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75-418: The hypothesis has proved controversial. Jürgen Schmidhuber argues that it is not possible to assign an equal weight or probability to all mathematical objects a priori due to there being infinitely many of them. Physicists Piet Hut and Mark Alford have suggested that the idea is incompatible with Gödel's first incompleteness theorem . Tegmark replies that not only is the universe mathematical, but it

150-632: A hyperplane at infinity for general dimensions , each consisting of points at infinity . In complex analysis the symbol ∞ {\displaystyle \infty } , called "infinity", denotes an unsigned infinite limit . The expression x → ∞ {\displaystyle x\rightarrow \infty } means that the magnitude  | x | {\displaystyle |x|} of  x {\displaystyle x} grows beyond any assigned value. A point labeled ∞ {\displaystyle \infty } can be added to

225-562: A consistent and unified picture, so math is converging in this sense." In his 2014 book on the MUH, Tegmark argues that the resolution is not that we invent the language of mathematics, but that we discover the structure of mathematics. Don Page has argued that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it

300-405: A halting program, due to the undecidability of the halting problem . In response, Tegmark notes that a constructive mathematics formalized measure of free parameter variations of physical dimensions, constants, and laws over all universes has not yet been constructed for the string theory landscape either, so this should not be regarded as a "show-stopper". It has also been suggested that

375-410: A limit, infinity can be also used as a value in the extended real number system. Points labeled + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us

450-506: A mathematico-philosophic address given in 1930 with: Mathematics is the science of the infinite. The infinity symbol ∞ {\displaystyle \infty } (sometimes called the lemniscate ) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY ( ∞ ) and in LaTeX as \infty . It

525-561: A nested hierarchy of increasing diversity, with worlds corresponding to different sets of initial conditions (level 1), physical constants (level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4). Andreas Albrecht of Imperial College in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct

600-449: A philosophical concept. The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron , which means "unbounded", "indefinite", and perhaps can be translated as "infinite". Aristotle (350 BC) distinguished potential infinity from actual infinity , which he regarded as impossible due to

675-428: A physically 'real' world". The theory can be considered a form of Pythagoreanism or Platonism in that it proposes the existence of mathematical entities; a form of mathematicism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism . Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it

750-578: 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. 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. Piet Hut Piet Hut (born September 26, 1952)

825-456: 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

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900-482: A theory where everything we see is all there is". Jürgen Schmidhuber argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight,' there is no way of assigning equal non-vanishing probability to all (infinitely many) mathematical structures." Schmidhuber puts forward a more restricted ensemble which admits only universe representations describable by constructive mathematics , that is, computer programs ; e.g.,

975-544: A tortoise, giving the latter a head start. 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

1050-465: A tree-based data structure, the Barnes–Hut method significantly speeds up the calculation of the gravitational motion of large numbers of stars, making accessible such problems as collisions between galaxies. Barnes–Hut simulation algorithm, which has become a standard in n-body problems , reduces its complexity to N log N. Hut introduced the concept of pseudo-synchronicity, which is now widely cited in

1125-714: Is a Dutch astrophysicist, who divides his time between research in computer simulations of dense stellar systems and broadly interdisciplinary collaborations, ranging from other fields in natural science to computer science, cognitive psychology and philosophy. He is currently the Head of the Program in Interdisciplinary Studies at the Institute for Advanced Study (IAS) in Princeton, New Jersey, United States. Asteroid 17031 Piethut

1200-585: Is also computable . Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics — specifically, a mathematical structure . Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in

1275-481: Is an aberration from the general trend of this period. Zeno of Elea ( c.  495 – c.  430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, 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". Achilles races

1350-544: Is based on the radical Platonist view that math is an external reality. However, Jannes argues that "mathematics is at least in part a human construction", on the basis that if it is an external reality, then it should be found in some other animals as well: "Tegmark argues that, if we want to give a complete description of reality, then we will need a language independent of us humans, understandable for non-human sentient entities, such as aliens and future supercomputers". Brian Greene argues similarly: "The deepest description of

1425-730: 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 one-to-one 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". Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers . Ordinal numbers characterize well-ordered 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

1500-401: Is just another metaphysical theory like solipsism... In the end the metaphysics just demands that we use a different language for saying what we already knew." Tegmark responds that "The notion of a mathematical structure is rigorously defined in any book on Model Theory ", and that non-human mathematics would only differ from our own "because we are uncovering a different part of what is in fact

1575-580: Is logical nonsense to talk of Level 4 in the sense of the co-existence of all mathematical structures." This means there can only be one mathematical corpus. Tegmark responds that "This is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified." Alexander Vilenkin comments that "The number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. This seems to be in conflict with

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1650-650: Is named after him, in honor of his work in planetary dynamics and for co-founding the B612 Foundation , which focuses on prevention of asteroid impacts on Earth. In the Netherlands, Hut did a double PhD program, at Utrecht University, in particle physics under Martinus Veltman and in Amsterdam in astrophysics under Ed van den Heuvel , resulting in a PhD at the University of Amsterdam . Previously an assistant professor at

1725-511: Is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, c = ℵ 1 = ℶ 1 {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}} . This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory , even assuming the Axiom of Choice . Cardinal arithmetic can be used to show not only that

1800-411: Is preferred over other theories-of-everything by Occam's Razor . Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis ( CUH ), which says that the mathematical structure that is our external physical reality is defined by computable functions . The MUH is related to Tegmark's categorization of four levels of the multiverse . This categorization posits

1875-436: Is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter 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 The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable , innumerable, and infinite. Each of these

1950-569: Is still used). In particular, in modern mathematics, lines are infinite sets . 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

2025-454: Is the case of iterated loop spaces . 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 . Leopold Kronecker was skeptical of

2100-464: Is to offer a new hypothesis "that only Gödel-complete ( fully decidable ) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe." Tegmark goes on to note that although conventional theories in physics are Gödel-undecidable,

2175-504: The Global Digital Mathematics Library and Digital Library of Mathematical Functions , linked open data representations of formalized fundamental theorems intended to serve as building blocks for additional mathematical results. He explicitly includes universe representations describable by non-halting programs whose output bits converge after finite time, although the convergence time itself may not be predictable by

2250-611: The University of California, Berkeley , Hut was in 1985, at the age of 32, appointed as a full professor at the Institute for Advanced Study. At the time, he was the youngest professor appointed there. Hut became a corresponding member of the Royal Netherlands Academy of Arts and Sciences in 1996. An accomplished astrophysicist, Hut is best known for the Barnes–Hut simulation algorithm, developed with Joshua Barnes. By using

2325-684: The World Economic Forum in Davos, Switzerland, and he has been invited as a member of the Husserl Circle. Hut is one of the founders of the Kira Institute . In July, 2000, IAS sued Hut in federal district court, seeking to enforce a 1996 agreement in which Hut had promised to resign by mid-2001. According to IAS Director Phillip Griffiths , Hut had been hired in expectation of his eventually succeeding professor John N. Bahcall as leader of

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2400-413: The cardinality of the line) is larger than the number of integers . 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

2475-438: The extended real numbers . We can also treat + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } as the same, leading to the one-point compactification of the real numbers, which is the real projective line . Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and

2550-438: The mirror test of self-consciousness . But a few surprising examples of mathematical abstraction notwithstanding (for example, chimpanzees can be trained to carry out symbolic addition with digits, or the report of a parrot understanding a "zero-like concept"), all examples of animal intelligence with respect to mathematics are limited to basic counting abilities. He adds, "non-human intelligent beings should exist that understand

2625-533: 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 non-standard calculus is fully developed in Keisler (1986) . A different form of "infinity" is the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor . In this system,

2700-412: The MUH is inconsistent with Gödel's incompleteness theorem . In a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford, the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is 'out there' is incompatible with the idea that it consists of formal systems." Tegmark's response

2775-518: The MUH, stating that an infinite ensemble of completely disconnected universes is "completely untestable, despite hopeful remarks sometimes made, see, e.g., Tegmark (1998)." Tegmark maintains that MUH is testable , stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is". The MUH

2850-512: The Riemann sphere taking the value of ∞ {\displaystyle \infty } at the poles. The domain of a complex-valued 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 ). The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In

2925-707: The actual mathematical structure describing our world could still be Gödel-complete, and "could in principle contain observers capable of thinking about Gödel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Gödel-incomplete formal systems like Peano arithmetic ." In he gives a more detailed response, proposing as an alternative to MUH the more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecidable or uncomputable theorems. Tegmark admits that this approach faces "serious challenges", including (a) it excludes much of

3000-473: The argument. Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1 , a + a x + a x 2 + a x 3 + a x 4 + a x 5 + ⋯ = a 1 − x . {\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.} Suppose that Achilles

3075-406: The astrophysics group, but "was not performing" at the required level. Hut's rebuttal was first, that his work was not inferior but only (to some eyes) unfashionable, and second, that he had been coerced into signing any agreement. Many prominent astrophysicists defended the quality of Hut's work, while others based their support on the importance of academic tenure to creative scholarship. The case

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3150-443: 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. 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

3225-450: The beauty and simplicity of the theories describing our world". He goes on to note that Tegmark's solution to this problem, the assigning of lower "weights" to the more complex structures seems arbitrary ("Who determines the weights?") and may not be logically consistent ("It seems to introduce an additional mathematical structure, but all of them are supposed to be already included in the set"). Tegmark has been criticized as misunderstanding

3300-451: The complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold , or Riemann surface , called the extended complex plane or the Riemann sphere . Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in

3375-507: The demise of the dinosaurs, when he edited a review article for Nature with four paleontologists, two geologists and one other astrophysicist. He has also widely engaged in joint research with computer scientists and philosophers and cognitive psychologists. In recognition of his work, he was invited to participate in various conferences, spanning a range from a workshop with the 14th Dalai Lama and five physicists in Dharamsala , India to

3450-525: The existence of Grothendieck universes , very large infinite sets, for solving a long-standing problem that is stated in terms of elementary arithmetic . In physics and cosmology , whether the universe is spatially infinite or not , is an open question. 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

3525-409: The first transfinite cardinal is aleph-null ( 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. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as

3600-434: 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. 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. For example, if a line is viewed as the set of all of its points, their infinite number (i.e.,

3675-464: The integers is countably infinite . If a set is too large to be put in one-to-one 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. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes. One of Cantor's most important results

3750-466: The language of advanced mathematics. However, none of the non-human intelligent beings that we know of confirm the status of (advanced) mathematics as an objective language." In the paper "On Math, Matter and Mind" the secularist viewpoint examined argues that math is evolving over time, there is "no reason to think it is converging to a definite structure, with fixed questions and established ways to address them", and also that "The Radical Platonist position

3825-574: The literature on tidal evolution of exoplanets . He co-authored a graduate textbook The Gravitational Million Body Problem , invented a mathematical sequence called Piet Hut's "coat-hanger" sequence, and has pioneered the use of virtual worlds for research and education in (astro)physics. Hut is one of the founders of the B612 Foundation , MODEST, MICA, ACS, the GRAPE (Gravity Pipe) project, and AMUSE. Hut's broadly interdisciplinary research started with his study of an asteroid impact to explain

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3900-581: The mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH". Stoeger, Ellis, and Kircher note that in a true multiverse theory, "the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one. This lack of any causal connection in such multiverses really places them beyond any scientific support". Ellis specifically criticizes

3975-437: The nature and application of Occam's razor ; Massimo Pigliucci reminds that "Occam's razor is just a useful heuristic , it should never be used as the final arbiter to decide which theory is to be favored". Infinitely Infinity is something which is boundless, endless, or larger than any natural number . It is often denoted by the infinity symbol ∞ {\displaystyle \infty } . From

4050-640: 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 . 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. The first published proposal that

4125-485: 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 finite-dimensional space. The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval ( − ⁠ π / 2 ⁠ , ⁠ π / 2 ⁠ ) and R . The second result

4200-569: The order of 1 ∞ . {\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (1656), 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." In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas . Hermann Weyl opened

4275-485: The original straight line] that the [sum of the internal angles] is less than two right angles. Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", 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"

4350-402: 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

4425-420: The prime numbers , Euclid "was the first to overcome the horror of the infinite". 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

4500-918: The same properties in accordance with the Law of continuity . In real analysis , the symbol ∞ {\displaystyle \infty } , called "infinity", is used to denote an unbounded limit . The notation x → ∞ {\displaystyle x\rightarrow \infty } means that  x {\displaystyle x} increases without bound, and x → − ∞ {\displaystyle x\to -\infty } means that  x {\displaystyle x} decreases without bound. For example, if f ( t ) ≥ 0 {\displaystyle f(t)\geq 0} for every  t {\displaystyle t} , then Infinity can also be used to describe infinite series , as follows: In addition to defining

4575-403: 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

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4650-421: 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 z / 0 = ∞ {\displaystyle z/0=\infty } for any nonzero complex number   z {\displaystyle z} . In this context, it is often useful to consider meromorphic functions as maps into

4725-441: The square. 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

4800-494: 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 and the infinitesimal calculus , mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli ) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with

4875-518: 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 two-dimensional 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

4950-692: The universe is infinite came from Thomas Digges in 1576. 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." Cosmologists have long sought to discover whether infinity exists in our physical universe : Are there an infinite number of stars? Does

5025-419: The universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn't, in any fundamental way, depend on ideas of our making." However, there are many non-human entities, plenty of which are intelligent, and many of which can apprehend, memorise, compare and even approximately add numerical quantities. Several animals have also passed

5100-411: 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

5175-469: The various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" 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." It has also been maintained, that, in proving the infinitude of

5250-675: 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 non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry. Before

5325-448: 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

5400-430: Was further subdivided into three orders: 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 ∞ {\displaystyle \infty } for such a number in his De sectionibus conicis , and exploited it in area calculations by dividing the region into infinitesimal strips of width on

5475-447: Was introduced in 1655 by John Wallis , and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology . Gottfried Leibniz , one of the co-inventors 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

5550-399: Was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves , curved lines that twist and turn enough to fill the whole of any square, or cube , or hypercube , or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in

5625-556: Was that the cardinality of the continuum c {\displaystyle \mathbf {c} } is greater than that of the natural numbers ℵ 0 {\displaystyle {\aleph _{0}}} ; that is, there are more real numbers R than natural numbers N . Namely, Cantor showed that c = 2 ℵ 0 > ℵ 0 {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}} . The continuum hypothesis states that there

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