Infinity

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol ${\displaystyle \infty }$.

Since the time of the

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 long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the universe is spatially infinite is an open question.

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 pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".^[1][6]

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

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,^[17]

$${\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}$$

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 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

$${\displaystyle 10+0.1+0.001+0.00001+\cdots ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}$$

Early Indian

The

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,^[19] and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\displaystyle {\tfrac {1}{\infty }}.}$^[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 mathematico-philosophic address given in 1930 with:^[24]

Mathematics is the science of the infinite.

Symbol

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 (&infin;)^[25] and in LaTeX as \infty.^[26]

It was introduced in 1655 by

In real analysis, the symbol ${\displaystyle \infty }$, called "infinity", is used to denote an unbounded limit.^[32] The notation ${\displaystyle x\rightarrow \infty }$ means that ${\displaystyle x}$ increases without bound, and ${\displaystyle x\to -\infty }$ means that ${\displaystyle x}$ decreases without bound. For example, if ${\displaystyle f(t)\geq 0}$ for every ${\displaystyle t}$, then^[33]

• ${\displaystyle \int _{a}^{b}f(t)\,dt=\infty }$ means that ${\displaystyle f(t)}$ does not bound a finite area from ${\displaystyle a}$ to ${\displaystyle b.}$
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }$ means that the area under ${\displaystyle f(t)}$ is infinite.
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}$ means that the total area under ${\displaystyle f(t)}$ is finite, and is equal to ${\displaystyle a.}$

Infinity can also be used to describe

• ${\displaystyle \sum _{i=0}^{\infty }f(i)=a}$ means that the sum of the infinite series converges to some real value ${\displaystyle a.}$
• ${\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }$ 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 ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ can be added to the

We can also treat ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ as the same, leading to the

In complex analysis the symbol ${\displaystyle \infty }$, called "infinity", denotes an unsigned infinite limit. The expression ${\displaystyle x\rightarrow \infty }$ means that the magnitude ${\displaystyle |x|}$ of ${\displaystyle x}$ grows beyond any assigned value. A point labeled ${\displaystyle \infty }$ can be added to the complex plane as a

, namely ${\displaystyle z/0=\infty }$ for any nonzero

${\displaystyle z}$

${\displaystyle \infty }$

Nonstandard analysis

The original formulation of

A different form of "infinity" are the

One of Cantor's most important results was that the cardinality of the continuum ${\displaystyle \mathbf {c} }$ is greater than that of the natural numbers ${\displaystyle {\aleph _{0}}}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}$.^[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, ${\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}$.

Further information:

Axiom of Choice.^[42]

finite-dimensional space.^{[citation needed}

]

The first of these results is apparent by considering, for instance, the

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 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

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 to be distinguished in projective geometry.

Before the use of

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

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

constructivism and intuitionism.^[44]

Physics

In

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 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 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

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 two-dimensional 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 three-dimensional 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

arithmetic overflow, division by zero, and other exceptional operations.^[56]

Some

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 floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.^{[citation needed]}

In programming, an

loop

whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.^{[citation needed}

]

Variations of chess played on an unbounded board are called infinite chess.^[60][61]

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 ever-increasing sequence <1,2,3,...>.^[62]

See also

• 0.999...
• Aleph number
• Ananta
• Exponentiation
• Indeterminate form
• Names of large numbers
• Infinite monkey theorem
• Infinite set
• Infinitesimal
• Paradoxes of infinity
• Supertask
• Surreal number

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Sources

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External links

Look up infinity in Wiktionary, the free dictionary.

Wikibooks has a book on the topic of: Infinity is not a number

Wikimedia Commons has media related to Infinity.

Wikiquote has quotations related to Infinity.

• "The Infinite". Internet Encyclopedia of Philosophy.
• Infinity on In Our Time at the BBC
• A Crash Course in the Mathematics of Infinite Sets Archived 2010-02-27 at the Wayback Machine, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
• Infinite Reflections Archived 2009-11-05 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 2017-10-22. Retrieved 2013-04-06.
• Hotel Infinity
• John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 2006-09-16 at the
  MacTutor History of Mathematics archive
  .
• John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 2008-12-20 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)