Infinity Totally Explained

Everything about Infinity totally explained

Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.
In mathematics, "infinity" is often used in contexts where it's treated as if it were a number (for example, it counts or measures things: "an infinite number of terms") but it's a different type of "number" from the real numbers. Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.

History

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". ť Pūrṇam adaḥ pūrṇam idam

   Pūrṇāt pūrṇam udacyate ť Pūrṇasya pūrṇam ādāya

   Pūrṇam evāvasiṣyate. ť That is full, this is full

   From the full, the full is subtracted ť When the full is taken from the full

   The full still will remain — Isha Upanishad.
   The Indian mathematical text Surya Prajnapti (c. 400 BC) 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

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).
According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null

aleph_0

(the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.
In the Jaina 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 [[Asaṃkhyeya|]] ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

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 (for example, of justification) that it's supposed to play."

Infinity symbol

The precise origin of the infinity symbol ∞ is unclear. One possibility is suggested by the name it's sometimes called—the lemniscate, from the Latin lemniscus, meaning "ribbon."
   A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. However, this explanation isn't plausible, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858.
   It is also possible that it's inspired by older religious/alchemical symbolism. For instance, it has been found in Tibetan rock carvings, and the ouroboros, or infinity snake, is often depicted in this shape. John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicis. 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.
   Another possibility is that the symbol was chosen because it was easy to rotate an "8" character by 90° when typesetting was done by hand. The symbol is sometimes called a "lazy eight", evoking the image of an "8" lying on its side.
   Another popular belief is that the infinity symbol is a clear depiction of the hourglass turned 90°. Obviously, this action would cause the hourglass to take infinite time to empty thus presenting a tangible example of infinity. The invention of the hourglass predates the existence of the infinity symbol allowing this theory to be plausible.
   The infinity symbol is represented in Unicode by the character ∞ (U+221E).

Mathematical infinity

Infinity is used in various branches of mathematics.

Calculus

In real analysis, the symbol

infty

, called "infinity", denotes an unbounded limit.

x 
ightarrow infty

means that x grows without bound, and

x 
ightarrow -infty

means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then

int_.

Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought, in the philosophy of mathematics called finitism which influenced the philosophical and mathematical school of mathematical constructivism.

Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (for example counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value , for instance by taking an infinite value in an extended real number system (see also: hyperreal number), 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. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things.
   It should be pointed out that this practice of refusing infinite values for measurable quantities doesn't come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations. 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 isn't what we can observe in reality.
   This point of view doesn't mean that infinity can't 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. One application where infinities arise is the quantification of thermodynamic temperatures.
   However, there are some currently-accepted circumstances where the end result is infinity. One example is black holes. Physicists have verified that, when a star experiences gravitational collapse, it'll eventually shrink down to a point of zero size, and thus have infinite density. This is an example of what is called a mathematical singularity, or a point where the laws of mathematics, and therefore of physics, break down. Some physicists now believe the singularity may be physically real, and have since turned their attention to finding new mathematics where infinities are possible.

Infinity in cosmology

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important 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 walking/sailing/driving/flying straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you'd eventually revisit your starting point. If, however, the universe is ever expanding then you could never get back to your starting point even on an infinite time scale.

Computer representations of infinity

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, for example 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're useful as sentinel values in algorithms involving sorting, searching or windowing. In languages that don't have greatest and least elements, but do allow overloading of relational operators, it's possible to create greatest and least elements (with some overhead, and the risk of incompatibility between implementations).

Perspective and points at infinity in the arts

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' depict distance and foreshortening of objects. Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

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