We know that the number of radio and television impulses, sound waves, ultra-violet rays, etc., that may occupy the very same space, each solitary upon its own frequency, is infinite.

** Every infinite game in which is a Borel subset of Baire space is determined.

Particles cannot be restricted to a geometric point in space, since this would require an infinite particle momentum.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

This is not true for infinite dimensions ; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

The Cantor space is the collection of all infinite sequences of 0s and 1s.

* a primeval abyss, an infinite expanse of waters or space

In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space.

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite ( see below ) and what its topology is.

Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space ( as in elliptic geometry ), and all five axioms are consistent with a variety of topologies ( e. g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry ).

The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite.

In electromagnetic radiation ( such as microwaves from an antenna, shown here ) the term applies only to the parts of the electromagnetic field that radiate into infinite space and decrease in intensity by an inverse-square law of power, so that the total radiation energy that crosses through an imaginary spherical surface is the same, no matter how far away from the antenna the spherical surface is drawn.

In these problems the estimates are functions that can be thought of as point estimates in an infinite dimensional space, and there are corresponding interval estimation problems.

He taught that pleasure and pain are the measures of what is good and evil ; death is the end of both body and soul and should therefore not be feared ; the gods do not reward or punish humans ; the universe is infinite and eternal ; and events in the world are ultimately based on the motions and interactions of atoms moving in empty space.

An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.

* Infinite dimensional groups, such as the additive group of an infinite dimensional real vector space.

It turns out that they mostly fall into four infinite families, the " classical Lie algebras " A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub > and D < sub > n </ sub >, which have simple descriptions in terms of symmetries of Euclidean space.

Even worse, it could keep radiating infinite amounts of energy because there were infinitely many negative energy states available.

Some sets are infinite ; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.

Infinitists typically take the infinite series to be merely potential, in the sense that an individual may have indefinitely many reasons available to him, without having consciously thought through all of these reasons when the need arises.

Later ancient commentators such as Proclus ( 410 – 485 CE ) treated many questions about infinity as issues demanding proof and, e. g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.

A linear equation with two variables has many ( i. e. an infinite number of ) solutions.

Or, according to Pascal's Wager, it may be to our advantage to have faith, because faith may promise infinite rewards, while the rewards of reason are seen by many as finite.

Alphabets can also be infinite ; e. g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, … that play the role of variables.

By proving that there are ( infinitely ) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.

Hinduism is a complex of various belief systems that sees many gods and goddesses as being representative of and / or emanative from a single source, Brahman, understood either as a formless, infinite, impersonal monad in the Advaita tradition or as a dual god in the form of Lakshmi-Vishnu, Radha-Krishna, Shiva-Shakti in Dvaita traditions.

Improv companies may have at their disposal some number of readily accessible props that can be called upon at a moment's notice, but many improvisers eschew props in favor of the infinite possibilities available through mime.

Gradually, and by dint of infinite patience and concentration, the young peasant became master of many languages, and began the scientific study of their structure.

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

Lie groups are often defined to be finite dimensional, but there are many groups that resemble Lie groups, except for being infinite dimensional.

However this is inadequate for many applications, because many natural examples of infinite dimensional Lie groups are not Banach manifolds.

# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.

This generates an infinite regress, but many argue that it is not vicious.

According to Harrison, the first to conceive of anything like the paradox was Thomas Digges, who was also the first to expound the Copernican system in English and may have been the first to postulate an infinite universe with infinitely many stars.

But if the universe were infinitely old, there would be infinitely many other stars in the same angular direction, with an infinite total impact.

There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.

The open sets in the product topology are unions ( finite or infinite ) of sets of the form, where each U < sub > i </ sub > is open in X < sub > i </ sub > and U < sub > i </ sub > ≠ X < sub > i </ sub > only finitely many times.

However, if I is infinite and the rings R < sub > i </ sub > are non-zero, then the converse is false ; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the R < sub > i </ sub >.

In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results.