Infinite

The symbol of infinite ∞ (Unicode U+221E), also called lemniscata, in different fonts.

The concept of infinity (symbol: ∞) appears in various branches of mathematics, philosophy and astronomy, in reference to a quantity without limit or without end, as opposed to the concept of finitude.

In mathematics, infinity appears in various forms: in geometry, the point at infinity in projective geometry and the vanishing point in descriptive geometry; in mathematical analysis, the infinite limits; and in set theory as transfinite numbers.

Set theory

Finite sets have an "intuitive" that characterizes them: "given a proper part of them, it contains a smaller number of elements than the entire set". That is to say, a bijection cannot be established between a proper part of the finite set and the whole set. However, that "intuitive" of the finite sets the infinite sets do not have it, and formally it is said that:

The idea of cardinality of a set is based on the earlier notion of bijection. Two sets between which a bijection can be established are said to have the same cardinality. For a finite set, its cardinality can be represented by a natural number. For example, the set {apple, pear, peach} has 3 elements. This means more formally that a bijection can be established between such a set and the number 3, which is the set {0,1,2}:

Manzana▪ ▪ 0Pera▪ ▪ 1Durazno▪ ▪ 2{displaystyle {begin{matrix}{mbox{Manzana}}}{leftrightarrow &0{mbox{Pera}}}{leftrightarrow &1{mbox{Durazno}}{leftrightarrow &2end{matrix}}}}

In other words, it is possible to pair (0, apple), (1, pear), (2, peach) so that each element of the two sets is used exactly once. When it is possible to establish such a "one to one" between two sets it is said that both sets have the same cardinality, which, for finite sets, is equivalent to having the same number of elements.

First positive definition of infinite set

The first positive definition of an infinite set was given by Georg Cantor and is based on the following observation: If a set S is finite and T is a proper subset, it is not possible to construct a bijection between S and T. For example, if S = {1,2,3,4,5,6,7,8} and T = {2,4,6,8} then no it is possible to construct a bijection between S and T, because if so they would have the same cardinality (the same number of elements).

A set is infinite if it is possible to find a subset of its own that has the same cardinality as the original set. Consider the set of natural numbers N={1,2,3,4,5,...}, which is an infinite set. To verify such an assertion it is necessary to find a subset of its own and build a bijection between the two. For this case, consider the set of positive integers pairs P={2,4,6,8,10,...}. The whole P is a subset of its own Nand the assignment rule n→ → 2n{displaystyle nto 2n} It's a bijection:

A=[chuckles]N▪ ▪ P1▪ ▪ 22▪ ▪ 43▪ ▪ 64▪ ▪ 8].{displaystyle mathbf {A} ={begin{bmatrix}N exposeleftrightarrow &P1 fakeleftrightarrow &22 exposeleftrightarrow &43 fakeleftrightarrow &6\4 supposedleftrightarrow &8end{bmatrix}}}}}. !

since every N element corresponds to a single P element and vice versa.

Infinite ordinal numbers

Ordinary numbers serve to notice a position in an orderly set (first, second, third element...). The most elementary example is that of natural numbers, which are rigorously defined as follows: Note 0{displaystyle 0,} the empty set:

0={!=∅ ∅ {displaystyle 0={{}=varnothing }

note 1{displaystyle 1,} the set containing only 0{displaystyle 0,}:

1={0!={∅ ∅ !{displaystyle 1={0}={varnothing }

Then you notice 2{displaystyle 2,} the set containing only 0{displaystyle 0,} and 1{displaystyle 1,}:

2={0,1!={0,{0!!={∅ ∅ ,{∅ ∅ !!{displaystyle 2={0,1}={0,{0}{varnothing{varnothing }{varnothing }{}}}}

And so on:

3={0,1,2!={∅ ∅ ,{∅ ∅ !,{∅ ∅ ,{∅ ∅ !!!,(n+1)=n {n!{displaystyle 3={0,1,2}={varnothing{varnothing },{varnothing{varnothing }},qquad (n+1)=nbigcup {n}}}

By construction, 0 is included in 1, which in turn is included in 2, since obviously:

n n {n!=(n+1){displaystyle nsubseq nbigcup {n}=(n+1)}

Inclusion makes it possible to convert ordinals into a well-ordered set (two distinct elements can always be compared, and adding equality would give a total order) among these sets that we prefer, out of habit, to write "< ", which gives the relations 0 < 1 < 2 < 3. To say that one ordinal is less (strictly) than another means, when both are considered as sets, that it is included in the other.

If a and b are ordinals, then aUb, the union of the sets, is also is an ordinary In particular, if there are finite ordinals (finite sets) corresponding to the naturals a and b, then aUb corresponds to the greater of the two, a or b. In general, if the sets a_i are ordinal, where i takes all the values of a set I, then a = Ua_i will also be. And if the set I is not finite, neither is a. Thus we will obtain infinite ordinals (that is, numbers).

In order to adequately formalize the discision, it is necessary to rigorously define the notion of "infinite", in order to apply it to ordinals. Two well-ordered sets A and B are isomorphic (with respect to order) if there is a bijection f between them that respects the order: if a < a' into A, so f(a) < f(a) in B. It is obvious to state that if A is an ordered set with n elements (n natural integer) then A is isomorphic a_n = {0, 1, 2,..., n-1}. Simply rename each element of A to get A = {a₀, a₁, a₂,..., a_{n -1}}. An isomorphism is merely a change of name. We will say that an ordinal is finite if each of its non-empty parts has a maximum element. Therefore all natural is a finite order. Intuition tells us that there are no other finite ordinals. Logically, we will say that an ordered set is finite if it is isomorphic to a finite ordinal, that is, to a natural.

To introduce infinite ordinals, it is now necessary to give the exact definition of an ordinal:

A fully ordained set A (by inclusion) is an ordinal if and only if each element of A is also a subset of A

We have already seen that it is the case for naturals: For example, the set 2 = {0, 1} admits 1= {0}, as an element and therefore also as a subset.

First infinite ordinal

We have already seen that any union of ordinals is an ordinal. If we take a finite union of finite ordinals, we make a finite ordinal. To obtain the first infinite ordinal we have to collect a non-finite number of finite ordinals. Doing so, we always fall into the same set, built by gathering all the finite ordinals, that is, the natural ones. The set of all naturals, ℕ, is thus the first infinite ordinal, which should not be surprising, and we note it in this context ω (omega).

To visualize ordinals, it is very practical to represent each one by a point of a converging increasing sequence, such as u_n = 1 - 1/(n+1). This gives something similar to:

X__________________________________________.

Let's choose a point of the sequence, and see how many points are further to the left. In the example, there are four, and therefore it is u₄, which corresponds to the ordinal 4. To represent the ordinal w, it is natural to add a point 'O' to the previous sequence; situated exactly on the limit of the sequence:

X__________X________________________X__X_X_

To the left of u_w there are infinitely many points, so w is infinite. But if we choose any other point in the sequence to its left, this is no longer the case, which proves that w is the first infinite ordinal. After w comes w+1, w+2... which are represented by adding two or more points to the right, initially distant, and then closer to each other:

X__________________________________________________________________________________________

The last point drawn corresponds to w+2.

More generally, to add two ordinals A and B, the names of the elements are changed so that they are all different, then the sets A and B are joined, putting B to the right of A, that is, imposing that each element of B is greater than all those of A. Thus we have built w+1,... and thus we can build 1+w: Let's note Y the element of 1, and X those of w:

X______________________X______________________________

It is obvious that w and 1+w are very similar. In fact the function x →x - 1 performs an isomorphism between them (1+w has two elements called 0: 0_A and 0_B. The first one plays the role of -1 in the function). Therefore they correspond to the same ordinal: 1+w = w. But it is not the case of w+1, which is different from w because the set w+1 has a maximum element (the 0 of the drawing) while the set w does not (the limit of the naturals is not a natural).

The point w (the O of the drawing) has no ancestor, that is, there is no n such that n+1=w: it is said that w is a limit ordinal. Zero also has this property but does not deserve this name. Since w+1 ≠ 1+w, addition is not commutative in ordinals.

W + w is constructed in the same way that we logically note 2w. Multiplication is defined from addition as for naturals.

Once nw has been represented, with n natural, it is not too difficult to imagine what will be w.w, written w². Then we can define wⁿ, with natural n, and, Taking the limit, w^w, has as many elements as the real line.

Succession ω ω ω ω ...... ω ω {displaystyle omega ^{omega ^{{{dots }^{omega }}}}} It's like a limit. ε ε 0{displaystyle varepsilon}.

Infinite cardinal numbers

The cardinal of a set is the number of elements it contains. This notion is therefore different from the ordinal, which characterizes the place of an element in a sequence. "Five" differs from "fifth" although there is obviously a relationship between the two. Two sets are said to have the same cardinal if there is a bijection between them. Contrary to ordinals, this bijection does not have to respect order (moreover, sets do not have to be ordered).

As we already have a range of sets—ordinales—let us see their respective sizes (or their cardinals). It is no surprise that finite ordinales are also cardinal: between two sets with n and m elements, m and n different, there can be no bijection, therefore they have different cardinals. But it is not the case with infinite ordinales: For example, ω ω {displaystyle omega } and ω ω +1{displaystyle omega +1} are in bijection by function:

ω ω +1→ → ω ω {displaystyle omega +1to omega }

x→ → x+1{displaystyle xto x+1} and ω ω → → 0{displaystyle omega to 0}such bijection does not respect the order, so two different ordinales can correspond to the same cardinal.

The cardinal of A is usually noted. His name is Русский Русский 0{displaystyle aleph _{0}} (alef₀) the cardinal of w, that is the whole of the natural (where alef is the first letter of the Hebrew alphabet.

If A and B are set, then 日本語A× × B日本語=日本語A日本語⋅ ⋅ 日本語B日本語{displaystyle scriptstyle ΔAtimes BUD = AUSBYwhere x designates the Cartesian product of the assemblies, and "·" is the product of the cardinals defined by this formula. The set of parts of a set A, P(A) is in bijection with the set of A functions towards {0.1}, set that writes as 2A, as a particular case of Y^X that denotes the set of X applications to Y.

The cardinal value of R, the set of real numbers, is therefore 2^alef₀, because R is in bijection with the parts of N, by means of the decimal writing of the real numbers.

It is not possible to decide, with the classical axioms (those of set theory, fundamentals of mathematics), if there is a cardinal greater than alef₀ and less than 2^{alef ₀}, that is, if there exists a set with more elements than N but with fewer elements than R. The continuum hypothesis, which is an additional axiom, says not.

Mathematical analysis

Standard or ordinary analysis

A set of real numbers S is collected superiorly if there is a number c (tape) such that c is greater than any element of S (For example, if S={π; 7; 2{displaystyle {sqrt {2},!}Then S is a set, since the number c=10 fulfills that π tariff10, 7 tax10, 2{displaystyle {sqrt {2},!}. When a set is not matched, for any number c it is possible to find x한 한 S{displaystyle xin S} so that c. x. The concept of infinity is introduced as a special boot for this type of set. This concept of infinity is represented with the symbol ∞ ∞ {displaystyle infty }.

It is also used in mathematical Analysis when you want to express that the terms of an orderly succession, or the values that take a function when taking the dependent variable values close to a previously fixed "diverge" ("tiende to infinite", or its limit is infinite). In this context, it is considered ∞ ∞ {displaystyle infty ,!} to represent the limit that tends to infinity and 0{displaystyle 0,!} to the limit when it tends to 0; and not to number 0).

To remember the limit rules, we usually resort to the following mnemonic rules: (here "x" represents any real number)

• ∞ ∞ =+(+∞ ∞ )=− − (− − ∞ ∞ ){displaystyle infty =+(+infty)=-(-infty)}

  − − ∞ ∞ =− − (+∞ ∞ )=+(− − ∞ ∞ ){displaystyle -infty =-(+infty)=+(-infty)}

• x+∞ ∞ =∞ ∞ {displaystyle x+infty =infty }

  x− − ∞ ∞ =− − ∞ ∞ {displaystyle x-infty =-infty}

• x∞ ∞ =0{displaystyle {x over infty}=0}, x− − ∞ ∞ =0{displaystyle {x over -infty}=0}

• Yeah. 0,,,xcdot infty =infty }" xmlns="http://www.w3.org/1998/Math/MathML">x▪0,x⋅ ⋅ ∞ ∞ =∞ ∞ {displaystyle xg0,,xcdot infty =infty }0,,,xcdot infty =infty " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0f52144b780e1db4ea6706d541064b3156d976" style="vertical-align: -0.671ex; width:18.153ex; height:2.509ex;"/> and x⋅ ⋅ (− − ∞ ∞ )=(− − ∞ ∞ ){displaystyle xcdot (-infty)}

  Yeah. <math alttext="{displaystyle xx.0,x⋅ ⋅ ∞ ∞ =− − ∞ ∞ {displaystyle x vis0,,xcdot infty =-infty }<img alt="x and x⋅ ⋅ (− − ∞ ∞ )=∞ ∞ {displaystyle xcdot (-infty)=infty }

• ∞ ∞ +∞ ∞ =∞ ∞ ,− − ∞ ∞ − − ∞ ∞ =− − ∞ ∞ {displaystyle infty +infty =infty,,-infty -infty =-infty }

• ∞ ∞ ⋅ ⋅ ∞ ∞ =(− − ∞ ∞ )(− − ∞ ∞ )=∞ ∞ =∞ ∞ {displaystyle infty cdot infty =(-infty)(-infty)=infty =infty }

  (− − ∞ ∞ )⋅ ⋅ ∞ ∞ =∞ ∞ (− − ∞ ∞ )=− − ∞ ∞ =− − ∞ ∞ {displaystyle (-infty)cdot infty =infty (-infty)=-infty =-infty }

The above identities are perfectly formalizable in the non-standard analysis associated with hyperreal numbers.

Undetermined limits (it is not possible to determine their value a priori as in the rest of the examples, there is no assigned value):

0⋅ ⋅ (± ± ∞ ∞ ),+∞ ∞ − − ∞ ∞ {displaystyle 0cdot (pm infty),qquad +infty -infty },

1± ± ∞ ∞ ,00,(± ± ∞ ∞ )0{displaystyle qquad 1^{pm infty },qquad 0^{0},qquad {(pm infty)}^{0},}

00,± ± ∞ ∞ ± ± ∞ ∞ {displaystyle {frac {0}{0}{0}},qquad {pm infty over pm infty },}

Non-standard analysis

Non-standard analysis extends real number theory. From the logical point of view, real numbers can be understood as a formal language in which the existence of certain objects is assumed and in which the existence of other objects can be deduced. In terms of formal languages, the non-standard analysis is a logical extension of the ordinary theory of real numbers that is also conservative (in the sense that its deducible theorems coincide with those deducible in the ordinary theory of real numbers). Although this extension seems wasteful from the point of view of Ockham's razor, since the complication introduced does not alter the class of basic theorems about ordinary real numbers, it actually allows for shorter proofs, more easily deriving results than in theory ordinary and often more intuitive in logical terms.

At the heart of non-standard parsing, a new predicate st(·) and three new axioms describing the use of said predicate are introduced. Thanks to this predicate, the set of numbers described by the form language can be divided into "standard elements" for which (r is standard if st(r) is true) and "non-standard elements" (r is non-standard if ¬st(r) is true.) The standard elements have essentially the same properties as the ordinary real numbers, while the non-standard elements include special numbers, some of which are either infinitesimal or unlimited (infinity) numbers. The advantage of the logical structure of non-standard analysis is that such numbers can be used and used in deductions without any inconsistency (unlike the traditional infinitesimal calculus heuristics before the formalization of the century XIX).

In non-standard parsing you can define numbers that intuitively behave like infinite numbers thanks to the predicate st() "· is standard". For example an unlimited number r satisfies that "for any number e of the set and any standard natural number it turns out that ne < r", formally:

<math alttext="{displaystyle forall ein {}^{*}mathbb {R}forall nin mathbb {N}left[mathrm {st} (n)land neРусский Русский e한 한 ↓ ↓ R,Русский Русский n한 한 N,[chuckles]st(n)∧ ∧ ne.r]{displaystyle forall ein {}^{*}mathbb {R}forall nin mathbb {N}left[mathrm {st} (n)land ne fitrright]}<img alt="forall ein {}^{*}mathbb {R}forall nin mathbb {N}left[mathrm {st} (n)land ne

Obviously the number r cannot be standard, since for standard numbers one has to "for any number e and any r exists a natural such that ne > r, formally:

rright]}" xmlns="http://www.w3.org/1998/Math/MathML">Русский Русский e한 한 ↓ ↓ R,consuming consuming n한 한 N,[chuckles]ne▪r]{displaystyle forall ein {}^{*}mathbb {R}exists nin mathbb {N}left[ne censorright]}}rright]" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af056747467a6ed87be2a0e6f008e85689ae9f77" style="vertical-align: -0.838ex; width:25.142ex; height:2.843ex;"/>

Note that the predicate "standard" st(·), and therefore it is formalizable in the ordinary theory, while the notion of unlimited number is not formalizable in the ordinary theory because this theory lacks the predicate st(·).

Similarly in non-standard analysis, infinitesimal numbers can be defined, smaller in absolute value than any positive standard number. In fact, the inverse of an unlimited number is always an infinitesimal number.

Infinity in computing

Similarly to infinity for real numbers, some programming languages support a special value called infinity: a value that can be obtained as a result of certain non-performable mathematical operations, such as like those described in the previous point or operations theoretically possible, but too complex for their work in the computer/language in question. In other languages it would simply produce an error.

Infinity in metaphysics

The infinite cannot admit any restriction, which supposes that it is absolutely unconditional and indeterminate, since all determination, whatever it may be, is necessarily a limitation, because it leaves something out of it. On the other hand, the limitation presents the character of a true negation: to put a limit, is to deny, for what is enclosed in it, everything that this limit excludes; Consequently, the denial of a limit is properly the negation of a negation, that is, logically and even mathematically an affirmation, in such a way that the denial of any limit is in reality equivalent to the total and absolute affirmation. What has no limits is that of which nothing can be denied, and consequently, that which contains everything, that outside of which there is nothing; and this idea of the Infinite, which is thus the most affirmative of all, since it includes or involves all particular affirmations, whatever they may be, is not expressed by a term in a negative form (in-finite) but in the very reason of its absolute indeterminacy.

Infinity according to Aristotelian physics

The finite concept according to Aristotelian physics denies that the infinite exists in acto. against the existence of a finite body. The infinite exists only as potentiality or potentiality. Potentially infinite is, for example, the number, because it is always possible to add another number to any number, without ever reaching an extreme limit after the which cannot be advanced further; or potentially infinite is also space, because it is divisible to infinity, insofar as the result of the division is always a magnitude that, as such, is further divisible; finally, infinite potential is also time, which cannot exist in its entirety at once, but develops and grows without end.

Aristotle did not come to glimpse the idea that the immaterial could be infinite, because he associated the concept of infinity to the category of quantity, which can only be applied to the sensible. And it is also explained that the philosopher concluded by definitively seal the Pythagorean idea, and, in general, typical of almost all Greek culture, according to which the finite is perfect and the infinite is imperfect.

This is the reason why Aristotle had to necessarily deny God's attribute of infinity. After this conception of the infinite as potentiality and imperfection, the ancient intuition of the Milesians, Melissus and Anaxagoras had to be eliminated., who considered the Absolute to be infinite: such an intuition was eccentric with respect to the thought of the entire Greek culture and, in order to be reborn, would have to wait for the discovery of further metaphysical horizons.

History

John Wallis was the first mathematician to use the symbol of infinite in his works.

The infinity symbol

The symbol ∞ ∞ {displaystyle infty } with which the infinite is expressed was introduced to mathematical notation by the English mathematician John Wallis (1616-1703) in one of his most important works: Arithmetica Infinitorum in 1656. In 1694 the graphical representation lemniscata by Jacob Bernoulli (1655-1705) was created.

It is also believed possible that the shape comes from other alchemical or religious symbols, such as certain representations of the uroboros serpent.^{[citation needed]}

Another hypothesis defends that the symbol seems the graphic representation of the phenomenon known as solar analemma. This theory makes more sense if we give importance to the formal part of the design and the chronology of its origin.^{[citation required]}. The solar analemma is a photo or a graph of the sun, recorded every day for a year at the same time and in the same place, when gathering the points we will see that a lemniscate is drawn due to the inclination of the terrestrial axis that The angle at which we visualize the sun varies throughout a year, the summer solstice being at the highest point and the winter solstice at the lowest. Its shape varies slightly depending on the geographical point on earth from where it is record. This symbolizes an "Annual cycle" turn in the infinite cycle of the sun year after year.

It has also been wanted to see a Möbius band in its shape,^{[citation needed]} although the symbol was used for hundreds of years before August Möbius discovered the band that bears his name.

The infinity symbol is represented in Unicode by the character ∞^[1] (U +221E).

Chronology

Year	Development
350 a. C.	Aristotle rejects a real infinite.
1639	Gérard Desargues introduces the idea of infinite into geometry.
1655	John Wallis is attributed to be the first to use the symbol ∞ ∞ {displaystyle {infty}} for infinity.
1874	Georg Cantor specifies, in the theory of sets, different infinite orders.

More information

• Manolios, Panagiotis & Vroon, Daron. Algorithms for ordinal arithmetic. Baader, Franz (ed), 19th International Conference on Automated Deduction--CADE-19. Pages 243-257 of LNAI, vol. 2741. Springer-Verlag.