# Infinity: Wikis

### Did you know ...

More interesting facts on Infinity

# Encyclopedia

The Lemniscate, ∞, in several typefaces.
Infinity (symbolically represented by ) is a concept in mathematics and philosophy that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. In mathematics, infinity is defined in the context of set theory. The word comes from the Latin infinitas or "unboundedness."
.In mathematics, "infinity" is often used in contexts where it is 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.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Moreover, the real/Cambridge distinction is both vague and equivocal and is often used to mark several different distinctions.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

The German mathematician Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. .He also discovered that there are different "kinds" or "measures" of infinity, a concept called cardinality.^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

For example, the set of integers is countably infinite. However, the set of real numbers is uncountably infinite.
.A set of elements can be defined as infinite if the set has a seemingly paradoxical quality: a subset of elements in an infinite set can be matched up, one-to-one, to all of the elements in a set.^ One can define changes (i.e., all changes that exist) in terms of acquiring or losing an n -adic property by virtue of a causal event or process .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

[1] The paradoxical nature of infinity is illustrated by the idea of a grand hotel, with infinitely many rooms—all of which are occupied by guests—but can nevertheless manage to accommodate a new guest by moving each existing guest over, one by one, to other rooms.

## History

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 Indian

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".
The Indian mathematical text Surya Prajnapti (c. .400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite.^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ This line of reasoning proceeds infinitely, leading to the hypothesis that there are an infinite number of temporal series of order type omega-star plus omega, both before the series {.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.

### Buddhism

In some Buddhist imagery including Tibetan Buddhist thangka & vajrayana meditation deities such as Chenrezig the deity is often pictured holding a mala twisted in the middle to form a figure of 8. This represents the endless (infinite) cycle of existence, of birth, death & rebirth, i.e. the [infinity] of samsara.

### Early Greek

.In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinite; 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).^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There is a more general argument that shows that any object, concrete or abstract, exists in time if time exists.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.However, recent readings of the Archimedes Palimpsest have hinted that at least Archimedes had an intuition about actual infinite quantities.^ However, if we are not talking merely about finite intervals, but about all intervals, finite and infinite, then time does "begin" in the sense that it has a first infinite interval, namely, the maximal interval with the order type w*.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

## Mathematics

### Calculus

.Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics.^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties.[2][3]

#### Real analysis

In real analysis, the symbol $\infty$, called "infinity", denotes an unbounded limit. $x \rightarrow \infty$ means that x grows without bound, and $x o -\infty$ means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
• $\int_{a}^{b} \, f(t)\ dt \ = \infty$ means that f(t) does not bound a finite area from a to b
• $\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty$ means that the area under f(t) is infinite.
• $\int_{-\infty}^{\infty} \, f(t)\ dt \ = n$ means that the total area under f(t) is finite, and equals n
Infinity is also used to describe infinite series:
• $\sum_{i=0}^{\infty} \, f(i) = a$ means that the sum of the infinite series converges to some real value a.
• $\sum_{i=0}^{\infty} \, f(i) = \infty$ 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.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Moreover, the real/Cambridge distinction is both vague and equivocal and is often used to mark several different distinctions.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

Points labeled $+\infty$ and $-\infty$ 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 the extended real numbers. .We can also treat $\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line.^ This line of reasoning proceeds infinitely, leading to the hypothesis that there are an infinite number of temporal series of order type omega-star plus omega, both before the series {.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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 $\infty$, called "infinity", denotes an unsigned infinite limit. $x \rightarrow \infty$ means that the magnitude | x | of x grows beyond any assigned value. A point labeled $\infty$ 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).^ Similar arguments hold for other real n -adic properties in terms of which temporal existence may (allegedly) be defined, such as light-connectability, movement and the like.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ One instantaneous event is not caused by any event that occupies the immediately preceding instant, since there is no immediately preceding instant.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

On the other hand, this kind of infinity enables division by zero, namely $z/0 = \infty$ for any complex number z except for zero. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of $\infty$ 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.

### Nonstandard analysis

The original formulation of infinitesimal calculus by Newton and 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 whole field; there is no equivalence between them as with the Cantorian transfinites.^ Since there is no spatial midpoint between S and p and no spatial place relative to which the correspondence relation obtains, the present event or state of p's standing in the correspondence relation to S is absolutely present.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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 H. Jerome Keisler's book (see below).

### Set theory

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 $(\aleph_0)$, the cardinality of the set of natural numbers.^ But Cantor has shown us that there are "counterintuitive" truths about transfinite cardinals, ordinals and other order types.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ One event consists in Socrates' exemplification of dying and the other event consists in this first event's exemplifying corresponding to a part of the proposition q .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ This implies that p acquires and loses the property of being a part of a conjunctive proposition one of whose conjuncts is transiently corresponding to a state of affairs.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.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.^ In STR, the laws have the same form in each frame (are covariant), but the infinite multiplicity of different contents makes its postulation of laws infinitely less simple than the Lorentzian postulate of one content.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.Cantor defined two kinds of infinite numbers, ordinal numbers and cardinal numbers.^ But Cantor has shown us that there are "counterintuitive" truths about transfinite cardinals, ordinals and other order types.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.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.^ In fact, the principle of discrete temporal infinitude implies that time's order type cannot be any ordinal number (recall that not all order types are ordinals).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There is no difference between a sequence of finite equal-lengthed intervals (e.g., hours) beginning or ending arbitrarily at some interval and a sequence of infinite intervals beginning or ending arbitrarily at some interval, apart from the finite/infinite difference .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ The principle of nonarbitrary duration, applied to time, is properly generalized to intervals of any length, finite or infinite.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ It may be objected that there is a relevant difference between the finite/infinite cases.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

#### Cardinality of the continuum

.One of Cantor's most important results was that the cardinality of the continuum $\mathbf c$ is greater than that of the natural numbers ${\aleph_0}$; that is, there are more real numbers R than natural numbers N.^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

Namely, Cantor showed that $\mathbf{c} = 2^{\aleph_0} > {\aleph_0}$ (see Cantor's diagonal argument).
.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, $\mathbf{c} = \aleph_1 = \beth_1$ (see Beth one).^ One instantaneous event is not caused by any event that occupies the immediately preceding instant, since there is no immediately preceding instant.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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 segment 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).^ Two distant events E 1 and E 2 emit light signals that are observed to arrive simultaneously at the apparent spatial midpoint between E 1 and E 2 , relative to the reference frame R, but the signal from E 1 is observed to arrive first at the apparent midpoint relative to the reference frame R'.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ One event consists in Socrates' exemplification of dying and the other event consists in this first event's exemplifying corresponding to a part of the proposition q .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Consider one argument for the nonexistence of these properties: .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ "Time" is reductively definable in terms of an observable spatial relationship between one physical thing (the hands of a clock) and another physical thing (points on the clock dial).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ In the present case, no relevant difference between numbers and times has ever been established, i.e., a difference that would show times cannot correspond to these numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

### Geometry and topology

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 an infinite perimeter resp. an infinite surface area. An example for a fractal curve of infinite length 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.[4]

## Physics

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 (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. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

### Theoretical applications of physical infinity

It should be pointed out that this 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.^ I shall briefly present one argument for absolute simultaneity that is based solely on physicalism, with its sets, after I have presented the detailed argument in terms of platonic realism.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ But we need not rely on Lorentz's theory to show this; any world in which time has a topology and metric that cannot be known by observations of physical clocks (but in which Lorentz's laws also do not obtain) is a world that shows (7) is false.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ (Anti-platonism includes Aristotelian realism, conceptualism, physicalism, trope theory and the many varieties of nominalism.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.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.^ If time stopped at noon, August 22, 1999, this would be arbitrary since the current boundary conditions of the universe (the current amount and arrangement of mass-energy, etc.,), in conjunction with the laws of nature, imply that it is probable that the universe will continue to exist after this time.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions.^ It may be objected that there is a relevant difference between the finite/infinite cases.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.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.^ It may be objected that there is a relevant difference between the finite/infinite cases.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ But we need not think that it is logically or metaphysical necessary that concrete objects or real changes exist at some time in order for there to be time.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Reality would contain fewer entities if we supposed that time ended then, rather than lasted longer or had an infinite future.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.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.^ If we adopt a substantival theory of time or spacetime, as many philosophers of physics have done since the 1970s, then (it may be alleged) my above criticisms of these theories do not hold.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ The verificationism of these theories has sometimes been noted in a general way in the philosophical literature, [1] but the verificationist arguments need to be pinpointed precisely and their unsoundness made plain to view.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ But we need not rely on Lorentz's theory to show this; any world in which time has a topology and metric that cannot be known by observations of physical clocks (but in which Lorentz's laws also do not obtain) is a world that shows (7) is false.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.However, there are some theoretical circumstances where the end result is infinity.^ There is no difference between a sequence of finite equal-lengthed intervals (e.g., hours) beginning or ending arbitrarily at some interval and a sequence of infinite intervals beginning or ending arbitrarily at some interval, apart from the finite/infinite difference .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ For example, he writes in his seminal 1916 paper "The Foundation of the General Theory of Relativity": .
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Thus, we reject the "received wisdom" that the following conditionals are true: (a) "if STR is false, a neo-Lorentzian theory is true", (b) "if time appears frame-relative but is really absolute, a neo-Lorentzian theory is true".
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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.^ If we adopt a substantival theory of time or spacetime, as many philosophers of physics have done since the 1970s, then (it may be alleged) my above criticisms of these theories do not hold.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Perhaps his argument may be more charitably construed as the argument that some objects exist in time, but that "an object O's exemplification of an n-adic property F is an event simultaneous with other events" does not imply "the object O exists in time".
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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

.An intriguing question is whether infinity exists in our physical universe: Are there an infinite number of stars?^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ Given that the possibility of there being time before the big bang can be derived from our preceding arguments, the important question to address in this section is whether it is probable or improbable that there is time before the big bang.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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 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.^ For example, if our local temporal series has the order type w* + w, and consists of a universe that contracts for an infinite amount of time (corresponding to the {.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

.The Planck spacecraft launched in 2009 is expected to record the Cosmic Background Radiation with ten times higher precision, and will give more insight into the question whether the universe is infinite or not.^ "Considering now the measurement of time, we give an indication that one cannot measure a time t for which ct is less than the Planck length [the Planck length is 10 -33 centimeter.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ For example, if our local temporal series has the order type w* + w, and consists of a universe that contracts for an infinite amount of time (corresponding to the {.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ I shall not here address the question of whether it is possible for P1 and P2 to be satisfied by a time with a different structure.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

## 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."[5]

## 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.^ The number zero corresponds to the present hour, the negative numbers correspond to past hours, and the positive numbers to future hours.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

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 (with some overhead, and the risk of incompatibility between implementations).^ If the (possible) causal relations between events is relative to a frame of reference, but there is nonetheless an absolute temporal ordering of events, then relative to one of these frames of reference, some event can causally influence another that is absolutely earlier.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

^ There is no spatial midpoint between Socrates' death and the set of which his death is the sole member that could make a relative B-relation possible.
• ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]

## The 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' 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.
From the perspective of cognitive scientists George Lakoff and Nunez, 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,...>.

## The infinity symbol

John Wallis introduced the infinity symbol to mathematical literature.
The precise origin of the infinity symbol, $\infty$, is unclear. One possibility is suggested by the name it is sometimes called—the lemniscate, from the Latin lemniscus, meaning "ribbon".
John Wallis is usually credited with introducing $\infty$ 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.[6] Also, before typesetting machines were invented, ∞ was easily made in printing by typesetting an 8 type on its side.
The infinity symbol is available in standard HTML as &infin; and in LaTeX as \infty. In Unicode, it is the character at code point U+221E (∞), or 8734 in decimal notation.

## References

1. ^ http://mathworld.wolfram.com/InfiniteSet.html
2. ^ "Continuity and Infinitesimals" article by John Lane Bell in the Stanford Encyclopedia of Philosophy
3. ^ 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.
4. ^ Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 1197-1198. ISBN 77-170263.
5. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
6. ^ 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.

### Other references

• Amir D. Aczel (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 0-7434-2299-6.
• D. P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
• Bell, J. L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
• L. C. Jain (1982). Exact Sciences from Jaina Sources.
• L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
• George G. Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd edition ed.). Penguin Books. ISBN 0-14-027778-1.
• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 0-691-02511-8.
• John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive.
• John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive.
• Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
• Rudy Rucker (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
• N. Singh (1988). 'Jaina Theory of Actual Infinity and Transfinite Numbers', Journal of Asiatic Society, Vol. 30.
• David Foster Wallace (2004). Everything and More: A Compact History of Infinity. Norton, W. W. & Company, Inc.. ISBN 0-393-32629-2.

# Simple English

Infinity, also written $\infty$, is the name for a group of ideas about things which never end. The term is from a Latin word meaning "without end". Infinity goes on forever, so sometimes space, numbers, and other things are said to be 'infinite', because they never come to a stop.

Infinity is really not an ordinary number, but it is sometimes used as one.

Infinity can be broken up in two different kinds; potential and actual infinity. Potential infinity is a process that never stops. For example, adding 10 to a number. No matter how many times 10 is added, 10 more can still be added. Actual infinity is a more abstract idea. For example, there are infinitely many numbers as it is impossible to write them all down.

Infinity has various properties that are not normally found in numbers:-

Infinity added to any number is infinity. Infinity times any number is infinity

Division only works if the number you divide by is not infinity (except for 0 divided by infinity, which is 0).

# Citable sentences

Up to date as of December 21, 2010

Here are sentences from other pages on Infinity, which are similar to those in the above article.