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262:. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical
164:
used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the
250:; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF. Moreover, NFU remains
829:
188:
objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally).
1504:
228:
151:
125:
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implies that there is a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's
270:, urelements are mapped to the lowest-level components of the target phenomenon, such as atomic constituents of a physical object or members of an organisation.
224:
1587:
728:
305:. ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist (unless it is inconsistent, in which case it will
208:, the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus, standard expositions of the canonical
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An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set.
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184:: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are
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493:(December 1968). "On the Consistency of a Slight (?) Modification of Quine's 'New Foundations'".
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and
Lawrence Moss, use the latter term to denote the larger class of sets with the property
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of 1908 included urelements, and hence is a version now called ZFA or ZFCA (i.e. ZFA with
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One way is to work in a first-order theory with two sorts, sets and urelements, with
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There are several different but essentially equivalent ways to treat urelements in a
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242:(NF) to produce NFU has surprising consequences. In particular, Jensen proved the
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286:) are sets that only contain themselves, that is, sets that satisfy the formula
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235:, an object of type 0 can be called an urelement; hence the name "atom".
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204:). It was soon realized that in the context of this and closely related
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This situation is analogous to the treatments of theories of sets and
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must be formulated to apply only to objects that are not urelements.
638:, CSLI Lecture Notes, vol. 60, CSLI Publications, p. 306,
368:
663:, CSLI Lecture Notes, vol. 60, CSLI Publications, p. 57,
297:
Quine atoms cannot exist in systems of set theory that include the
1560:
906:
751:
661:
Vicious circles. On the mathematics of non-wellfounded phenomena
636:
Vicious circles. On the mathematics of non-wellfounded phenomena
710:
462:(4th ed.). London: Chapman & Hall. pp. 297â304.
216:
309:), but it is compatible with the existence of Quine atoms.
219:
do not mention urelements (for an exception, see Suppes).
70:. Ur-elements are also not identical with the empty set.
317:, which implies that the distinct Quine atoms form a
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371:, on: ncatlab.org: nLab, revised on July 16, 2016.
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328:, which allows more than one such set to exist.
223:of set theory that do invoke urelements include
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565:. Cambridge University Press. p. 199.
369:ZFA: ZermeloâFraenkel set theory with atoms
54:, 'primordial') is an object that is not a
1548:
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729:
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546:Elementary Set Theory with a Universal Set
101:is an urelement, it makes no sense to say
392:. Mineola, New York: Dover Publ. p.
132:
106:
659:Barwise, Jon; Moss, Lawrence S. (1996),
634:Barwise, Jon; Moss, Lawrence S. (1996),
225:KripkeâPlatek set theory with urelements
360:
176:. Indeed, urelements are in some sense
62:of a set. It is also referred to as an
58:(has no elements), but that may be an
331:Quine atoms are the only sets called
7:
229:Von NeumannâBernaysâGödel set theory
428:( ed.). New York: Dover Publ.
324:Quine atoms also appear in Quine's
459:Introduction to Mathematical Logic
25:
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238:Adding urelements to the system
339:, although other authors, e.g.
1:
2409:History of mathematical logic
311:Aczel's anti-foundation axiom
307:prove any arbitrary statement
2334:Primitive recursive function
156:Another way is to work in a
97:is a set. In this case, if
456:Mendelson, Elliott (1997).
303:non-well-founded set theory
231:described by Mendelson. In
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1398:SchröderâBernstein theorem
1125:Monadic predicate calculus
784:Foundations of mathematics
501:(1/2). Springer: 250â264.
315:axiom of superuniversality
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2431:Philosophy of mathematics
2380:Automated theorem proving
1551:
1505:Von NeumannâBernaysâGödel
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562:Logic, Induction and Sets
153:is perfectly legitimate.
301:, but they can exist in
192:Urelements in set theory
2081:Self-verifying theories
1902:Tarski's axiomatization
853:Tarski's undefinability
848:incompleteness theorems
559:Thomas Forster (2003).
543:Holmes, Randall, 1998.
284:Willard Van Orman Quine
254:when augmented with an
167:axiom of extensionality
2455:Mathematics portal
2066:Proof of impossibility
1714:propositional variable
1024:Propositional calculus
210:axiomatic set theories
206:axiomatic set theories
147:
146:{\displaystyle U\in X}
121:
120:{\displaystyle X\in U}
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2277:Computably enumerable
2177:Model complete theory
1969:Principia Mathematica
1029:Propositional formula
858:BanachâTarski paradox
592:Non-well-founded sets
589:Aczel, Peter (1988),
252:relatively consistent
148:
122:
27:Concept in set theory
2272:ChurchâTuring thesis
2259:Computability theory
1468:continuum hypothesis
986:Square of opposition
844:Gödel's completeness
549:. Academia-Bruylant.
491:Jensen, Ronald Björn
424:Axiomatic Set Theory
367:Dexter Chua et al.:
131:
105:
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2317:P versus NP problem
2282:Computable function
2076:Reverse mathematics
2002:Logical consequence
1879:primitive recursive
1874:elementary function
1647:Free/bound variable
1500:TarskiâGrothendieck
1019:Logical connectives
949:Logical equivalence
799:Logical consequence
388:The Axiom of Choice
299:axiom of regularity
268:finitist set theory
246:of NFU relative to
227:and the variant of
2224:Transfer principle
2187:Semantics of logic
2172:Categorical theory
2148:Non-standard model
1662:Logical connective
789:Information theory
738:Mathematical logic
690:Weisstein, Eric W.
507:10.1007/bf00568059
198:Zermelo set theory
143:
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93:only defined when
80:first-order theory
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2007:Rule of inference
1997:Natural deduction
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1039:Many-valued logic
1014:Boolean functions
897:Russell's paradox
872:diagonal argument
769:First-order logic
572:978-0-521-53361-4
256:axiom of infinity
16:(Redirected from
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2452:
2404:History of logic
2399:Category of sets
2292:Decision problem
2071:Ordinal analysis
2012:Sequent calculus
1910:Boolean algebras
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333:reflexive sets
290: = {
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182:proper classes
162:unary relation
160:theory with a
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2092:
2088:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2061:
2060:from ZFC
2057:
2054:
2052:
2049:
2043:
2040:
2039:
2038:
2035:
2033:
2030:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1989:
1987:
1985:
1981:
1971:
1970:
1966:
1965:
1960:
1959:non-Euclidean
1957:
1953:
1950:
1948:
1945:
1943:
1942:
1938:
1937:
1935:
1932:
1931:
1929:
1925:
1921:
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1907:
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1877:
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1872:
1870:
1867:
1865:
1862:
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1859:
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1852:
1847:
1841:
1836:Example
1833:
1825:
1820:
1819:
1818:
1815:
1813:
1810:
1806:
1803:
1801:
1798:
1796:
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1734:
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1712:
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1701:
1700:
1697:
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1618:
1616:
1613:
1611:
1608:
1606:
1603:
1599:
1596:
1594:
1593:by definition
1591:
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1579:
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1574:
1572:
1569:
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1564:
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1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1480:KripkeâPlatek
1478:
1476:
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1469:
1466:
1464:
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1401:
1399:
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1389:
1385:
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1378:
1375:
1373:
1370:
1368:
1365:
1364:
1363:
1359:
1356:
1355:
1353:
1351:
1347:
1343:
1335:
1332:
1330:
1327:
1325:
1324:constructible
1322:
1321:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
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1234:
1231:
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1226:
1224:
1221:
1219:
1216:
1214:
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1210:
1208:
1204:
1201:
1199:
1196:
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1191:
1189:
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1123:
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1111:
1108:
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1103:
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1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1006:
1004:
1002:
1001:Propositional
998:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
960:
957:
956:
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952:
950:
947:
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937:
935:
932:
930:
929:Logical truth
927:
925:
922:
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919:
917:
913:
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908:
904:
898:
895:
893:
890:
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869:
865:
861:
859:
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851:
849:
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841:
839:
837:
831:
826:
820:
817:
815:
812:
810:
807:
805:
802:
800:
797:
795:
792:
790:
787:
785:
782:
780:
777:
775:
772:
770:
767:
765:
762:
758:
755:
754:
753:
750:
749:
747:
743:
739:
732:
727:
725:
720:
718:
713:
712:
709:
700:
699:
694:
691:
686:
685:
681:
672:
666:
662:
655:
652:
647:
641:
637:
630:
627:
616:
612:
608:
606:0-937073-22-9
602:
598:
594:
593:
585:
583:
579:
574:
568:
564:
563:
555:
552:
548:
547:
540:
537:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
486:
483:
471:
465:
461:
460:
452:
449:
437:
431:
426:
425:
419:
413:
410:
405:
399:
395:
390:
389:
383:
377:
374:
370:
364:
361:
354:
352:
350:
347: â
346:
342:
338:
334:
329:
327:
322:
320:
316:
312:
308:
304:
300:
295:
293:
289:
285:
282:(named after
281:
273:
271:
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265:
261:
257:
253:
249:
245:
241:
236:
234:
230:
226:
222:
218:
214:
211:
207:
203:
199:
191:
189:
187:
183:
179:
175:
170:
168:
163:
159:
154:
140:
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134:
114:
111:
108:
100:
96:
92:
88:
83:
81:
73:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
18:Reflexive set
2446:
2244:Ultraproduct
2091:Model theory
2056:Independence
1992:Formal proof
1984:Proof theory
1967:
1940:
1897:real numbers
1869:second-order
1780:Substitution
1657:Metalanguage
1598:conservative
1571:Axiom schema
1515:Constructive
1485:MorseâKelley
1451:Set theories
1430:Aleph number
1423:inaccessible
1329:Grothendieck
1213:intersection
1168:
1100:Higher-order
1088:Second-order
1034:Truth tables
991:Venn diagram
774:Formal proof
696:
660:
654:
635:
629:
618:, retrieved
591:
561:
554:
544:
539:
498:
494:
485:
475:17 September
473:. Retrieved
458:
451:
441:17 September
439:. Retrieved
423:
412:
387:
376:
363:
348:
344:
332:
330:
323:
319:proper class
296:
291:
287:
279:
277:
237:
195:
171:
155:
98:
94:
90:
86:
84:
77:
67:
63:
51:
43:
39:
29:
2354:Type theory
2302:undecidable
2234:Truth value
2121:equivalence
1800:non-logical
1413:Enumeration
1403:Isomorphism
1350:cardinality
1334:Von Neumann
1299:Ultrafilter
1264:Uncountable
1198:equivalence
1115:Quantifiers
1105:Fixed-point
1074:First-order
954:Consistency
939:Proposition
916:Traditional
887:Lindström's
877:Compactness
819:Type theory
764:Cardinality
693:"Urelement"
341:Jon Barwise
337:Peter Aczel
280:Quine atoms
274:Quine atoms
244:consistency
233:type theory
127:, although
36:mathematics
2475:Urelements
2165:elementary
1858:arithmetic
1726:Quantifier
1704:functional
1576:Expression
1294:Transitive
1238:identities
1223:complement
1156:hereditary
1139:Set theory
670:1575860090
645:1575860090
620:2016-10-17
435:0486616304
403:0486466248
355:References
158:one-sorted
68:individual
46:(from the
44:ur-element
32:set theory
2436:Supertask
2339:Recursion
2297:decidable
2131:saturated
2109:of models
2032:deductive
2027:axiomatic
1947:Hilbert's
1934:Euclidean
1915:canonical
1838:axiomatic
1770:Signature
1699:Predicate
1588:Extension
1510:Ackermann
1435:Operation
1314:Universal
1304:Recursive
1279:Singleton
1274:Inhabited
1259:Countable
1249:Types of
1233:power set
1203:partition
1120:Predicate
1066:Predicate
981:Syllogism
971:Soundness
944:Inference
934:Tautology
836:paradoxes
698:MathWorld
515:0039-7857
138:∈
112:∈
40:urelement
2469:Category
2421:Logicism
2414:timeline
2390:Concrete
2249:Validity
2219:T-schema
2212:Kripke's
2207:Tarski's
2202:semantic
2192:Strength
2141:submodel
2136:spectrum
2104:function
1952:Tarski's
1941:Elements
1928:geometry
1884:Robinson
1805:variable
1790:function
1763:spectrum
1753:Sentence
1709:variable
1652:Language
1605:Relation
1566:Automata
1556:Alphabet
1540:language
1394:-jection
1372:codomain
1358:Function
1319:Universe
1289:Infinite
1193:Relation
976:Validity
966:Argument
864:theorem,
531:46960777
523:20114640
495:Synthese
420:(1972).
384:(1973).
264:universe
258:and the
2363:Related
2160:Diagram
2058: (
2037:Hilbert
2022:Systems
2017:Theorem
1895:of the
1840:systems
1620:Formula
1615:Grammar
1531: (
1475:General
1188:Forcing
1173:Element
1093:Monadic
868:paradox
809:Theorem
745:General
615:0940014
186:minimal
174:classes
60:element
50:prefix
2126:finite
1889:Skolem
1842:
1817:Theory
1785:Symbol
1775:String
1758:atomic
1635:ground
1630:closed
1625:atomic
1581:ground
1544:syntax
1440:binary
1367:domain
1284:Finite
1049:finite
907:Logics
866:
814:Theory
667:
642:
613:
603:
569:
529:
521:
513:
466:
432:
400:
74:Theory
48:German
2116:Model
1864:Peano
1721:Proof
1561:Arity
1490:Naive
1377:image
1309:Fuzzy
1269:Empty
1218:union
1163:Class
804:Model
794:Lemma
752:Axiom
527:S2CID
519:JSTOR
266:. In
38:, an
2239:Type
2042:list
1846:list
1823:list
1812:Term
1746:rank
1640:open
1534:list
1346:Maps
1251:sets
1110:Free
1080:list
830:list
757:list
665:ISBN
640:ISBN
601:ISBN
567:ISBN
511:ISSN
477:2012
464:ISBN
443:2012
430:ISBN
398:ISBN
215:and
196:The
178:dual
64:atom
1926:of
1908:of
1856:of
1388:Sur
1362:Map
1169:Ur-
1151:Set
503:doi
335:by
294:}.
217:ZFC
180:to
66:or
56:set
52:ur-
42:or
30:In
2471::
2312:NP
1936::
1930::
1860::
1537:),
1392:Bi
1384:In
695:.
611:MR
609:,
599:,
597:57
581:^
525:.
517:.
509:.
499:19
497:.
396:.
394:45
351:.
321:.
213:ZF
89:â
82:.
2392:/
2307:P
2062:)
1848:)
1844:(
1741:â
1736:!
1731:â
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1687:â
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1672:âš
1667:ÂŹ
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1386:/
1360:/
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1167:(
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730:e
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674:.
649:.
624:.
575:.
533:.
505::
479:.
445:.
406:.
349:x
345:x
292:x
288:x
141:X
135:U
115:U
109:X
99:U
95:b
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87:a
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