Pepe Silvia
Occasional Pre-Show
I was probably trying to mask how pissed off I was that you guys decided to stand in the packed room that smelled like armpits and shame, I don't know how you guys were able to stand that for more than a minute
The Difference Between PPVs with NJexus, Charlie, and my roommates
- Thread starter JGlass
- Start date
I was probably trying to mask how pissed off I was that you guys decided to stand in the packed room that smelled like armpits and shame, I don't know how you guys were able to stand that for more than a minute
I went to college and slept with a few fat chicks before. Standing in that room was a walk in the park.
Pepe Silvia
Occasional Pre-Show
How charitable
I'd never tell a poster "tl;dr", for several reasons.
First, I'll never avoid reading something simply because I find the length challenging, especially as what qualifies here as "too long" usually comes out to about a printed paperback page.
Second, reflexive sarcasm is just too easy. I look down on posters who won't read something because they think it's too long; why would I reduce myself to the level of people I look down on in order to attack people I don't respect?
Third, I don't feel the need to express my offhand disapproval with the community at large; if I disapprove of something, I will give the target a reasonable explanation of why.
Fourth, I hadn't even paid attention to it until I heard my name mentioned.
You're an alright guy, Nate, but I'm not about to go picking fights with JGlass every time he posts a reflexively sarcastic post aimed in my direction. I've no real interest in these bar room cockwaving contests that are de rigueur around here. My name in the title again might help next time, I didn't even know about this thread until I had it pointed out to me.
Oh yeah? Can you read this?
or other uses, see Infinity (disambiguation).
Infinity (symbol: ∞
is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness".
In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.
Contents [hide]
1 History
1.1 Early Greek
1.2 Early Indian
2 Mathematics
2.1 Infinity symbol
2.2 Calculus
2.2.1 Real analysis
2.2.2 Complex analysis
2.3 Nonstandard analysis
2.4 Set theory
2.4.1 Cardinality of the continuum
2.5 Geometry and topology
2.6 Fractals
2.7 Mathematics without infinity
3 Physics
3.1 Theoretical applications of physical infinity
3.2 Cosmology
4 Logic
5 Computing
6 Arts and cognitive sciences
7 See also
8 References
8.1 Notes
8.2 Bibliography
9 External links
History
Main article: Infinity (philosophy)
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Greek
The earliest attestable accounts of mathematical infinity come from Zeno of Elea (ca. 490 BCE? – ca. 430 BCE?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20).
However, recent readings of the Archimedes Palimpsest have hinted that at least Archimedes had an intuition about actual infinite quantities.
Early Indian
The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The Indian mathematical text Surya Prajnapti (c. 400 BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
Mathematics
Infinity symbol
John Wallis introduced the infinity symbol to mathematical literature.
John Wallis is credited with introducing the infinity symbol, , in 1655 in his De sectionibus conicis.[2][3] One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.[4]
The infinity symbol is also sometimes depicted as a special variation of the ancient ouroboros snake symbol. The snake is twisted into the horizontal eight configuration while engaged in eating its own tail, a uniquely suitable symbol for endlessness.
The symbol is encoded in Unicode at U+221E ∞ infinity (HTML: ∞ &infin and in LaTeX as \infty.
Also, but less available in fonts, are encoded: U+29DC ⧜ incomplete infinity (HTML: ⧜ ISOtech entity &iinfin, U+29DD ⧝ tie over infinity (HTML: ⧝ and U+29DE ⧞ infinity negated with vertical bar (HTML: ⧞ in block Miscellaneous Mathematical Symbols-B.[5]
Calculus
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 the same properties.[6][7]
Real analysis
In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows without bound, and means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
means that f(t) does not bound a finite area from a to b
means that the area under f(t) is infinite.
means that the total area under f(t) is finite, and equals a
Infinity is also used to describe infinite series:
means that the sum of the infinite series converges to some real value a.
means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled and 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 the extended real numbers. We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.
Complex analysis
As in real analysis, in complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. means that the magnitude | x | of x grows beyond any assigned value. A point labeled can be added to 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 below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is 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 z. 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 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.
Nonstandard analysis
The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the twentieth 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, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Howard Jerome Keisler's book (see below).
Set theory
Main articles: Cardinality and Ordinal number
A different form of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null , the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work 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 a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. 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.
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and 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 one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Cardinality of the continuum
Main article: 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 (see Cantor's diagonal argument or Cantor's first uncountability proof).
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, (see Beth one). However, this hypothesis can neither be proved nor 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 the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
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 (see also Hilbert's paradox of the Grand Hotel). The second result 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 in the side of a square and those in the square.
Geometry and topology
Main article: Dimension (vector space)
Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
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—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.
Mathematics without infinity
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism.[8]
Physics
This section does not cite any references or sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009)
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value,[citation needed] for instance by taking an infinite value in an extended real number system, or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.[citation needed]
Theoretical applications of physical infinity
The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations.[citation needed] One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.
However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.
Cosmology
In 1584, 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 the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. Note that 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 one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. As 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. The Planck spacecraft launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question of whether the universe is infinite or not.
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."[9]
Computing
The IEEE floating-point standard specifies positive and negative infinity values; these can be the result of arithmetic overflow, division by zero, or other exceptional operations.
Some programming languages (for example, J and UNITY) specify greatest and least elements, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed top and bottom, or plus infinity and minus infinity; they are useful as sentinel values in algorithms involving sorting, searching or windowing. In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible to create greatest and least elements.
Arts and cognitive sciences
Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[10] Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
From the perspective of cognitive scientists George Lakoff, concepts of infinity in mathematics and the sciences are metaphors, based on what they term the Basic Metaphor of Infinity (BMI), namely the ever-increasing sequence <1,2,3,...>.
See also
0.999...
Aleph number
Infinite monkey theorem
Paradoxes of infinity
Surreal number
References
Notes
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (June 2009)
^ Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616. ISBN 0-691-11880-9. http://books.google.com/books?id=LmEZMyinoecC., Extract of page 616
^ Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2 ed.), AMS Bookstore, p. 24, ISBN 0-828-40314-7, Chapter 1, page 24
^ COLOG-88: International Conference on Computer Logic Tallinn, USSR, December 12–16, 1988: proceedings, Springer, 1990, p. 147, ISBN 3-540-52335-9, page 147
^ The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.
^ Unicode chart (odf)
^ Continuity and Infinitesimals entry by John Lane Bell in the Stanford Encyclopedia of Philosophy
^ Jesseph, Douglas Michael (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science 6 (1&2): 6–40. ISSN 1063-6145. OCLC 42413222. Archived from the original on 16 February 2010. Retrieved 16 February 2010.
^ Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 1197–1198. ISBN 0195061357.
^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
^ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 0-486-24823-2., Section 10-7, p. 229
Bibliography
or other uses, see Infinity (disambiguation).
Infinity (symbol: ∞
is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness".In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.
Contents [hide]
1 History
1.1 Early Greek
1.2 Early Indian
2 Mathematics
2.1 Infinity symbol
2.2 Calculus
2.2.1 Real analysis
2.2.2 Complex analysis
2.3 Nonstandard analysis
2.4 Set theory
2.4.1 Cardinality of the continuum
2.5 Geometry and topology
2.6 Fractals
2.7 Mathematics without infinity
3 Physics
3.1 Theoretical applications of physical infinity
3.2 Cosmology
4 Logic
5 Computing
6 Arts and cognitive sciences
7 See also
8 References
8.1 Notes
8.2 Bibliography
9 External links
History
Main article: Infinity (philosophy)
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Greek
The earliest attestable accounts of mathematical infinity come from Zeno of Elea (ca. 490 BCE? – ca. 430 BCE?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20).
However, recent readings of the Archimedes Palimpsest have hinted that at least Archimedes had an intuition about actual infinite quantities.
Early Indian
The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The Indian mathematical text Surya Prajnapti (c. 400 BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
Mathematics
Infinity symbol
John Wallis introduced the infinity symbol to mathematical literature.
John Wallis is credited with introducing the infinity symbol, , in 1655 in his De sectionibus conicis.[2][3] One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.[4]
The infinity symbol is also sometimes depicted as a special variation of the ancient ouroboros snake symbol. The snake is twisted into the horizontal eight configuration while engaged in eating its own tail, a uniquely suitable symbol for endlessness.
The symbol is encoded in Unicode at U+221E ∞ infinity (HTML: ∞ &infin
and in LaTeX as \infty.Also, but less available in fonts, are encoded: U+29DC ⧜ incomplete infinity (HTML: ⧜ ISOtech entity &iinfin
, U+29DD ⧝ tie over infinity (HTML: ⧝ and U+29DE ⧞ infinity negated with vertical bar (HTML: ⧞ in block Miscellaneous Mathematical Symbols-B.[5]Calculus
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 the same properties.[6][7]
Real analysis
In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows without bound, and means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
means that f(t) does not bound a finite area from a to b
means that the area under f(t) is infinite.
means that the total area under f(t) is finite, and equals a
Infinity is also used to describe infinite series:
means that the sum of the infinite series converges to some real value a.
means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled and 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 the extended real numbers. We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.
Complex analysis
As in real analysis, in complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. means that the magnitude | x | of x grows beyond any assigned value. A point labeled can be added to 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 below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is 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 z. 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 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.
Nonstandard analysis
The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the twentieth 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, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Howard Jerome Keisler's book (see below).
Set theory
Main articles: Cardinality and Ordinal number
A different form of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null , the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work 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 a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. 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.
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and 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 one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Cardinality of the continuum
Main article: 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 (see Cantor's diagonal argument or Cantor's first uncountability proof).
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, (see Beth one). However, this hypothesis can neither be proved nor 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 the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
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 (see also Hilbert's paradox of the Grand Hotel). The second result 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 in the side of a square and those in the square.
Geometry and topology
Main article: Dimension (vector space)
Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
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—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.
Mathematics without infinity
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism.[8]
Physics
This section does not cite any references or sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009)
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value,[citation needed] for instance by taking an infinite value in an extended real number system, or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.[citation needed]
Theoretical applications of physical infinity
The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations.[citation needed] One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.
However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.
Cosmology
In 1584, 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 the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. Note that 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 one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. As 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. The Planck spacecraft launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question of whether the universe is infinite or not.
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."[9]
Computing
The IEEE floating-point standard specifies positive and negative infinity values; these can be the result of arithmetic overflow, division by zero, or other exceptional operations.
Some programming languages (for example, J and UNITY) specify greatest and least elements, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed top and bottom, or plus infinity and minus infinity; they are useful as sentinel values in algorithms involving sorting, searching or windowing. In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible to create greatest and least elements.
Arts and cognitive sciences
Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[10] Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
From the perspective of cognitive scientists George Lakoff, concepts of infinity in mathematics and the sciences are metaphors, based on what they term the Basic Metaphor of Infinity (BMI), namely the ever-increasing sequence <1,2,3,...>.
See also
0.999...
Aleph number
Infinite monkey theorem
Paradoxes of infinity
Surreal number
References
Notes
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (June 2009)
^ Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616. ISBN 0-691-11880-9. http://books.google.com/books?id=LmEZMyinoecC., Extract of page 616
^ Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2 ed.), AMS Bookstore, p. 24, ISBN 0-828-40314-7, Chapter 1, page 24
^ COLOG-88: International Conference on Computer Logic Tallinn, USSR, December 12–16, 1988: proceedings, Springer, 1990, p. 147, ISBN 3-540-52335-9, page 147
^ The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.
^ Unicode chart (odf)
^ Continuity and Infinitesimals entry by John Lane Bell in the Stanford Encyclopedia of Philosophy
^ Jesseph, Douglas Michael (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science 6 (1&2): 6–40. ISSN 1063-6145. OCLC 42413222. Archived from the original on 16 February 2010. Retrieved 16 February 2010.
^ Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 1197–1198. ISBN 0195061357.
^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
^ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 0-486-24823-2., Section 10-7, p. 229
Bibliography
Pepe Silvia
Occasional Pre-Show
GD, stop being asian
Lee
Is it a bird? Is it a plane? No it's Supermod!
TL;DR
This thread has nothing to do with you.
Yes, if you define charitable as looking to shove my junk in anything that walked, that night... sure.
Stormtrooper
Championship Contender
tl;dr
Yes, but I come and go as I please, and as I thought we've firmly established, your opinion on me is totally (literal usage, not figurative) unimportant to my thought process.
BTW, I saw you call me "too cool to care" in another post. You just don't fucking get it. It's not an issue of coolness. I want to impress none of you. If I do, that's great, but it is not a driving concern of mine. I'm not telling people here to fuck off because I'm trying to fit in and be an internet bad ass. I'm telling people to fuck off. There is absolutely no desire on my behalf to appear popular to anyone here.
Board at large: If you think it's you I'm telling to fuck off, you're probably right. If you don't think it's you, you're also probably right. If you think I'm telling people to fuck off because I'm hoping to trade a couple of friendly tough guy blows and then be your friend, you could not be more wrong.
To whoever posted the encyclopedia post, interest is also a factor. If I don't give a shit what you have to say, I won't read it, regardless of if it's one sentence or one thousand. I also won't feel the need to tell the board at large that I am uninterested, because that observation does nothing for a conversation besides puffing out one's chest.
If something's uninteresting, people tend not to pay attention to it, rather than have to tell someone they aren't paying attention to it. Fuck, it's like none of you have ever courted a woman.
Goliath Uterus
Getting Noticed By Management
Jesus dude, nobody cares.
Is there going to be a SummerSlam gathering? I'd try to attend, even if it would be sans JGlass.
Oh I'll be in NJ for SummerSlam this year, you can count on dat!
Then why the comment about how your name should have been in the title if I wanted to get your attention? I didn't want your attention, and never really have.
No, it is you who doesn't get that I was being very sarcastic there. Really, I care as little about you as little as care about me.
Muffin Top Merkley
Be a man!
I'd love to meet up with nearly anyone on the forum for some wrestling watching & general hang time (clothing and alcohol optional).
The closest guy to me that I know of is Arkham Noir, who lives about an hour away.
For some reason, I've always said (read: hoped) that WrestleMania would be coming to Toronto next year & that I was going to go no matter what. At first, I was upset when I first read the rumours of it being in New Jersey, but lately I've been reading that nearly everyone plans to attend & it makes me want to make the 8 hour drive myself.
The closest guy to me that I know of is Arkham Noir, who lives about an hour away.
For some reason, I've always said (read: hoped) that WrestleMania would be coming to Toronto next year & that I was going to go no matter what. At first, I was upset when I first read the rumours of it being in New Jersey, but lately I've been reading that nearly everyone plans to attend & it makes me want to make the 8 hour drive myself.
Goliath Uterus
Getting Noticed By Management
I would totally meet up with you, we haven't even fucked yet.
Ok... show of hands...
Does anybody... ANYBODY... care about this?
<crickets>
Ok, good. then it's not just me.
Muffin Top Merkley
Be a man!
The second half of your post is going to make this seem weird but...
Where do you live?
Goliath Uterus
Getting Noticed By Management
Don't worry, I'd lay you down and cum in your asshole too.
Muffin Top Merkley
Be a man!
I don't think I knew you were from Canada. You want in on this too?
Goliath Uterus
Getting Noticed By Management
And I live in Kansas.
Pepe Silvia
Occasional Pre-Show
Please let me suck your cock. It must be huge like Cloud's sword.
Please?
(Don't let them know homosocial isn't sexual in the slightest)
I was a fan of wrestling years ago, not anymore. Homosocial intimacy comment was directed at JGlass btw, not you & bieber if there was any confusion.
Lmao please make more gay comments when you respond to where I said homosocial intimacy, the irony is hysterical hahaha.