88:
829:
have our idea of infinity... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression.
150:
613:
47:
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precisely in the disproportion between the idea of infinity and the infinity of which it is the idea that this exceeding of limits is produced. The idea of infinity is the mode of being, the infinition, of infinity... All knowing qua intentionality already presupposes the idea of infinity, which is preeminently non-adequation.
828:
ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we
416:
This is often called potential infinity; however, there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction.
893:
correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite
525:
Aristotle's views on the continuum foreshadow some topological aspects of modern mathematical theories of the continuum. Aristotle's emphasis on the connectedness of the continuum may have inspiredâin different waysâmodern philosophers and mathematicians such as
Charles Sanders Peirce, Cantor, and
977:
The idea of infinity is not an incidental notion forged by a subjectivity to reflect the case of an entity encountering on the outside nothing that limits it, overflowing every limit, and thereby infinite. The production of the infinite entity is inseparable from the idea of infinity, for it is
747:
So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the
1774:
A Dissertation on the
Philosophy of Aristotle, in Four Books. In which his principle physical and metaphysical dogmas are unfolded, and it is shown, from undubitable evidence, that his philosophy has not been accurately known since the destruction of the Greeks. The insufficiency also of the
948:... I can see in space the possibility of any finite experience... we recognize essential infinity of space in its smallest part." " is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room.
847:
Modern discussion of the infinite is now regarded as part of set theory and mathematics. Contemporary philosophers of mathematics engage with the topic of infinity and generally acknowledge its role in mathematical practice. Although set theory is now widely accepted, this was not always so.
742:
It appeared by this reasoning as though a "set" (Galileo did not use the terminology) which is naturally smaller than the "set" of which it is a part (since it does not contain all the members) is in some sense the same "size". Galileo found no way around this problem:
516:
But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually
403:... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number.
521:
The parts are actually there, in some sense. However, in this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more."
974:...infinity is produced in the relationship of the same with the other, and how the particular and the personal, which are unsurpassable, as it were magnetize the very field in which the production of infinity is enacted...
374:, alive for the period 384â322 BCE, is credited with being the root of a field of thought, in his influence of succeeding thinking for a period spanning more than one subsequent millennium, by his rejection of the idea of
496:
303:
first placed infinity into a coherent mathematical framework. Keenly aware of his departure from traditional wisdom, Cantor also presented a comprehensive historical and philosophical discussion of infinity. In
511:
Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter
1536:
T.Hurst 1802 (ed. Locke writes: And hence it is, that in disputes and reasonings concerning eternity, or any other infinite, we are apt to blunder, and involve ourselves in manifest absurdities...)
340:
who considered infinity to be a foundational and primitive basis of reality. Anaximander was the first in the Greek philosophical tradition to propose that the universe was infinite.
363:) all similarly considered matter to be made of an infinite number of structures as considered by imagining dividing or separating matter from itself an infinite number of times.
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or "impressions," and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative.
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The Jains were the first to discard the idea that all infinities were the same or equal. They recognized different types of infinities: infinite in length (one
966:(January 12, 1906 â December 25, 1995) uses infinity to designate that which cannot be defined or reduced to knowledge or power. In Levinas' magnum opus
598:(c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
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291:, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (áŒÏΔÎčÏÎżÎœ,
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He considered that in considerations on the subject of eternity, which he classified as an infinity, humans are likely to make mistakes.
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metaphysics and mathematics were the first to define and delineate different "types" of infinities. The work of the mathematician
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1908:
312:, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity.
1704:
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n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as
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philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from
193:
765:, although Hume, like Galileo, believed the principle could not be applied to the infinite. The same concept, applied by
1954:
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Infinity: New
Research Frontiers - Chapter 1 : Infinity as a Transformative concept in Science and Theology (p.22)
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1740:
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781:(April 5, 1588 â December 4, 1679) tried to defend the idea of a potential infinity in light of the discovery, by
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35:
182:
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102:
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669:. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number
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351:(500â428 BCE) was of the opinion that matter of the universe had an innate capacity for infinite division.
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attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.
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Aristotle and modern mathematical theories of the continuum, in
Aristotle and Contemporary Science II
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1699:, D. Sfendoni-Mentzou, J. Hattiangadi, and D.M. Johnson, eds. Frankfurt: Peter Lang, 2001, 113â129,
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correlates a class with its subclass, we merely have yet another case of ambiguous grammar.
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According to Singh (1987), Joseph (2000) and
Agrawal (2000), the highest enumerable number
540:, in Book 7 of the same work, the reasoning of which was later studied and commented on by
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1934:
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philosophy that has been substituted by the moderns for that of
Aristotle, is demonstrated
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Unlike the traditional empiricists, he thought that the infinite was in some way given to
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process itself in some sense or other such a pair of classes... In the superstition that
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also argued against the idea that infinity could be in any sense complete or a totality.
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1980:
1975:
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A. Newstead (2001). "Aristotle and Modern
Mathematical Theories of the Continuum", in
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This article is about the philosophical concept. For the mathematical concept, see
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The idea that size can be measured by one-to-one correspondence is today known as
1710:
A. Newstead (2009). "Cantor on
Infinity in Nature, Number, and the Divine Mind",
1510:
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1944:
696:("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
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Simplicius, alive circa 490 to 560 AD, thought the concept "Mind" was infinite.
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1630:(Translated by A. Lingis) Springer Science & Business Media, 31 March 1987
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680:, two basic types of infinite numbers are distinguished. On both physical and
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Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
348:
718:(February 15, 1564 â January 8, 1642) discussed the example of comparing the
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1752:
N. Singh (1988). 'Jaina Theory of Actual
Infinity and Transfinite Numbers',
1372:. A&C Black, 10 Apr 2014, 202 pages, Ancient Commentators on Aristotle.
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having infinite length yet bounding finite areas were known much before.
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1282:. SUNY Press, 1995, 270 pages, SUNY series in ancient Greek philosophy.
860:, and upon the idea of the actual infinite, during his "middle period".
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953:... what is infinite about endlessness is only the endlessness itself.
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considered infinity, while he was alive, during the 3rd century A.D.
1665:
L. C. Jain (1973). "Set theory in the Jaina school of mathematics",
1534:
Philosophical beauties selected from the works of John Locke - p.237
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is finite. Not reported, this motivation of Hobbes came too late as
491:{\displaystyle \forall n\in \mathbb {Z} (\exists m\in \mathbb {Z} )}
856:(April 26, 1889 â April 29, 1951) made an impassioned attack upon
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of the infinite set of integers 1, 2, ...), the smallest cardinal
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Jain theory of numbers (See IIIrd section for various infinities)
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812:(August 29, 1632 â October 28, 1704) in common with most of the
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Infinite: nearly infinite, truly infinite, infinitely infinite
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thought infinity to be "incomprehensible for the human mind".
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143:
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The Crest of the
Peacock: Non-European Roots of Mathematics
1059:"Cantor on Infinity in Nature, Number, and the Divine Mind"
359:
A group of thinkers of ancient Greece (later identified as
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An early engagement with the idea of infinity was made by
1190:. School of Computer Science - University of St Andrews
1097:
Reforming the Doctrine of God (footnote 4. of page 99)
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1241:. Cambridge University Press, 7 Feb 2011, edited by
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Aristotle deals with infinity in the context of the
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1407:
Greek Philosophers as Theologians: The Divine Arche
268:is explored in articles under headings such as the
174:. Unsourced material may be challenged and removed.
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631:of the Jains corresponds to the modern concept of
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1130:The Cambridge companion to Early Greek philosophy
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1431:J.J. O'Connor and E.F. Robertson (April 1999).
1184:J.J. O'Connor, E.F. Robertson (February 2002).
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1369:Simplicius: On Aristotle Physics 7 (page 1.)
602:Enumerable: lowest, intermediate and highest
1447:: CS1 maint: numeric names: authors list (
1133:. Cambridge University Press. p. 127.
75:Learn how and when to remove these messages
1808:
1794:
1786:
1655:Ancient Jaina Mathematics: an Introduction
1548:"Ludwig Wittgenstein, British philosopher"
1340:. Frankfurt: Peter Lang. pp. 113â129.
700:Views from the Renaissance to modern times
1712:American Catholic Philosophical Quarterly
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1163:The History of Philosophy: A Short Survey
1100:. Wm. B. Eerdmans Publishing, 326 pages.
1066:American Catholic Philosophical Quarterly
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252:Learn how and when to remove this message
234:Learn how and when to remove this message
132:Learn how and when to remove this message
769:, is used in relation to infinite sets.
684:grounds, a distinction was made between
95:This article includes a list of general
1746:MacTutor History of Mathematics archive
1729:MacTutor History of Mathematics Archive
1318:. University of California - Santa Cruz
1279:Aristotle's Theory of Actuality (p.119)
1166:. The University of Tennessee at Martin
1022:
834:Essay, II. xvii. 7., author's emphasis
1462:J.J. O'Connor, E.F. Robertson (2002).
1440:
391:of infinity in terms of his notion of
1697:Aristotle and Contemporary Science II
1628:Collected Philosophical Papers (p.47)
1572:Asenjo, F. G.; Tamburino, J. (1975).
1486:"Thomas Hobbes (English philosopher)"
308:theology, for example in the work of
7:
1858:Hilbert's paradox of the Grand Hotel
1667:Indian Journal of History of Science
1366:R. Sorabji (C. Hagen) (2014-04-10).
172:adding citations to reliable sources
1033:Infinity: A Very Short Introduction
992:, which was published during 1957.
990:Philosophy and the Idea of Infinity
988:Levinas also wrote a work entitled
1578:Notre Dame Journal of Formal Logic
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441:
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101:it lacks sufficient corresponding
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2053:Differential geometry of surfaces
1511:"John Locke, English philosopher"
56:This article has multiple issues.
1848:Controversy over Cantor's theory
848:Influenced by L.E.J Brouwer and
148:
86:
45:
1909:Synthetic differential geometry
1605:Bergo, Bettina (23 July 2006).
1353:The Continuous and the Discrete
777:Famously, the ultra-empiricist
692:("countless, innumerable") and
381:In Book 3 of his work entitled
159:needs additional citations for
64:or discuss these issues on the
1404:Dr Adam Drozdek (2013-05-28).
1094:F. LeRon Shults (2005-11-01).
726:{1, 2, 3, 4, ...} as follows:
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438:
1:
1484:T. Sorell (30 October 2014).
1955:Cardinality of the continuum
1509:G.A.J. Rogers (2015-12-14).
1012:Philosophy of space and time
722:{1, 4, 9, 16, ...} with the
264:In philosophy and theology,
1410:. Ashgate Publishing, Ltd.
1036:. Oxford University Press.
1007:Measure problem (cosmology)
686:
654:{\displaystyle \aleph _{0}}
387:, Aristotle deals with the
2156:
1918:Formalizations of infinity
1755:Journal of Asiatic Society
1679:Princeton University Press
1355:. Oxford University Press.
843:Modern philosophical views
708:
319:
183:"Infinity" philosophy
29:
2094:Gottfried Wilhelm Leibniz
1673:George G. Joseph (2000).
1336:Newstead, A.G.J. (2001).
1214:. EncyclopĂŠdia Britannica
676:In the Jaina work on the
36:Infinity (disambiguation)
1734:University of St Andrews
1591:10.1305/ndjfl/1093891610
1546:R. Monk (8 April 2016).
1127:A.A. Long (1999-05-28).
498:, which reads, "for any
2135:Metaphysical properties
2099:August Ferdinand Möbius
1882:Branches of mathematics
1873:Paradoxes of set theory
1466:. St Andrews University
1351:White, Michael (1992).
1002:Infinite monkey theorem
793:is infinite, but whose
529:Among the scholastics,
116:more precise citations.
1243:Revd Dr Michael Heller
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955:
950:
938:
917:
887:
837:
783:Evangelista Torricelli
759:
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34:. For other uses, see
2063:Möbius transformation
1960:Dedekind-infinite set
1868:Paradoxes of infinity
1863:Infinity (philosophy)
1609:. Stanford University
1574:"Logic of antinomies"
1312:Aristotelian Infinity
1211:Infinity: Mathematics
1160:James Fieser (2008).
1078:10.5840/acpq200983444
1057:Newstead, A. (2009).
1030:Stewart, Ian (2017).
968:Totality and Infinity
933:Philosophical Grammar
929:Philosophical Remarks
918:
888:
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583:Early Indian thinking
493:
320:Further information:
27:Philosophical concept
1899:Nonstandard analysis
1780:Robert Wilks, London
1720:Robertson, Edmund F.
916:{\displaystyle m=2n}
898:
886:{\displaystyle m=2n}
868:
858:axiomatic set theory
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421:
168:improve this article
2068:Riemannian manifold
2037:Transfinite numbers
1894:Internal set theory
1739:Ian Pearce (2002).
1724:"Jaina mathematics"
1718:O'Connor, John J.;
1660:Infinity Foundation
1276:Z. Bechler (1995).
754:On two New Sciences
367:Aristotle and after
322:History of infinity
2140:Physical cosmology
2021:Sphere at infinity
1972:(Complex infinity)
1714:, 83 (4), 533â553.
1607:"Emmanual Levinas"
913:
883:
864:Does the relation
667:transfinite number
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618:
488:
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2011:Point at infinity
1991:Hyperreal numbers
1965:Directed infinity
1930:Absolute infinite
1853:Galileo's paradox
1838:Ananta (infinite)
1688:978-0-14-027778-4
1464:"Galileo Galilei"
1247:Dr W. Hugh Woodin
1043:978-0-19-875523-4
711:Galileo's paradox
673:is the smallest.
504:William of Ockham
287:, for example in
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2016:Regular cardinal
1970:Division by zero
1950:Cardinal numbers
1889:Complex analysis
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1981:Gimel function
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526:LEJ Brouwer.
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417:For example,
410:
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184:
180:
179:Find sources:
173:
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157:This article
155:
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122:February 2012
115:
111:
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83:
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69:
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63:
62:
57:
52:
43:
42:
37:
33:
19:
2084:Georg Cantor
2058:Möbius plane
1996:Infinite set
1940:Aleph number
1862:
1779:
1773:
1753:
1744:
1727:
1711:
1696:
1674:
1666:
1653:
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1611:. Retrieved
1600:
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1457:
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1383:. Retrieved
1368:
1361:
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1331:
1320:. Retrieved
1311:
1309:John Bowin.
1304:
1293:. Retrieved
1278:
1271:
1260:. Retrieved
1237:
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1210:
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1129:
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1096:
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987:
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863:
854:Wittgenstein
846:
838:
825:
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808:
791:surface area
776:
767:Georg Cantor
760:
753:
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741:
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687:
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670:
628:
626:
619:
586:
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528:
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520:
510:
509:
415:
408:
402:
397:potentiality
383:
380:
370:
361:the Atomists
358:
355:The Atomists
347:
335:
301:Georg Cantor
292:
282:
265:
263:
248:
230:
221:
211:
204:
197:
190:
178:
166:Please help
161:verification
158:
128:
119:
100:
72:
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59:
58:Please help
55:
1945:Beth number
682:ontological
538:prime mover
512:existentes.
338:Anaximander
332:Anaximander
310:Duns Scotus
289:Anaximander
114:introducing
2124:Categories
2046:Geometries
1904:Set theory
1758:, Vol. 30.
1705:0820441473
1644:References
1636:9024733952
1613:2016-04-21
1557:2016-04-21
1552:Britannica
1520:2016-04-21
1515:Britannica
1495:2016-04-21
1490:Britannica
1470:2016-04-21
1434:Simplicius
1385:2015-06-25
1322:2015-06-24
1295:2015-06-21
1262:2015-06-21
1218:2016-03-13
1194:2016-03-13
1170:2016-03-14
1146:2016-03-18
1113:2015-06-26
931:§ 141, cf
818:sense data
814:empiricist
805:John Locke
688:asaáčkhyÄta
633:aleph-null
564:Simplicius
542:Simplicius
349:Anaxagoras
344:Anaxagoras
194:newspapers
97:references
61:improve it
2027:Supertask
1741:'Jainism'
1584:: 17â44.
1570:See also
1443:cite book
852:in part,
824:Whatever
643:ℵ
622:dimension
577:Augustine
572:Augustine
517:existent.
468:∧
448:∈
442:∃
431:∈
425:∀
393:actuality
372:Aristotle
306:Christian
224:June 2007
67:talk page
2130:Infinity
1925:0.999...
1817:Infinity
1652:(2000).
1187:Infinity
996:See also
983:p. 26-27
981:â
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751:â
558:Plotinus
553:Plotinus
406:â
270:Absolute
266:infinity
32:Infinity
18:Finitude
1843:Apeiron
1831:History
705:Galileo
591:upanga
531:Aquinas
500:integer
409:Physics
395:and of
389:concept
384:Physics
295:). The
293:apeiron
208:scholar
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274:God
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