36:
611:, but used implicitly a further axiom that implies the existence of very large sets. The requirement of this further axiom has been later dismissed, but infinite sets remains used in a fundamental way. This was not an obstacle for the recognition of the correctness of the proof by the community of mathematicians.
158:, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an
705:
is the most significant mathematician who defended actual infinities. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of
Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one
289:
Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility,
255:
place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is
442:
I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.
491:
Accordingly I distinguish an eternal uncreated infinity or absolutum, which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm
1144:
Or, the "tape" may be fixed and the reading "head" may move. Roger
Penrose suggests this because: "For my own part, I feel a little uncomfortable about having our finite device moving a potentially infinite tape backwards and forwards. No matter how lightweight its material, an
506:
One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor
284:
Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody â not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.
482:, opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3) the transfinite numbers and sets of mathematics.
666:
onwards, reject the claim that there are actually infinite mathematical objects or sets. Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand,
354:"As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed."
492:
conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (Georg Cantor) (G. Cantor )
264:"Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." (Aristotle)
1149:
tape might be hard to shift!" Penrose's drawing shows a fixed tape head labelled "TM" reading limp tape from boxes extending to the visual vanishing point. (Cf page 36 in Roger
Penrose, 1989,
698:
along it in finitely many steps: the tape is therefore only "potentially" infinite, sinceâââwhile there is always the ability to take another stepâââinfinity itself is never actually reached.
725:, within which statements may be made. All such statements are necessarily finite in length. The soundness of the manipulations is founded only on the basic principles of a formal language:
486:
A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set is only a part of it, I will call an infinite multitude. (B. Bolzano )
358:
With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed.
741:
offer the needed tools to work with infinities. One does not have to "believe" in infinity in order to write down algebraically valid expressions employing symbols for infinity.
329:"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different."
372:
Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the infinite and do not use it" (
362:"For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately."
281:
Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
585:
are defined as the set of their points. Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example
977:
909:
404:
It is well known that in the Middle Ages all scholastic philosophers advocate
Aristotle's "infinitum actu non datur" as an irrefutable principle. (
604:
260:
The theme was brought forward by
Aristotle's consideration of the apeironâin the context of mathematics and physics (the study of nature):
228:
is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues
1170:
Actual infinity follows from, for example, the acceptance of the notion of the integers as a set, see J J O'Connor and E F Robertson,
139:
form a set (necessarily infinite). A great discovery of Cantor is that, if one accept infinite sets, then there are different sizes (
1372:
1135:
Kleene 1952/1971:48 p. 357; also "the machine ... is supplied with a tape having a (potentially) infinite printing ..." (p. 363).
456:
Actual infinity is now commonly accepted in mathematics, although the term is no longer in use, being replaced by the concept of
806:
694:
tape as "a linear 'tape', (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a
759:
that states that there exist infinite sets, and in particular that the natural numbers form an infinite set. However, some
1362:
1654:
1322:
461:
17:
108:, the concept of actual infinity has been the objects of debates between philosophers. Also, the question whether the
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1270:
1158:
1100:
1026:
961:
934:
893:
866:
839:
1649:
460:. This drastic change was initialized by Bolzano and Cantor in the 19th century, and was one of the origins of the
278:
If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
1423:
752:
563:
290:
because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude.
220:
was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance.
128:
46:
749:
The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.
511:
Cantor distinguished two types of actual infinity, the transfinite and the absolute, about which he affirmed:
1469:
777:
651:
form definite sets is therefore independent of the question of whether infinite things exist physically in
417:
During the
Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
1064:
Stephen Kleene 1952 (1971 edition):48 attributes the first sentence of this quote to (Werke VIII p. 216).
1187:
1171:
1613:
1608:
431:
144:
82:
1200:
596:
1577:
1387:
1161:). Other authors solve this problem by tacking on more tape when the machine is about to run out.
131:. This theory, which is presently commonly accepted as a foundation of mathematics, contains the
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1241:
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1016:
951:
924:
883:
856:
1474:
1382:
1377:
668:
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275:
From the division of magnitudes â for the mathematicians also use the notion of the infinite.
1413:
1367:
1190:, treating the history of the notion of infinity, including the problem of actual infinity.
444:
8:
1582:
1408:
772:
582:
578:
186:
179:
167:
53:
991:
438:
However, the majority of pre-modern thinkers agreed with the well-known quote of Gauss:
1551:
1535:
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903:
831:
574:
1021:. The Dialogues of Plato. Vol. 4. New Haven: Yale University Press. p. 256.
1644:
1525:
1479:
1444:
1352:
1308:
1266:
1258:
1154:
1106:
1096:
1032:
1022:
957:
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889:
862:
835:
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663:
567:
515:
These concepts are to be strictly differentiated, insofar the former is, to be sure,
132:
1623:
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1530:
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1484:
1403:
1290:
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163:
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1336:
1204:
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718:
608:
586:
467:
422:
159:
105:
1572:
339:
Aristotle distinguished between infinity with respect to addition and division.
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730:
714:
691:
687:
644:
268:
Belief in the existence of the infinite comes mainly from five considerations:
136:
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is presently the standard foundation of mathematics. One of its axioms is the
1638:
1603:
1515:
1500:
1110:
707:
385:
140:
1036:
763:
philosophers of mathematics and constructivists still object to the notion.
325:
is never complete: elements can be always added, but never infinitely many.
256:
present not only in the objects of sense but in the Forms also." (Aristotle)
1598:
1510:
1454:
1221:
738:
734:
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is that they consist of a collection of special symbols, and an associated
702:
659:
648:
590:
555:
457:
405:
252:
124:
120:
1459:
1297:, Vol. 2, W.A.J. Luxemburg, S. Koerner (Hrsg.), North Holland, Amsterdam.
213:
148:
116:
86:
554:
Actual infinity is now commonly accepted in mathematics under the name "
392:. This means there is only a (developing, improper, "syncategorematic")
190:(unlimited or indefinite), in contrast to the actual or proper infinite
1418:
559:
479:
413:
Actual infinity exists in number, time and quantity. (J. Baconthorpe )
1541:
1193:
826:. Handbook of the Philosophy of Science. Elsevier. pp. 523â585.
671:
does accept the existence of the completed infinity of the integers.
603:, which has been proved only more than 350 years later. The original
533:
379:
321:
is completed and definite, and consists of infinitely many elements.
298:
244:
1439:
1331:
1076:
Gesammelte abhandlungen: Mathematischen und philosophischen inhalts
1052:
Gesammelte abhandlungen: Mathematischen und philosophischen inhalts
760:
109:
98:
1226:
Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
573:
All mathematics has been rewritten in terms of ZF. In particular,
543:
Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
539:Ăber verschiedene Standpunkte in bezug auf das aktuelle Unendliche
184:
The ancient Greek term for the potential or improper infinite was
1357:
799:
Infinite powers: how calculus reveals the secrets of the universe
531:
as a mathematical concept. This mistake we find, for example, in
421:
The continuum actually consists of infinitely many indivisibles (
1300:
652:
570:, that essentially says that the natural numbers form a set.
247:
sums up the views of his predecessors on infinity as follows:
221:
151:
is strictly larger than the cardinal of the natural numbers.
293:
950:
Thomas, Kenneth W.; Thomas, Thomas, Aquinas (2003-06-01).
882:
Thomas, Kenneth W.; Thomas, Thomas, Aquinas (2003-06-01).
614:
400:. There were exceptions, however, for example in England.
1188:"Infinity" at The MacTutor History of Mathematics archive
61:
1265:, North-Holland Publishing Company, Amsterdam New York.
713:
The present-day conventional finitist interpretation of
57:
1122:
1120:
343:
But Plato has two infinities, the Great and the Small.
115:
The concept of actual infinity has been introduced in
1251:
Adolf
Abraham Fraenkel, Y. Bar-Hillel, A. Levy 1984,
1095:. Fairleigh Dickinson University Press. p. 271.
1117:
992:"Logos Virtual Library: Aristotle: Physics, III, 7"
701:Mathematicians generally accept actual infinities.
112:is infinite is still a debate between physicists.
380:Scholastic, Renaissance and Enlightenment thinkers
101:entities as given, actual and completed objects.
619:The mathematical meaning of the term "actual" in
1636:
549:
497:The numbers are a free creation of human mind. (
1092:The Greek Mode of Thought in Western Philosophy
1255:, 2nd edn., North Holland, Amsterdam New York.
607:, used not only the full power of ZF with the
1316:
926:Proportion: Science, Philosophy, Architecture
272:From the nature of time â for it is infinite.
56:. Consider transferring direct quotations to
1074:Cantor, Georg (1966). Zermelo, Ernst (ed.).
1050:Cantor, Georg (1966). Zermelo, Ernst (ed.).
976:: CS1 maint: multiple names: authors list (
949:
908:: CS1 maint: multiple names: authors list (
881:
858:Arresting Language: From Leibniz to Benjamin
733:, and so on. More abstractly, both (finite)
674:For intuitionists, infinity is described as
202:(limit). These notions are today denoted by
143:) of infinite sets, and, in particular, the
1089:Kohanski, Alexander Sissel (June 6, 2021).
396:but not a (fixed, proper, "categorematic")
1323:
1309:
861:. Stanford University Press. p. 331.
599:is a theorem that was stated in terms of
154:Actual infinity is to be contrasted with
1283:H. Meschkowski, W. Nilson (Hrsg.) 1991,
1088:
821:
796:
678:; terms synonymous with this notion are
294:Aristotle's potentialâactual distinction
27:Concept in the philosophy of mathematics
922:
615:Opposition from the Intuitionist school
430:I am so in favour of actual infinity. (
14:
1637:
1153:, Oxford University Press, Oxford UK,
1073:
1049:
854:
605:Wiles's proof of Fermat's Last Theorem
334:Aristotle, Physics, book 3, chapter 6.
1304:
1278:Georg Cantor: Leben, Werk und Wirkung
1014:
929:. Taylor & Francis. p. 123.
801:. Boston: Houghton Mifflin Harcourt.
566:(ZF). One of the axioms of ZF is the
47:too many or overly lengthy quotations
1373:Hilbert's paradox of the Grand Hotel
1261:1952 (1971 edition, 10th printing),
822:Fletcher, Peter (2007). "Infinity".
744:
478:), and Georg Cantor, who introduced
119:near the end of the 19th century by
29:
1236:Was sind und was sollen die Zahlen?
198:stands opposed to that which has a
24:
462:foundational crisis of mathematics
25:
1666:
1568:Differential geometry of surfaces
1078:. Georg Olms Verlag. p. 399.
1054:. Georg Olms Verlag. p. 174.
953:Commentary on Aristotle's Physics
885:Commentary on Aristotle's Physics
301:handled the topic of infinity in
1363:Controversy over Cantor's theory
1330:
832:10.1016/b978-044451541-4/50017-8
34:
1424:Synthetic differential geometry
1263:Introduction to Metamathematics
1164:
1138:
1129:
1082:
1067:
1058:
923:Padovan, Richard (2002-09-11).
470:, who introduced the notion of
367:Metaphysics, book 9, chapter 6.
216:(610â546 BC) held that the
1043:
1008:
984:
943:
916:
888:. A&C Black. p. 163.
875:
848:
815:
790:
173:
13:
1:
1246:Einleitung in die Mengenlehre
783:
639:, but not to be mistaken for
550:Current mathematical practice
451:
384:The overwhelming majority of
1470:Cardinality of the continuum
855:Fenves, Peter David (2001).
797:Strogatz, Steven H. (2019).
778:Cardinality of the continuum
239:
7:
1015:Allen, Reginald E. (1998).
766:
753:ZermeloâFraenkel set theory
564:ZermeloâFraenkel set theory
562:has been formalized as the
348:Physics, book 3, chapter 4.
309:. He distinguished between
129:ZermeloâFraenkel set theory
10:
1671:
1433:Formalizations of infinity
1224:in E. Zermelo (ed.) 1966,
1209:Paradoxien des Unendlichen
1181:
690:describes the notion of a
643:. The question of whether
177:
1655:Philosophy of mathematics
1609:Gottfried Wilhelm Leibniz
1591:
1560:
1432:
1396:
1345:
1280:(2. Aufl.), BI, Mannheim.
1253:Foundations of Set Theory
145:cardinal of the continuum
83:philosophy of mathematics
523:, whereas the latter is
390:Infinitum actu non datur
127:, later formalized into
54:summarize the quotations
1650:Concepts in metaphysics
1614:August Ferdinand Möbius
1397:Branches of mathematics
1388:Paradoxes of set theory
1238:, Vieweg, Braunschweig.
706:is not involved in any
386:scholastic philosophers
135:, which means that the
1242:Adolf Abraham Fraenkel
1214:Bernard Bolzano 1837,
1151:The Emperor's New Mind
547:
509:
503:
494:
488:
449:
436:
427:
415:
410:
370:
356:
351:
337:
266:
258:
1578:Möbius transformation
1475:Dedekind-infinite set
1383:Paradoxes of infinity
1378:Infinity (philosophy)
1285:Georg Cantor â Briefe
1276:H. Meschkowski 1981,
669:constructive analysis
601:elementary arithmetic
597:Fermat's Last Theorem
525:incapable of increase
513:
504:
495:
489:
484:
440:
428:
419:
411:
402:
388:adhered to the motto
360:
352:
341:
327:
262:
249:
123:, with his theory of
1414:Nonstandard analysis
1126:Kleene 1952/1971:48.
204:potentially infinite
1583:Riemannian manifold
1552:Transfinite numbers
1409:Internal set theory
1287:, Springer, Berlin.
1248:, Springer, Berlin.
1228:, Olms, Hildesheim.
824:Philosophy of Logic
773:Limit (mathematics)
641:physically existing
623:is synonymous with
180:Apeiron (cosmology)
1536:Sphere at infinity
1487:(Complex infinity)
1216:Wissenschaftslehre
1211:, Reclam, Leipzig.
1018:Plato's Parmenides
394:potential infinity
323:Potential infinity
156:potential infinity
95:completed infinity
18:Potential infinity
1632:
1631:
1526:Point at infinity
1506:Hyperreal numbers
1480:Directed infinity
1445:Absolute infinite
1368:Galileo's paradox
1353:Ananta (infinite)
1259:Stephen C. Kleene
956:. A&C Black.
808:978-1-328-87998-1
757:axiom of infinity
745:Modern set theory
568:axiom of infinity
527:and is therefore
519:, yet capable of
224:'s notion of the
208:actually infinite
133:axiom of infinity
79:
78:
16:(Redirected from
1662:
1624:Abraham Robinson
1619:Bernhard Riemann
1538:(Kleinian group)
1531:Regular cardinal
1485:Division by zero
1465:Cardinal numbers
1404:Complex analysis
1339:
1325:
1318:
1311:
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1301:
1291:Abraham Robinson
1232:Richard Dedekind
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719:cardinal numbers
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210:, respectively.
164:infinite product
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1547:Surreal numbers
1521:Ordinal numbers
1450:Actual infinity
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1341:
1335:
1329:
1295:Selected Papers
1205:Bernard Bolzano
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723:formal language
686:. For example,
621:actual infinity
617:
609:axiom of choice
587:finite geometry
552:
545:, pp. 375, 378)
468:Bernard Bolzano
454:
398:actual infinity
382:
369:
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350:
347:
336:
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319:Actual infinity
296:
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160:infinite series
137:natural numbers
106:Greek antiquity
91:actual infinity
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60:or excerpts to
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731:term rewriting
692:Turing machine
688:Stephen Kleene
658:Proponents of
616:
613:
581:, all sort of
551:
548:
537:. (G. Cantor,
529:indeterminable
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381:
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376:III 2079 29).
364:
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331:
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178:Main article:
175:
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93:, also called
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1639:Categories
1561:Geometries
1419:Set theory
1172:"Infinity"
1001:2017-11-14
784:References
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452:Modern era
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