Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol^{[1]} and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli)^{[2]} regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.^{[1]} At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.^{[1]}^{[3]} For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers.^{[4]} In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.^{[1]} The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of Grothendieck universes, very large infinite sets,^{[5]} for solving a longstanding problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the universe is spatially infinite is an open question.
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How big is infinity?  Dennis Wildfogel
Transcription
When I was in fourth grade, my teacher said to us one day: "There are as many even numbers as there are numbers." "Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them. But, on the other hand, even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole numbers than even numbers, right? To see what my teacher was getting at, let's first think about what it means for two sets to be the same size. What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it's actually simpler than that. I don't have to count, I only need to see that I can match them up, one to one. In fact, we think that some ancient people who spoke languages that didn't have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and then putting those stones back one by one when the sheep return, so you know if any are missing without really counting. As another example of matching being more fundamental than counting, if I'm speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don't know how many there are of either. So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. So my fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double. As you can see, the bottom row contains all the even numbers, and we have a onetoone match. That is, there are as many even numbers as there are numbers. But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don't have the same number of fingers on my right hand as I do on my left? Of course not. It doesn't matter if you try to match the elements in some way and that doesn't work, that doesn't convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements. Can you make a list of all the fractions? This might be hard, there are a lot of fractions! And it's not obvious what to put first, or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg Cantor, in the late eighteen hundreds. First, we put all the fractions into a grid. They're all there. For instance, you can find, say, 117/243, in the 117th row and 223rd column. Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we've already picked. And so we get a list of all the fractions, which means we've created a onetoone match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions. Ok, here's where it gets really interesting. You may know that not all real numbers —that is, not all the numbers on a number line— are fractions. The square root of two and pi, for instance. Any number like this is called irrational. Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are called rationals; meaning the rest are nonrational, that is, irrational. Irrationals are represented by infinite, nonrepeating decimals. So, can we make a onetoone match between the whole numbers and the set of all the decimals, both the rationals and the irrationals? That is, can we make a list of all the decimal numbers? Candor showed that you can't. Not merely that we don't know how, but that it can't be done. Look, suppose you claim you have made a list of all the decimals. I'm going to show you that you didn't succeed, by producing a decimal that is not on your list. I'll construct my decimal one place at a time. For the first decimal place of my number, I'll look at the first decimal place of your first number. If it's a one, I'll make mine a two; otherwise I'll make mine a one. For the second place of my number, I'll look at the second place of your second number. Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one. See how this is going? The decimal I've produced can't be on your list. Why? Could it be, say, you 143rd number? No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number. I made it that way. Your list is incomplete. It doesn't contain my decimal number. And, no matter what list you give me, I can do the same thing, and produce a decimal that's not on that list. So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity that the infinity of whole numbers. So, even though we're familiar with only a few irrationals, like square root of two and pi, the infinity of all irrationals is actually greater that the infinity of fractions. Someone once said that the rationals —the fractions— are like the stars in the night sky; the irrationals are like the blackness. Cantor also showed that, for any infinite set, forming a new set made of all the subsets of the original set represents a bigger infinity than that original set. This means that, once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on. And so, there are an infinite number of infinities of different sizes. If these ideas make you unconfortable, you are not alone. Some of the greatest mathematicians of Cantor's day were very upset with this stuff. They tried to make this different infinities irrelevant, to make mathematics work without them somehow. Cantor was even vilified personally, and it got so bad for him that he suffered severe depression, and spent the last half of his life in and out of mental institutions. But eventually his ideas won out. Today, they're considered fundamental and magnificent. All research mathematicians accept these ideas, every college math major learns them, and I've explained them to you in a few minutes. Some day, perhaps, they'll be common knowledge. There's more. We just pointed out that the set of decimal numbers —that is, the real numbers— is a bigger infinity than the set of whole numbers. Candor wondered wether there are infinities of different sizes between these two infinities. He didn't believe there would, but could prove it. Candor's conjecture became known as the continuum hypothesis. In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics. The 20th century saw a resolution of this problem, but in a completely unexpected, paradigmshattering way. In the 1920s, Kurt Gödel showed that you can never prove that the continuum hypothesis is false. Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true. Taken together, these results mean that there are unanswerable questions in mathematics. A very stunning conclusion. Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, mathematics has some truly amazing things for us to think about.
History
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
Early Greek
The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a preSocratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".^{[1]}^{[6]}
Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce.^{[7]} It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite"^{[8]}^{[9]} which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."^{[10]} It has also been maintained, that, in proving the infinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite".^{[11]} There is a similar controversy concerning Euclid's parallel postulate, sometimes translated:
If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.^{[12]}
Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",^{[13]} thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.^{[14]}
Zeno: Achilles and the tortoise
Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,^{[15]} especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".^{[16]}
Achilles races a tortoise, giving the latter a head start.
 Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
 Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
 Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
 Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
Etc.
Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
Finally, in 1821, AugustinLouis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,^{[17]}
Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
Early Indian
The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:^{[18]}
 Enumerable: lowest, intermediate, and highest
 Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
 Infinite: nearly infinite, truly infinite, infinitely infinite
17th century
In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation for such a number in his De sectionibus conicis,^{[19]} and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ^{[20]} But in Arithmetica infinitorum (1656),^{[21]} he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."^{[22]}
In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.^{[23]}
Mathematics
Hermann Weyl opened a mathematicophilosophic address given in 1930 with:^{[24]}
Mathematics is the science of the infinite.
Symbol
The infinity symbol (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (∞)^{[25]} and in LaTeX as \infty
.^{[26]}
It was introduced in 1655 by John Wallis,^{[27]}^{[28]} and since its introduction, it has also been used outside mathematics in modern mysticism^{[29]} and literary symbology.^{[30]}
Calculus
Gottfried Leibniz, one of the coinventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of continuity.^{[31]}^{[2]}
Real analysis
In real analysis, the symbol , called "infinity", is used to denote an unbounded limit.^{[32]} The notation means that increases without bound, and means that decreases without bound. For example, if for every , then^{[33]}
 means that does not bound a finite area from to
 means that the area under is infinite.
 means that the total area under is finite, and is equal to
Infinity can also be used to describe infinite series, as follows:
 means that the sum of the infinite series converges to some real value
 means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.^{[34]}
In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled and can be added to the topological space of the real numbers, producing the twopoint compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers.^{[35]} We can also treat and as the same, leading to the onepoint compactification of the real numbers, which is the real projective line.^{[36]} Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in threedimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.^{[37]}
Complex analysis
In complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. The expression means that the magnitude of grows beyond any assigned value. A point labeled can be added to the complex plane as a topological space giving the onepoint compactification of the complex plane. When this is done, the resulting space is a onedimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere.^{[38]} Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely for any nonzero complex number . In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of at the poles. The domain of a complexvalued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
Nonstandard analysis
The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to nonstandard calculus is fully developed in Keisler (1986).
Set theory
A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is alephnull (ℵ_{0}), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.^{[1]}
Dedekind's approach was essentially to adopt the idea of onetoone correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a onetoone manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".^{[citation needed]}
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize wellordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in onetoone correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.^{[39]}^{[40]}^{[page needed]} Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.^{[citation needed]}
Cardinality of the continuum
One of Cantor's most important results was that the cardinality of the continuum is greater than that of the natural numbers ; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that .^{[41]}
The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, .
This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.^{[42]}
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finitedimensional space.^{[citation needed]}
The first of these results is apparent by considering, for instance, the tangent function, which provides a onetoone correspondence between the interval (−π/2, π/2) and R.
The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the spacefilling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finitedimensional space. These curves can be used to define a onetoone correspondence between the points on one side of a square and the points in the square.^{[43]}
Geometry
Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and nonparallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
In particular, in modern mathematics, lines are infinite sets.
Infinite dimension
The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
Fractals
The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.^{[citation needed]}
Mathematics without infinity
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.^{[44]}
Physics
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.^{[45]}
Cosmology
The first published proposal that the universe is infinite came from Thomas Digges in 1576.^{[46]} Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."^{[47]}
Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is still an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The twodimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.^{[48]}
The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.^{[49]}^{[50]}^{[51]}
However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider twodimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for threedimensional space.^{[52]}
The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.^{[53]} Also, cyclic models posit an infinite amount of Big Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.^{[54]}
Logic
In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."^{[55]}
Computing
The IEEE floatingpoint standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.^{[56]}
Some programming languages, such as Java^{[57]} and J,^{[58]} allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.^{[citation needed]}
In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floatingpoint data type, the infinity values may still be accessible and usable as the result of certain operations.^{[citation needed]}
In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.
Arts, games, and cognitive sciences
Perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.^{[59]} Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.^{[60]}
Variations of chess played on an unbounded board are called infinite chess.^{[61]}^{[62]}
Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the everincreasing sequence <1,2,3,...>.^{[63]}
See also
References
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 ^ Bell, John Lane. "Continuity and Infinitesimals". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
 ^ Taylor 1955, p. 63
 ^ These uses of infinity for integrals and series can be found in any standard calculus text, such as, Swokowski 1983, pp. 468–510
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 ^ Gemignani 1990, p. 177
 ^ Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, p. 27, ISBN 9780521483643
 ^ Rao, Murali; Stetkær, Henrik (1991). Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory. World Scientific. p. 113. ISBN 9789810203757.
 ^ "Infinity". math.dartmouth.edu. Retrieved 20191116.
 ^ Moore, A.W. (1991). The Infinite. Routledge.
 ^ Dauben, Joseph (1993). "Georg Cantor and the Battle for Transfinite Set Theory" (PDF). 9th ACMS Conference Proceedings: 4.
 ^ Cohen 1963, p. 1143
 ^ Sagan 1994, pp. 10–12
 ^ Kline 1972, pp. 1197–1198
 ^ Doric Lenses Archived 20130124 at the Wayback Machine – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
 ^ John Gribbin (2009), In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality, ISBN 9780470613528. p. 88
 ^ Brake, Mark (2013). Alien Life Imagined: Communicating the Science and Culture of Astrobiology (illustrated ed.). Cambridge University Press. p. 63. ISBN 9780521491297.
 ^ Koupelis, Theo; Kuhn, Karl F. (2007). In Quest of the Universe (illustrated ed.). Jones & Bartlett Learning. p. 553. ISBN 9780763743871. Extract of p. 553
 ^ "Will the Universe expand forever?". NASA. 24 January 2014. Archived from the original on 1 June 2012. Retrieved 16 March 2015.
 ^ "Our universe is Flat". FermiLab/SLAC. 7 April 2015. Archived from the original on 10 April 2015.
 ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. LXXIV1: 30.
 ^ Weeks, Jeffrey (2001). The Shape of Space. CRC Press. ISBN 9780824707095.
 ^ Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
 ^ McKee, Maggie (25 September 2014). "Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang". Nautilus. No. 17. NautilusThink Inc. Retrieved 31 March 2017.
 ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
 ^ "Infinity and NaN (The GNU C Library)". www.gnu.org. Retrieved 20210315.
 ^ Gosling, James; et al. (27 July 2012). "4.2.3.". The Java Language Specification (Java SE 7 ed.). California: Oracle America, Inc. Archived from the original on 9 June 2012. Retrieved 6 September 2012.
 ^ Stokes, Roger (July 2012). "19.2.1". Learning J. Archived from the original on 25 March 2012. Retrieved 6 September 2012.
 ^ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 9780486248233., Section 107, p. 229 Archived 20160516 at the Wayback Machine
 ^ Schattschneider, Doris (2010). "The Mathematical Side of M. C. Escher" (PDF). Notices of the AMS. 57 (6): 706–718.
 ^ Infinite chess at the Chess Variant Pages Archived 20170402 at the Wayback Machine An infinite chess scheme.
 ^ "Infinite Chess, PBS Infinite Series" Archived 20170407 at the Wayback Machine PBS Infinite Series, with academic sources by J. Hamkins (infinite chess: Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess". arXiv:1302.4377 [math.LO]. and Evans, C.D.A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ω^4$". arXiv:1510.08155 [math.LO].).
 ^ Elglaly, Yasmine Nader; Quek, Francis. "Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez" (PDF). CHI 2009. Archived from the original (PDF) on 20200226. Retrieved 20210325.
Bibliography
 Cajori, Florian (1993) [1928 & 1929], A History of Mathematical Notations (Two Volumes Bound as One), Dover, ISBN 9780486677668
 Gemignani, Michael C. (1990), Elementary Topology (2nd ed.), Dover, ISBN 9780486665221
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 Wallace, David Foster (2004), Everything and More: A Compact History of Infinity, Norton, W.W. & Company, Inc., ISBN 9780393326291
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 Aczel, Amir D. (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 9780743422994.
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 Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
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External links
 "The Infinite". Internet Encyclopedia of Philosophy.
 Infinity on In Our Time at the BBC
 A Crash Course in the Mathematics of Infinite Sets Archived 20100227 at the Wayback Machine, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The standalone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
 Infinite Reflections Archived 20091105 at the Wayback Machine, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
 Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran. Archived from the original on 20171022. Retrieved 20130406.
 Hotel Infinity
 John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 20060916 at the Wayback Machine, MacTutor History of Mathematics archive.
 John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 20081220 at the Wayback Machine, MacTutor History of Mathematics archive.
 Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
 The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
 Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)