model theory
4989:Independence
4925:Formal proof
4917:Proof theory
4900:
4873:
4830:real numbers
4802:second-order
4713:Substitution
4590:Metalanguage
4531:conservative
4504:Axiom schema
4448:Constructive
4418:MorseâKelley
4384:Set theories
4363:Aleph number
4356:inaccessible
4262:Grothendieck
4146:intersection
4033:Higher-order
4021:Second-order
3967:Truth tables
3924:Venn diagram
3707:Formal proof
3629:Chaos theory
3624:Kaleidoscope
3615:
3607:
3599:
3525:Gaston Julia
3505:Georg Cantor
3330:Escape-time
3262:Gosper curve
3210:LĂ©vy C curve
3195:Dragon curve
3107:
3074:Box-counting
2979:. Springer.
2976:
2957:
2937:
2917:
2898:
2874:
2871:Barwise, Jon
2852:
2831:
2805:
2780:
2757:
2729:
2725:
2714:Bibliography
2701:December 23,
2699:. Retrieved
2685:
2673:. Retrieved
2663:
2643:
2636:
2626:16 September
2624:. Retrieved
2614:
2606:
2601:24 September
2599:. Retrieved
2589:
2570:
2564:
2550:
2515:
2509:
2499:
2490:
2484:
2475:
2469:
2457:. Retrieved
2447:
2434:
2423:. Retrieved
2419:
2410:
2398:. Retrieved
2385:
2373:. Retrieved
2359:
2339:
2310:
2304:
2299:. Blackwell.
2296:
2292:
2287:
2267:
2260:
2249:the original
2228:
2224:
2211:
2200:. Retrieved
2196:
2186:
2143:
2139:
2133:
2124:
2118:
2107:. Retrieved
2103:
2094:
2083:. Retrieved
2079:
2070:
2042:
2035:
1999:Strange loop
1867:
1861:
1859:
1849:ad infinitum
1847:
1837:
1834:M. C. Escher
1832:
1817:
1811:
1804:
1786:
1767:Zvyozdochkin
1720:
1695:
1684:
1680:
1678:
1665:
1656:
1646:
1638:
1623:
1539:
1523:
1477:
1473:
1469:
1465:
1462:
1452:
1448:
1444:
1440:
1433:
1429:
1425:
1421:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1337:
1333:
1329:
1325:
1320:
1310:
1303:
1299:
1295:
1290:
1277:
1273:
1269:
1265:
1258:
1247:
982:
952:
945:
785:
781:
777:
758:
745:optimization
739:
715:
705:
704:(0) = 0 and
701:
697:
693:
689:
685:
681:
677:
667:
650:
617:
600:Peano axioms
597:
567:
541:
507:
446:
438:
430:
427:
410:
406:
402:
398:
392:
378:
372:
354:
334:
328:
324:
318:
303:
298:
296:
290:
286:
284:
280:Noam Chomsky
277:
268:
264:
261:
257:
254:
231:
204:
199:Peano axioms
192:
184:
180:
176:
168:
138:
132:
126:
122:
114:
112:
106:
99:
93:
72:
70:
44:
43:
25:
5287:Type theory
5235:undecidable
5167:Truth value
5054:equivalence
4733:non-logical
4346:Enumeration
4336:Isomorphism
4283:cardinality
4267:Von Neumann
4232:Ultrafilter
4197:Uncountable
4131:equivalence
4048:Quantifiers
4038:Fixed-point
4007:First-order
3887:Consistency
3872:Proposition
3849:Traditional
3820:Lindström's
3810:Compactness
3752:Type theory
3697:Cardinality
3620:(1987 book)
3612:(1986 book)
3604:(1982 book)
3590:Fractal art
3510:Bill Gosper
3474:LĂ©vy flight
3220:Peano curve
3215:Moore curve
3101:Topological
3086:Correlation
2891:corecursion
2596:"Recursion"
1882:Corecursion
1844:recursively
1830:technique.
1727:hierarchies
1711:In business
1681:recursivity
1051:such that:
371:Recursion,
323:. The word
274:In language
183:â 1) + Fib(
57:mathematics
49:linguistics
5402:Categories
5098:elementary
4791:arithmetic
4659:Quantifier
4637:functional
4509:Expression
4227:Transitive
4171:identities
4156:complement
4089:hereditary
4072:Set theory
3428:Orbit trap
3423:Buddhabrot
3416:techniques
3404:Mandelbulb
3205:Koch curve
3138:Cantor set
2920:. Mit Pr.
2518:(1): 171.
2425:2019-10-24
2202:2019-10-24
2109:2019-10-24
2085:2019-10-24
2027:References
1948:Infinitism
1856:In culture
1795:See also:
1737: via
1662:In biology
1323:: Suppose
761:set theory
676:sequence:
654:Cantor set
570:+ 1 is in
502:See also:
207:factorials
156:Fib(1) = 1
148:Fib(0) = 0
63:, where a
5408:Recursion
5369:Supertask
5272:Recursion
5230:decidable
5064:saturated
5042:of models
4965:deductive
4960:axiomatic
4880:Hilbert's
4867:Euclidean
4848:canonical
4771:axiomatic
4703:Signature
4632:Predicate
4521:Extension
4443:Ackermann
4368:Operation
4247:Universal
4237:Recursive
4212:Singleton
4207:Inhabited
4192:Countable
4182:Types of
4166:power set
4136:partition
4053:Predicate
3999:Predicate
3914:Syllogism
3904:Soundness
3877:Inference
3867:Tautology
3769:paradoxes
3535:Paul LĂ©vy
3414:Rendering
3399:Mandelbox
3345:Julia set
3257:Hexaflake
3188:Minkowski
3108:Recursion
3091:Hausdorff
3003:Recursion
2746:127891023
2542:229461433
2534:1354-0661
2197:ThoughtCo
2148:CiteSeerX
2140:Cognition
1863:Inception
1860:The film
1690:reflexive
1634:base case
1592:factorial
1556:factorial
1542:factorial
1493:∈
1354:∈
1342:for some
1291:Base Case
1036:→
1002:→
808:→
708:(1) = 1.
431:recursion
278:Linguist
211:functions
123:base case
100:base case
98:A simple
88:Ouroboros
73:recursive
45:Recursion
5423:Feedback
5354:Logicism
5347:timeline
5323:Concrete
5182:Validity
5152:T-schema
5145:Kripke's
5140:Tarski's
5135:semantic
5125:Strength
5074:submodel
5069:spectrum
5037:function
4885:Tarski's
4874:Elements
4861:geometry
4817:Robinson
4738:variable
4723:function
4696:spectrum
4686:Sentence
4642:variable
4585:Language
4538:Relation
4499:Automata
4489:Alphabet
4473:language
4327:-jection
4305:codomain
4291:Function
4252:Universe
4222:Infinite
4126:Relation
3909:Validity
3899:Argument
3797:theorem,
3445:fractals
3332:fractals
3300:L-system
3242:T-square
3050:Fractals
2828:(2001).
2802:(2000).
2778:(1999).
2756:(2004).
2695:Archived
2459:19 April
2245:16915455
2225:Language
2170:15694646
2062:62093042
1876:See also
1869:-ception
1771:Malyutin
1700:â
1692:efforts:
1482:for all
1438:implies
670:function
516:0 is in
391:'s book
227:fractals
179:) = Fib(
165:integers
163:For all
115:ancestor
65:function
5296:Related
5093:Diagram
4991: (
4970:Hilbert
4955:Systems
4950:Theorem
4828:of the
4773:systems
4553:Formula
4548:Grammar
4464: (
4408:General
4121:Forcing
4106:Element
4026:Monadic
3801:paradox
3742:Theorem
3678:General
3394:Tricorn
3247:n-flake
3096:Packing
3079:Higuchi
3069:Assouad
2977:Algebra
2178:1599505
1642:parsers
1626:(n - 1)
1447:+ 1) =
1382:+ 1) =
823:(where
692:â 1) +
566:, then
225:), and
221:(e.g.,
213:(e.g.,
197:by the
5059:finite
4822:Skolem
4775:
4750:Theory
4718:Symbol
4708:String
4691:atomic
4568:ground
4563:closed
4558:atomic
4514:ground
4477:syntax
4373:binary
4300:domain
4217:Finite
3982:finite
3840:Logics
3799:
3747:Theory
3493:People
3443:Random
3350:Filled
3318:H tree
3237:String
3125:system
2983:
2964:
2945:
2924:
2905:
2881:
2859:
2840:
2814:
2788:
2764:
2744:
2675:5 July
2651:
2577:
2540:
2532:
2400:7 July
2375:7 July
2347:
2275:
2243:
2176:
2168:
2150:
2060:
2050:
1814:Giotto
1783:Giotto
1773:, 1892
1753:In art
1685:social
1616:return
1583:return
1548:code:
1546:Python
1419:Hence
1298:(0) =
1248:where
544:is in
464:SPARQL
171:> 1
5049:Model
4797:Peano
4654:Proof
4494:Arity
4423:Naive
4310:image
4242:Fuzzy
4202:Empty
4151:union
4096:Class
3737:Model
3727:Lemma
3685:Axiom
3569:Other
2742:S2CID
2538:S2CID
2394:(PDF)
2369:(PDF)
2252:(PDF)
2241:S2CID
2221:(PDF)
2174:S2CID
1406:)) =
1394:)) =
1373:Then
1263:that
381:index
339:is a
53:logic
38:cocoa
5172:Type
4975:list
4779:list
4756:list
4745:Term
4679:rank
4573:open
4467:list
4279:Maps
4184:sets
4043:Free
4013:list
3763:list
3690:list
2981:ISBN
2962:ISBN
2943:ISBN
2922:ISBN
2903:ISBN
2879:ISBN
2857:ISBN
2838:ISBN
2812:ISBN
2786:ISBN
2762:ISBN
2703:2016
2677:2020
2649:ISBN
2628:2015
2603:2015
2575:ISBN
2530:ISSN
2461:2020
2402:2023
2377:2023
2345:ISBN
2273:ISBN
2166:PMID
2058:OCLC
2048:ISBN
1805:The
1799:and
1769:and
1610:else
1574:>
1472:) =
1455:+ 1)
1428:) =
1414:+ 1)
1332:) =
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1017:and
684:) =
478:The
456:WINE
387:and
219:sets
187:â 2)
175:Fib(
139:The
59:and
4859:of
4841:of
4789:of
4321:Sur
4295:Map
4102:Ur-
4084:Set
2734:doi
2520:doi
2315:doi
2233:doi
2158:doi
1836:'s
1816:'s
1785:'s
1765:by
1745:in
1733:to
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1553:def
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771:of
759:In
540:if
460:GNU
452:PHP
359:or
329:and
325:and
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51:to
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5245:NP
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1143:+
1140:n
1137:(
1134:F
1113:a
1110:=
1107:)
1104:0
1101:(
1098:G
1077:a
1074:=
1071:)
1068:0
1065:(
1062:F
1039:X
1032:N
1028::
1025:G
1005:X
998:N
994::
991:F
962:N
948:n
931:)
928:)
925:n
922:(
919:F
916:(
913:f
910:=
907:)
904:1
901:+
898:n
895:(
892:F
871:a
868:=
865:)
862:0
859:(
856:F
832:N
811:X
804:N
800::
797:F
786:X
782:X
778:f
773:X
769:a
765:X
706:F
702:F
698:n
696:(
694:F
690:n
688:(
686:F
682:n
680:(
678:F
579:N
568:n
553:N
542:n
525:N
375:.
189:.
185:n
181:n
177:n
169:n
23:.
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