The actual infinity in Cantor’s set theory
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The actual infinity in Cantor’s theory of sets( G.Mpantes )
. George Mpantes
www. mpantes gr
. The actual infinity Aristotle-Cantor
, potential infinity .
The origins of Cantor’s infinity, aleph null, the diagonal argument The natural infinity
, continuum
The mathematical infinity A first classification of sets Three notable examples of countable sets The 1-1 correspondence, equivalent sets, cardinality
.
The theory of transfinite numbers Existence and construction, existence proofs
The theory of Cantor, introduces us to the mathematical study of infinity . The definition of the Cantor set is : Definition 1 . A set is a gathering together into a whole of definite distinct objects of our perception or of our thought which are called elements of the set…G Cantor 1. The actual infinity, Aristotle-Cantor . The above definition in the case of infinite sets , establishes the concept of active or actual infinity, namely the existence of an infinite set as a mathematical object (in whole), on par with the numbers and finite sets . The concept of actual infinite reaches from Greek antiquity and was rejected
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by Aristotle . " ..... There are no infinite objects , wrote Aristotle , and the infinite totalities of objects do not form objects of study . Infinite totalities of objects can be studied only internally ... ( potential infinity) there is no actual infinity, viz infinity as a whole , whose parts would exist simultaneously , otherwise we would not comprehend nor the continuity of time (the time would have start and end) , nor divisibility of sizes ", which means that for Aristotle the only meaningful study is the study of potential infinity ( Anapolitanos 2005 ) He wrote that: …infinity does not exist in the form of an infinite solid or an infinite magnitude perceived from senses ... infinity exists only potentially, while actual infinity as a subject , is the result of a mental leap , which is a process not allowed….
Gauss later (1831) said that : in mathematics infinite magnitude may never be used as something final:infinity is only a façon de parler, meaning a limit to which certain ratios may approach…..
Let's look at the mathematical version of the above : We consider the natural numbers ( 0,1,2,3 , .... n, n +1, ..... ) . Potential infinite in this case means that one can continue to write terms infinitely without never reaching to an end of the succession, the infinity of numbers is tested internally. So for Aristotle, the collection of natural numbers is potential infinity as there is not a greater last natural , but not an actual infinity as it does not exist as an integrated entity " ( Aristotle) But in mathematics of Cantor, set
is a certain object whose clarity
arises from the fact that it has exactly the elements that indicates, and if the natural numbers are a set N , then the above procedure is considered finished , and N is actual infinity the same for rational numbers , so for real numbers . Here we say : the natural numbers is potential infinity because there is no greater number , but the Cantor’s set of natural numbers is actual infinity because it exists as an integrated entity . Thus an expression “actual infinite set " is redundant , saying only infinite set . The concept of Cantor’s set is a development exceeded the physiology of the human brain , which in the route of it’s growth has learned thinking linearly (especially regarding time ) , the only way to think infinity is a
The actual infinity in Cantor’s set theory
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line that never ends , or a thing that exists forever , how to conceive a line without end but finished ! We forget intuition , we did and other times , let us remember the space-time, the continuum , etc. Descardes said that infinity is recognizable but not comprehensible. In Aristotle's question " where the human being can sustain the controversial conceptual leap of objectification the infinity ? " The answer is : in the concept of Cantor’s set. In the sequence 1,2,3,4,5…we can write a billion, a trillion,…..terms but all of them are aleph (null) terms. 2.The origins of Cantor’s infinity
.
The origin of infinity has to do with it’s trading.
Cantor makes a
tripartite distinction of infinity in three contexts : "….first when it is realized in its most complete form, in a fully independent supernal being , in Deo, whereat I call Absolute Infinity , (that means God ,) the highest perfection of God is the ability to create an infinite set , and His boundless magnanimity prompted him (Cantor) to create it ." We are in the heart of metaphysics ! The infinite set created by God , and him (Cantor) discovered it ! secondly when displayed in the eventual world of creation ( ie the natural infinity) , third when the mind conceives it in abstracto, as a mathematical magnitude , cardinal or ordinal number. ( the mathematical infinity ). I want strongly to dissociate the Absolute Infinite from what I call transfinite ( will
see the
meaning of the term) , ie integrated infinite of the second and third kind, which are clearly delimited , can grow more, and therefore are associated with the finite ….Cantor ) .
3. The natural infinity. Is there the infinite "out there" in nature ? Here things are clearer to the reader since neither the individuality of matter , nor later of electricity and energy advocate with the idea of infinite divisibility as it is contradicted by the experience of physics and chemistry ( Hilbert , for infinity ). However, there are three areas in which our world appears to be unblocked and therefore infinite . It seems that the time is not possible to
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have an end , also the space , and yet it seems that every spatial or time interval can be divided indefinitely . But, recent opinions in physics verify the views of the operational definitions of physical concepts , ( The Logic of Modern Physics, Bridgman) , without exception , even in mental (i.e mathematical concepts, as the mathematical continuity), so the assertion that time is infinite is a mere statement, such as the statement of classical mechanics, that time is global… the concepts of science are defined by sets of operations , and not by arbitrary definitions of philosophical type , they are in a Platonic world only in our minds , Bridgman is the modern Aristotle for the new realities of physics (relativity). What experiment has proved that time is infinitely divisible? It is readily apparent that experiments and infinity are incompatible since the life of the experimenter is finite . In all theories of physics the problem of infinity is responsible for creating impasses . Even Zeno 's paradoxes arise from trying to connect infinity by the motion of natural bodies . These ( theories ) start to accept infinity initially and then slowly pooping out this concept to be able to produce equations to correspond to reality. The vision of the Dirac was just that: to throw out the infinities of the equations studied ( ' unification of forces” Abdus Salam). That seems as the second infinity of Cantor , in the eventual world of creation, the natural infinity, doesn’t exist just as Aristotle considered .. 4.The mathematical infinity : Until the nineteenth century mathematicians systematically abstained from the concept of actual infinity . So when Cantor introduced actually infinite sets he had to advance his creations against conceptions held by the greatest mathematicians of the past. The infinitesimals which founded calculus is an example of the contradiction between the two concepts: they were treated as both potential and actual infinity. Potential because they are constantly decreasing quantities without end . Actual because they take part in operations and behave as zero. The unending process of operation ( reduction ) that produces them, is considered finished . Restoring logical consequence in Calculus is done using only the potential infinity . This is basically the meaning of expulsion of infinitesimals from the scope of
The actual infinity in Cantor’s set theory
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Analysis : is dismissed the actual infinity. Indeed the great of analysis , as Cauchy, Weierstrass and others talked about the infinite small and the infinitely large just internally , through permissive acts of limits, ie as something that is born , formed , for an infinite process without ever achieved identification . They traded the potential infinity ( eliminate the infinitely large and infinitely small and reduced the proposals referred to them in relations between finite sizes , Hilbert ) , ie the rigorous foundation of calculus is on mathematics of potential infinity . Only in the work of Dedekind the infinite sets become self-consistent logical entities , ie the mathematical study of actual infinity has begun. This was completed by Cantor . 5. A first classification of sets . Two sets are called equal A = B if they contain the same elements. Sets are divided into finite and infinite, An infinite set called countable set if we can construct an 1-1 correspondence between the elements of the set and the natural numbers.
6. Three notable examples of countable sets . theorem 1 : the set of all rational is countable. The principle behind the above proof is called first method of diagonals (Kamke).
Consider first the positive rational numbers . We imagine lines as follows : in first line all integers in order of magnitude , ie all rational with denominator 1, second line the numbers of the first line divided by two, third likewise divided by 3 etc . Thus we have the sequence of numbers in the order that the line meets omitting the numbers we get, is clear that in the sequence will appear every positive rational only once 1, 2, 1 / 2, 1 / 3, 3 , 4 , 3 / 2, 2 / 3, 1 / 4 , .... And if we denote this sequence by { a1 , a2 , ...... } is obvious that { 0, - a1 , a1 , - a2 , a2 , ....... } are all the rational numbers and the set is countable . theorem 2 : the set of all algebraic numbers is countable .
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It is possible , however, that all infinite sets are countable . Separating sets in countable and non- countable acquires significance only after proving the existence of non countable sets . Their existence follows from the following theorem of Cantor theorem 3 : the set of real numbers in the interval [0,1 ] is not countable . ( uncountable ). The proof is based on a method the second method of diagonals of Cantor difficult and imaginative. So the infinite sets are subdivided into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets). So he proved the surprising result that the set of whole numbers is equivalent to the set of rational numbers but less than the set of all real numbers. We can’t enumetate the real numbers. Corollary : the set of transcendental numbers is not countable . (We presuppose the existence of transcendental numbers). 7. The 1-1 correspondence, equivalent sets, cardinality
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Now the next question is : is it possible a further division of the class of non- countable sets ? Cantor raised this question in the following manner truly intelligent (Kamke): « Is it possible to generalize the concept of number , so that in each set corresponds one of the generalized these “numbers” as somehow typical of the " number of elements "? If this could happen then it will automatically result a classification of infinite sets , depending on their " number of elements" . Cantor , needed new numbers to measure the infinity. The 1-1 corresponcence . Cantor’s basic idea to distinguish the infinite sets was the one-to-one correspondence . We can set up the following one-to-one correspondence between the whole numbers and the even numbers 1 2 3 4 5 …… 2 4 6 8 10….. Each whole number corresponds to precisely one even number its double and each even number corresponds to precisely one whole number its half.
The actual infinity in Cantor’s set theory
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Hence Cantor concluded that the two sets contain the same number of objects. Now an infinite set can be put into one-to-one correspondence with
a proper subset of itself. In this correspondence, Cantor saw that infinite sets could obey laws that did not apply to finite collections , as quaternions could obey new laws that did not hold for real numbers. This correspondence is a thought experiment , a process we saw several times in science (space –time, continuum). We get conclusions applicable equally to finite or infinite sets. Definition 2
two sets A and B have the same power or are
equivalent if and only if they can have one to one correspondence with each other eg A = {1,2,3 ....... 24} B = {letters of the alphabet} . The concepts: equivalence and correspondence are the drivers of Cantor in his research. We mention a few examples of equivalent sets. 1.For points any two finite intervals T1 and T2 always happens T1 ~ T2 2.A semi-line and an entire line are equivalent with an interval. 3.The sets of points [0,1], (0,1], [0,1), (0,1) are equivalent 4.The interval [0,1] is designated as the continuum. The intervals, the semi-lines and the lines are equivalent to each other, especially with the continuum.(fig1,2) 5. there are infinite sets that are not equivalent to each other, for example, the continuum and a countable set. But this does not preclude that all uncountable sets are equivalent to each other. Therefore, the following proposal acquires fundamental importance theorem 4 : there are infinite sets that are neither countable nor equivalent
to the 1continuum. (Second method the diagonal) The cardinality
.
Cantor gave the concept of cardinality ( cardinal number ) with a very sophisticated way ( Katerina Gikas N ) , in which if we take a random collection of discrete objects and remove all the physical properties of each object , then remains "something" called cardinality or cardinal number of the objects in the collection. A trio of trees and a trio of apples have a common 1
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property which we call three. The cardinality so appear as a characteristic of sets . All the sets that have the same cardinal number with { a, b } are said to have cardinal number two , all sets with the same cardinal number as the set { a, b , c } are said to have three cardinal number , etc. and we denote cardinal numbers one, two, three , ... with 1,2,3 .... The cardinality of the set, no matter how abstract seems this definition, agrees to the case of finite sets with the number expressing the number of their elements. Even we will get the sense of the crowd, when move from finite to infinite sets ( Transfinite numbers , an extension of R). .... and the mathematicians who incorporate various branches of mathematics within set theory are justified to identify the finite cardinality with natural numbers for the purpose of this activity ... (Moshe Machover).
8.The theory of transfinite numbers . Just as it is convenient to have the number symbols 2,4,8,etc to denote the number of the elements of finite sets , so Cantor used symbols to denote the number of elements in infinite sets (their cardinals). These symbols are known as transfinite numbers. The set of the whole number and sets that can be put into1-1 correspondence with it, have the same number of objects and this number is denoted by 0 ( אּaleph null, aleph the first letter of the Hebrew alphabet ). Since the set of all real numbers proved to be larger than the set of whole numbers , he denoted this set by a new number c, the cardinal number of continuum2.
To compare cardinalities, we introduce the relationship “is smaller than” by the definition: cardA
Before Cantor mathematicians accept a single infinite , denoted by the symbol ∞,
and implied the 'number ' of elements of sets such as the natural numbers or the real numbers
The actual infinity in Cantor’s set theory
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Note that subtraction and division are not defined in this arithmetic . There are not zero , unit , inverse and negative . The well-known theorem of Cantor tells us that the power
set (the set of subsets ) of a set is greater than the whole. If the set has n elements of the power set has 2 n Cantor by considering all the possible subsets of the set of integers, was able to prove that 2 = אּc. Also he proved
. Cantor's theory provides an infinite sequence of transfinite numbers and there is evidence that an unlimited number of cardinals larger than c, there is in reality. 1 = The size of the power set of a set with size aleph3 zero 2 = The size of the power set of a set with a size aleph one, 3 = The size of the power set of a set with a size aleph two .....
We take this way the sequence of the first infinite transfinite numbers 4 But are there transfinite numbers between 0
אּand c; the name ( aleph 1 ) was given for such a cardinal . The belief was
that there was not such a number, ie there is no set with cardinality between the two . This belief is known as the " continuum hypothesis ", and was not possible to be proved . Finally it has been shown that is not supported by the 3
More often the question has been discussed of whether Georg Cantor was of Jewish origin.
About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew . 4
“Some infinites are bigger than other infinites….Cantor”
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usual axioms of set theory , but is usually taken as an additional axiom (Paul Cohen). At this point the theory bifurcates, here applies the continuum hypothesis there is not true ! The situation is analogous to the parallel postulate of Euclidean geometry . In addition to the transfinite numbers already described
which are
called transfinite cardinal numbers , Cantor introduced transfinite ordinal numbers symbolized by ω. The distinction is rather delicate, it is another infinity different than אּ, is subject in different operations and laws i.e אּ+1=1+אּ
ω+1≠1+ω,
but it is beyond this description. Set Theory, as developed by G. Cantor, is often termed naive as it was based on the intuitive notion of sets and their properties. But this was the third crisis in the foundations of mathematics, after the irrational numbers and infinitesimals. Began to appear reasonably paradoxes and contradictions in the Cantor building, and to avoid such inconsistencies and keep Set Theory contradiction-free, mathematicians came up with several axiomatic systems, of which the one known as Zermelo-Fraenkel (ZF) became the most popular, there are many “set theories” as many geometries. This story, however, is beyond the limits of this article . 9. Existence and construction . The two schools of mathematical philosophy , constructivism and realism, carry in centuries the differences between Aristotle and Plato. The variety of paths that connect mathematics with natural science lead in both directions: …...Most areas of mathematics shed light / examine about directly on some part of nature. The geometry regards space. The theory of probability teaches us about the random processes. The group theory sheds light on symmetry. The logic describes the correct reasoning. Many parts of the analysis were created to study specific procedures ... is a fact that the best of our theorems give information for the specific world ………Nicolas Goodman
Cantor however believed in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena , as expressions within an internal reality . The only restrictions of this metaphysical system
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are that all mathematical concepts must be devoid of internal contradiction. This belief is summarized in his famous assertion that the “ essence of mathematics is it’s freedom”
In the case of actual infinity Aristotle does not admit infinite sets as a totality, in the same way that rejects a seven –sided regular polygon. The Platonists (Cantor was one) on the other hand believe that this exists in some objective word independent of man, who discovers it. …at all times there have been opposing tendencies in philosophy , and it would not seem that these differences are nearing any settlement. The reason is presumably that men have different minds, and that these minds cannot be changed. There is therefore no hope of expecting any agreement between the Pragmatists and the Cantorians. Men do not agree because they do not speak the same language and because some languages can never be learned….Poincare
But the problem of existence is transferred as problem in the value of existence proofs.
Cantor
proved that there were more real numbers than algebraic
numbers. Hence there must exist transcendental irrational numbers. But this existence proof did not enable one to name, much less calculate, even one
transcendental number. Also Gauss , proved that every nth degree polynomial equation with real or complex coefficients had at least one root. But the proof did not show how to calculate this root. Many mathematicians regarded the mere proof of existence as worthless. They wanted a proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object, say that the proof of existence should enable mathematicians to calculate the existing quantity. Such proofs they call constructive proof.
But if we do believe in mathematical reasoning we need not to distinguish between the existence and construction proofs. Existence or constructive proofs are finally mathematical proofs, and mathematical reality is the mathematical proof. But in actual infinity can not occur an constructive
proof, because
there is nothing to construct, the construction relates with physical reality, we can not construct some aleph number because is it not connected with anything in nature.. Actual infinity divides mathematics in two parts keeping for
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itself just the existential part of math, we can’t palpate actual infinity but only to imagine it. It is a mathematical reality on logic, a “monster of logic” just for Platonists who believe that true is everything we think, without a connection with the real world round us. George Mpantes
mpantes on scribd
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Sources
.
Amalia Christina N. Babili (διπλωματική εργασία): Το μαθηματικό άπειρο , τα παράδοξα και ο νους Anastasiadis: Γενικά μαθηματικά Howard Eves: foundations and fundamental concepts of mathematics, Dover . Moshe Machover: set theory and their limitations. James Stein How math explains the world Avgo Katerina Gikas N: Θεμελίωση του σώματος των πραγματικών αριθμών , ισχύς και διάταξη αυτού, διπλωματική εργασία), Robert L.Vaught :set theory: an introduction) E. Kamke: theory (versions Karavias 1962) Patric Suppes: Axiomatic set theory (Dover) Hilbert D’Abro: the rise of new physics Dover Morris Klein:mathematics the loss of certainty Dover Stewart Shapiro: thinking about mathematics (University of Patras)
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