About this topic
Summary The infinite has been an important topic in many branches of philosophy (and neighboring disciplines), including metaphysics, epistemology, the philosophy of physics, the philosophy of religion, and ethics.  But since at least the 19th century, when B. Bolzano, R. Dedekind, G. Cantor, and others made crucial contributions, the most central discussions about the infinite have taken place in the philosophy of mathematics and logic.  For a rich, historically grounded, but also opinionated introduction, see A.W. Moore, The Infinite (2nd edition, Routledge, 2001).  Many classic articles on the topic are contained in A.W. Moore, ed., Infinity (International Research Library of Philosophy, Dartmouth, 1993). For a more basic introduction, see P. Zellini's A Brief History of Infinity (Penguin, 2004); and on the mathematical side, see I. Stewart's Infinity. A Very Short Introduction (Oxford University Press, 2017) and E. Cheng's Beyond Infinity (Basic Books, 2017).  Finally, for advanced logico-mathematical aspects, see A. Kanamori, The Higher Infinite (2nd ed., Springer, 1994).
Key works Potential infinity, actual infinity, infinitesimals, paradoxes, the transfinite, set theory, cardinal numbers, ordinal numbers, space, time.
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  1. added 2018-12-17
    Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav D. Sergeyev - 2019 - Foundations of Science 24 (1):153-170.
    This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...)
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  2. added 2018-12-17
    Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  3. added 2018-12-17
    The Exact (Up to Infinitesimals) Infinite Perimeter of the Koch Snowflake and its Finite Area.Yaroslav Sergeyev - 2016 - Communications in Nonlinear Science and Numerical Simulation 31 (1-3):21–29.
    The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and (...)
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  4. added 2018-12-17
    The Olympic Medals Ranks, Lexicographic Ordering and Numerical Infinities.Yaroslav Sergeyev - 2015 - The Mathematical Intelligencer 37 (2):4-8.
    Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...)
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  5. added 2018-12-12
    Zeno's Paradoxes.Nicholas Huggett - 2002
    Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato's Parmenides. There we learn that Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Beyond this, really all we know is that he was close to Parmenides (Plato reports the gossip that they were lovers when Zeno was young), (...)
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  6. added 2018-12-08
    Existence Is Evidence of Immortality.Michael Huemer - manuscript
    Time may be infinite in both directions. If it is, then, if persons could live at most once in all of time, the probability that you would be alive now would be zero. Since you are alive now, with certainty, either the past is finite, or persons can live more than once.
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  7. added 2018-12-08
    Approaching Infinity.Michael Huemer - 2016 - New York: Palgrave Macmillan.
  8. added 2018-12-08
    Aristotelian Finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...)
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  9. added 2018-10-10
    Outer and Inner Surfaces of Bodies.Rush Rhees - 2017 - Philosophical Investigations 40 (1):10-31.
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  10. added 2018-10-03
    Margaret Cavendish on the Order and Infinitude of Nature.Michael Bennett McNulty - 2018 - History of Philosophy Quarterly 35 (3):219-239.
    In this paper, I develop a new interpretation of the order of nature, its function, and its implications in Margaret Cavendish’s philosophy. According to the infinite balance account, the order of nature consists in a balance among the infinite varieties of nature. That is, for Cavendish, nature contains an infinity of different types of matter: infinite species, shapes, and motions. The potential tumult implicated by such a variety, however, is tempered by the counterbalancing of the different kinds and motions of (...)
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  11. added 2018-09-06
    A Road Map of Dedekind’s Theorem 66.Ansten Klev - 2018 - Hopos: The Journal of the International Society for the History of Philosophy of Science 8 (2):241-277.
    Richard Dedekind’s theorem 66 states that there exists an infinite set. Its proof invokes such apparently nonmathematical notions as the thought-world and the self. This article discusses the content and context of Dedekind’s proof. It is suggested that Dedekind took the notion of the thought-world from Hermann Lotze. The influence of Kant and Bernard Bolzano on the proof is also discussed, and the reception of the proof in the mathematical and philosophical literature is covered in detail.
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  12. added 2018-08-14
    Zeno of Elea' Paradoxes (The Dialectic of Stability and Motion from a Contemporary Mathematical View) مفارقات زينون: جدل الثبات والحركة من منظور رياضي معاصر.Salah Osman - 2004 - Menoufia University, Faculty of Arts Journal, Egypt 58:99 - 139.
    لا شك أن مفارقات زينون في الحركة قد تم تناولها – تحليلاً ونقدًا – في كثيرٍ من أدبيات العلم والفلسفة قديمًا وحديثًا، حتى لقد ساد الظن بأن ملف المفارقات قد أغُلق تمامًا، لاسيما بعد أن نجح الحساب التحليلي في التعامل منطقيًا مع صعوبات الأعداد اللامتناهية، لكن الفرض الأساسي لهذا البحث يزعم عكس ذلك؛ أعني أن الملف مازال مفتوحًا وبقوة – خصوصًا على المستوى الرياضي الفيزيائي – وأن إغلاقه النهائي قد لا يتم في المستقبل القريب. من جهة أخرى، إذا كانت فكرة (...)
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  13. added 2018-02-18
    The Infinite.A. W. Moore - 1990 - Routledge.
    Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.
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  14. added 2018-02-18
    Aristotelian Infinity.Jonathan Lear - 1979 - Proceedings of the Aristotelian Society 80:187.
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  15. added 2018-02-17
    Cantor’s Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.
    Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...)
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  16. added 2018-02-17
    Topological Drinking Problems.Josh Parsons - 2006 - Analysis 66 (2):149-154.
    In my (2004), I argued that it is possible to drink any finite amount of alcohol without ever suffering a hangover by completing a certain kind of supertask. Assume that a drink causes drunkenness to ensue immediately and to last for a period proportional to the quantity of alcohol consumed; that a hangover begins immediately at the time the drunkenness ends and lasts for the same length of time as the drunkenness; and that at any time during which you are (...)
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  17. added 2017-11-09
    Infinite Cardinalities, Measuring Knowledge, and Probabilities in Fine-Tuning Arguments.Isaac Choi - 2018 - In Matthew A. Benton, John Hawthorne & Dani Rabinowitz (eds.), Knowledge, Belief, and God: New Insights in Religious Epistemology. Oxford: Oxford University Press.
  18. added 2017-09-28
    Das Unendliche und die Zahl.Hugo Bergmann - 1914 - Revue de Métaphysique et de Morale 22 (2):16-16.
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  19. added 2017-09-12
    Infini des Philosophes, Infini des Astronomes.Françoise Monnoyeur (ed.) - 1995, 1999, 2003 - Editions Belin.
    L'astronomie se confronte à la réalité d'un cosmos fini ou infini. Au gré des théories astronomiques, physiques, mathématiques, philosophiques ou esthétiques, l'infini s'impose ou disparaît.
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  20. added 2017-09-12
    Influence des astronomes sur les philosophes pour penser l'infini.Francoise Monnoyeur - 1995 - In Infini des philosophes, infini des astronomes. Belin. pp. 11-19.
    In book: Infini des mathématiciens, infini des philosophes, Edition: 1992, 1995, 1999, 2002, 2008, 2011 ebook, Chapter: Introduction, Publisher: Belin, Paris, Editors: F. Monnoyeur, pp.9-16.
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  21. added 2017-09-12
    Infini des mathématiciens, infini des philosophes.Francoise Monnoyeur (ed.) - 1992, 1995, 1999 - Paris: Editions Belin.
    Ce livre éclaire les étapes de la réflexion sur la notion philosophique et mathématique d'infini du XIVe au XIXe siècle. Un infini inaccessible qui n'a cessé de stimuler l'activité des mathématiciens, des théologiens, des philosophes et des artistes. Tous ont mis, à leur manière, leur art à son service, afin de nous permettre de mieux appréhender et transcender le fini. Ce livre éclaire, du xive au xixe siècle, les étapes de la réflexion sur la notion philosophique et mathématique d'infini.
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  22. added 2017-09-12
    L'infini et l'indéfini dans la théorie cartésienne de la connaissance.Francoise Monnoyeur - 1992 - In Infini des mathématiciens, infini des philosophes. Belin. pp. 83-94.
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  23. added 2017-07-12
    Discrete and Continuous: A Fundamental Dichotomy in Mathematics.James Franklin - 2017 - Journal of Humanistic Mathematics 7 (2):355-378.
    The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article (...)
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  24. added 2017-05-02
    A (Partially) Skeptical Response to Hart and Russell.Denys A. Turner - 2011 - In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press. pp. 290.
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  25. added 2017-04-30
    On Infinite Size.Bruno Whittle - 2015 - In Oxford Studies in Metaphysics: Volume 9. Oxford University Press. pp. 3-19.
    Cantor showed that there are infinite sets that do not have one-to-one correspondences between them. The standard understanding of this result is that it shows that there are different sizes of infinity. This paper challenges this standard understanding, and argues, more generally, that we do not have any reason to think that there are different sizes of infinity. Two arguments are given against the claim that Cantor established that there are different such sizes: one involves an analogy between Cantor’s result (...)
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  26. added 2017-02-15
    The Infinite and the Ethical.Richard Feist - 2004 - Maritain Studies/Etudes Maritainiennes 20:43-53.
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  27. added 2017-02-14
    A Secret Ethics of Infinity.Janet Borgerson - forthcoming - Levinas, Business Ethics.
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  28. added 2017-02-14
    Bolzano's Approach to the Paradoxes of Infinity: Implications for Teaching.Guillermina Waldegg - 2005 - Science & Education 14 (6):559-577.
  29. added 2017-02-14
    Reasoning with the Infinite: From the Closed World to the Mathematical Universe.Michel Blay - 1999 - University of Chicago Press.
    "One of Michael Blay's many fine achievements in Reasoning with the Infinite is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion. ...
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  30. added 2017-02-14
    Mathematical Intelligence, Infinity and Machines: Beyond Godelitis.Giuseppe Longo - 1999 - Journal of Consciousness Studies 6 (11-12):11-12.
    We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albeit very briefly, the mathematical intelligence both of God and of computers. We (...)
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  31. added 2017-02-14
    Infinity in Theology and Metaphysics.H. P. Owen - 1967 - In Paul Edwards (ed.), The Encyclopedia of Philosophy. New York: Macmillan. pp. 4--190.
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  32. added 2017-02-13
    The Realm of the Infinite.H. W. Woodin - 2011 - In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press.
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  33. added 2017-02-13
    Infinite Beliefs'.Infinite Regresses - 2003 - In Winfried Löffler & Weingartner Paul (eds.), Knowledge and Belief. Alws.
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  34. added 2017-02-13
    Scotus: Adumbrations of a New Concept of Infinity.S. Barbone - 1996 - Wissenschaft Und Weisheit 59 (1):35-43.
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  35. added 2017-02-13
    ''While Being as Infinite is Formless, Being as Infinite is Not Concrete: A Reply to Georges Hélal's' Pure Existence, Formless Infinite Being as Ultimate Reality and Meaning'(URAM 17: 70-83). [REVIEW]J. A. Bracken - 1996 - Ultimate Reality and Meaning 19 (2):156-157.
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  36. added 2017-02-13
    The Infinite.Janet Folina & A. W. Moore - 1990 - Philosophical Quarterly 41 (164):348.
    Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.
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  37. added 2017-02-12
    Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity.Peter K. Benbow - 2013 - Annals of Science 70 (3):1-3.
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  38. added 2017-02-12
    Cartesian Idea of God as the Infinite.Ksenija Puškarić - 2012 - Filozofia 67 (4):282.
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  39. added 2017-02-12
    Bernays, Dooyeweerd and Gödel – the Remarkable Convergence in Their Reflections on the Foundations of Mathematics.Dfm Strauss - 2011 - South African Journal of Philosophy 30 (1):70-94.
    In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such as (...)
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  40. added 2017-02-11
    New Zeno and Actual Infinity.Casper Storm Hansen - 2011 - Open Journal of Philosophy 1 (2):57.
    In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be resolved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno and the paradoxes (...)
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  41. added 2017-02-11
    The Infinite.Bernard Linsky - 1991 - Philosophical Books 32 (1):62-64.
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  42. added 2017-02-10
    The Infinity of God: Scientific, Theological, and Philosophical Perspectives.Benedikt Paul Goecke (ed.) - forthcoming - Notre Dame University Press.
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  43. added 2017-02-10
    Proof and Infinity: Response to André Porto.O. Chateaubriand - 2008 - Manuscrito 31 (1):45-49.
    The main issue André Porto raises in his paper concerns the use of dot notation to indicate an infinite set of hypotheses. Whereas I agree that one cannot extract a unique infinite expansion from a finite initial segment, in my response I argue that this holds for finite expansions as well. I further explain how my remarks on infinite proof structures are neither motivated by the impact of Gödel’s incompleteness theorems on Hilbert’s program, nor by a negative view of strict (...)
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  44. added 2017-02-10
    Essai de Représentation Par des Nombres Réels d'Une Analyse Infinite des Notions Individuelles Dans Une Infinité de Mondes Possibles.Miguel Sánchez-Mazas - 1989 - Argumentation 3 (1):75-96.
    The aim of this study is to try to make use of real numbers for representing an infinite analysis of individual notions in an infinity of possible worlds.As an introduction to the subject, the author shows, firstly, the possibility of representing Boole's lattice of universal notions by an associate Boole's lattice of rational numbers.But, in opposition to the universal notions, definable by a finite number of predicates, an individual notion, cannot admits this sort of definition, because each state of an (...)
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  45. added 2017-02-09
    Understanding the Infinite II: Coalgebra.David Corfield - 2011 - Studies in History and Philosophy of Science Part A 42 (4):571-579.
    In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such (...)
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  46. added 2017-02-08
    Infinite Reflections.Peter Suber - unknown
    Galileo's Paradox Contradictory or Counter-Intuitive? Imagination v. Conception Infinity as a Positive Idea Do We Experience Anything Infinite? The Sublimity of the Infinite Conclusion Bibliography Notes Appendix: A Crash Course in the Mathematics of Infinite Sets..
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  47. added 2017-02-08
    Review: Taming the Infinite. [REVIEW]Michael Potter - 1996 - British Journal for the Philosophy of Science 47 (4):609 - 619.
  48. added 2017-02-07
    Erik-Jon Gaizka, the Magician of Infinity.J. Perez Laraudogoitia - 2010 - Analysis 70 (3):451-456.
    (No abstract is available for this citation).
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  49. added 2017-02-07
    Erik-Jon Gaizka, the Magician of Infinity.Jon Pérez Laraudogoitia - 2010 - Analysis 70 (3):451 - 456.
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  50. added 2017-02-07
    Infinite War.Ellen Meiksins Wood - 2002 - Historical Materialism 10 (1):7-27.
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