Infinite

Definitions

In Mathematical

The mathematicians consider primarily two concepts of infinite, “the infinite potential” and “the infinite current one”. To define the infinite potential, one gives oneself means (axioms) to characterize the fact that what one considers does not have of limit of number or in the face. To define the infinite current one, one affirms the existence of one or more mathematical objects “infinite” which one gives the properties by axioms. To know if these objects have a reality returns in the middle of the Philosophie of mathematics.

In Physical

While the mathematicians define the infinite one by axioms, the physicists created a branch of physics called Cosmologie in which they wonder whether the universe is of finished or infinite size and put the question of the time and its finitude.

In Geometry

The painters of the Rebirth, seeking a realistic representation of reality, tackled (without the knowledge) the question of infinite when they developed the methods of representation Perspective. Parallel horizontal lines “are cut ad infinitum” in space and in a point on the table; on the one hand this point of the table as well as the horizon of the table correspond to a certain reality in two dimensions (2D). In addition with the Rebirth not more than today no one cannot affirm or to cancel only the points ad infinitum space with three dimensions (3D) correspond to the truth of the universe, from where certain a faintness. These problems do not recover exactly the nuance between infinite potential and current, but gave place to the projective Géométrie.

The projective Géométrie consists in adding with the Espace refines usual points called “ad infinitum” in each direction. The goal is more not to make distinction between secant and straight line right-hand sides parallel, these last having a common point ad infinitum. It is a remarkable tool for simplification. As example, in projective geometry, there exists one type of conical instead of three.

In Topology

The addition of an element ∞ to a topological Espace Localement compact makes it possible to make this space compact. It is about the compactification of Alexandroff.

Either $(E, U)$ a topological space locally compact, its compactifié is space $(E \ cup \ {\ infty \}, U')$ , where $\ infin$ is an element external with E, and U' is obtained from U by adding to him all the complementary ones in $E \ cup \ {\ infty \}$ of compact of $(E, U)$ .

One can then define the “vicinities of infinite”: it is about very part containing open of U' \ U.

Infinite cardinals

Countable infinite units

An infinite unit is known as countable if and only if there exists a bijection between him and $\ mathbb {NR}$ . Intuitively, an infinite unit is countable if and only if one can “enumerate” his elements: “the first” element, “the second” element, “the third” element, and so on without stopping.

For example, we can show that $\ mathbb {Q} ^+$ is countable: let us classify for that the irreducible fractions of numerator and denominator both positive in the following way:

for any fraction p/q, one calculates the sum p+q;
one classifies the fractions by order ascending of this sum p+q;
for the fractions having the same amount p+q (like 1/4 and 2/3), one classifies them by order ascending of p;
thus one can allot to each fraction a single entirety corresponding to its number of appearance in the list thus built, the beginning of this list would be:

1 → 0

2 → 1/1

3 → 1/2

4 → 2/1

5 → 1/3

…

We put well $\ mathbb {Q} ^+$ in bijection with $\ mathbb {NR}$ .

The cardinal of a finished unit is a natural integer. On the other hand, the cardinal of a countable infinite unit is known as “transfinite”.

In the example above the enumeration of rational positive is “effective”: the process of enumeration is a calculative process, an algorithm (described informellement). But one can have shown very well that a unit is infinite countable, for example by showing that it is subset of the entireties and cannot be finished, without being able to give an effective process of enumeration. This last concept is studied in the article Ensemble recursively énumérable.

Noncountable infinite units

A noncountable infinite unit cannot be put in bijection with

\ mathbb {NR}

. One cannot draw up a list of his elements.

For example, the whole of the realities ranging between 0 and 1 are noncountable: the demonstration rests on the Argument of the diagonal of Cantor.

It is said that $\ mathbb {R}$ with the power of continuous, its power (or its cardinal) is $2^ {\ aleph_0}$ (the cardinal of the Ensemble of the parts of $\ mathbb {\ NR}$ ). The diagonal argument of Cantor shows at the same time that $\ aleph_1$ , the smallest noncountable cardinal, is lower or equal to $2^ {\ aleph_0}$ (in ZFC). The equality of these two cardinals, whom one calls the Hypothèse of continuous the, is independent of the axioms of the set theory ZFC.

History

Infinite potential and the infinite current one

According to Ibicrate, the geometrician, raises Sophrotatos, the Greek philosophers always made the distinction between clearly the infinite potential - accepted by Aristote primarily with the use of the mathematicians - the “apeiron”, translated more exactly by “the unlimited one” - and infinite current the , for example the whole of the natural entireties as a completed totality, which he refuses to consider.

the infinite potential was conceived already in the ancient Greece. It is considered that one moves towards the infinite one without never reaching it. The infinite one is perceived like a potentiality. Let us notice that even potentials, the very great numbers can be difficult to conceive. Thus the series Goodstein are defined continuations very simply which give place to numbers which exceed the understanding, although they are still considerably smaller than those generated by the busy Castor.

infinite current the is a more contemporary design. With the Rebirth, the riding prospect and thereafter the projective Géométrie introduced break points ad infinitum perceptible on tables or drawings. That led the thinkers to imagine the infinite one like “atteignable” or having a close reality, they considered the infinite one as an intrinsic quality of what they studied, the infinite one being perceived like a reality, or more often, because representative God, therefore “unattainable”, “immontrable”, to hide it by a graphic artifice (building in the axis of the central break point).

Notations

The current Symbole of infinite was employed for the first time in 1655 by John Wallis, in his work: Of sectionibus conicis then shortly after in the Arithmetica Infinitorum :

esto enim ∞ foot-note numeri infiniti .

Two Hypothèse S exist as for the origin of this choice. Most commonly allowed is that it is about an evolution of the figure indicating “1000” in the Roman Numération: successively Ⓧ, then CIƆ, before becoming Mr. the graphic evolution of the second symbol would have given $\ infin$ . In parallel one notes the use of the Latin word thousand in the plural to indicate an arbitrarily large and unknown number. One will note the French Expression still used today “of the miles and the hundreds” pointing out this use. The current symbol would be thus simply the evolution of the tiny binding cıɔ in uncial manuscript writing.

A concurrent assumption is that the symbol would result from the Greek Lettre ω , last letter of the Greek alphabet, and current metaphor to indicate the final end (as in the expression alpha and the Omega ). Since Georg Cantor one uses Greek letters besides to indicate the ordinal numbers infinite. Smallest infinite Ordinal, which corresponds to the good usual order on the natural whole , is noted ω.

Variations on the infinite one

The Attracteur of Lorenz form the infinite symbol if he is repeated a certain number of times.

The infinite one was one of the topics Romantiques

Random links: