THE PARADOXES
OF INFINITY

"Mathematics is the science of infinity" (Hermann Weyl).

"Infinity is but a peculiar turn given to generality" (Charles Sanders Peirce).

"Infinity is not a quantity" (Wittgenstein).

"The infinite is always in potency, never in act" (Aristotle. Metaphysics).



The Infinite

Definition

Infinity is defined as "a number greater than any natural number". According to this definition, infinity is not a natural number, because if it were, the definition would be contradictory, since infinity would have to be greater than itself. Paradoxically, this mysterious concept plays a central role in mathematics.


Cardinal infinity and ordinal infinity

The infinite number initially appears in two forms: as cardinal (number of elements of an infinite set) or as ordinal (order number of an infinite element of an infinite ordered set). In the set of natural numbers N={1, 2, 3, ...} these two types of infinity appear. In both cases (cardinal and ordinal) we are dealing with imaginary numbers, in the sense that we have to appeal to the imagination to intuit them.


Potential and actual infinity

Traditionally, it is considered that there are two types of infinity: potential and actual. Since these concepts are difficult or impossible to define, one must necessarily resort to metaphors. For Aristotle and Kant, the infinite is always potential, that is, incomplete, never actual. For Gauss, "the use of infinity as something completed should not be allowed in mathematics."


Cantor's Theory of Infinity

Cantor was the great promoter of the subject of infinity. According to him, there are different orders or types of infinities. He based himself on the following ideas:


The cardinality of rational numbers

The cardinality of the set of rational numbers Q is infinite, like that of the natural numbers. The proof is simple, since it suffices to order the rational numbers in the form p/q, with q = 2, 3, 4,. .. and p = 1, 2, 3,..., q−1. There are some rational numbers that repeat (as in the case of 2/4 =1/2 ):

q2345....
p1121231234...
p/q1/21/32/31/42/43/41/52/53/54/5...
Natural no.12345678910...


Cardinality as equiparability

Two finite sets have the same cardinality (number of elements) if a biunivocal relationship can be established between the elements of both sets (i.e., what is called a bijection).

This property of equiparability is obvious for finite sets, but Cantor applied it to infinite sets as well. He discovered that an infinite set can be put in correspondence with a proper subset of it, so he deduced that both have the same cardinality. For example, even numbers have the same (infinite) cardinality as natural numbers:

Natural numbers12345...
Even natural numbers246810...

This paradox that the set of even numbers has the same number of elements as the set of natural numbers was already discovered by Galileo hence the name "Galileo's Paradox" and involved going against the common sense principle that states that "The whole is greater than any of its parts", one of the common notions (the eighth, specifically) of Euclid's Elements.

Although Cantor was the driving force behind the subject of infinity, Bolzano anticipated Cantor in the study of infinite sets and their paradoxes, defending (like Cantor) the actual infinite.


Recursive powers of a set

The power of a finite set C, denoted as P(C), is defined as the set of all subsets that can be formed from the elements of C. For example: P(C) has higher cardinality than C. It is usually symbolized by 2C, evoking the property that 2n is the number of elements of the power of a set of n elements. In the case of the set of natural numbers (N), the cardinality of the power of N (which is usually represented symbolically as 2N) is greater than the cardinality of N. Similarly Cantor recursively applied the concept of power to the set P(N), to the set P(P(N), etc., thus obtaining an infinite hierarchy of infinities, which he represented by ℵn and which he called "transfinite numbers": being ℵ0 the cardinality of N, ℵ1 = 2^ℵ0 the cardinality of P(N), ℵ2 = 2^ℵ1 the cardinality of P(P(N)), and so on. In general, ℵn = 2^ℵn−1. The symbol ℵ is "aleph", the first letter of the Hebrew alphabet.

Any set whose cardinality is ℵ0 is numerable, that is, it can be put in correspondence with the natural numbers. Examples of numberable sets are: Z (the integers), Q (the rational numbers), the even numbers, the odd numbers, the primes, the algebraic numbers (those satisfying a polynomial equation with integer coefficients), etc.


The set of real numbers R is not numerable

The set of real numbers is not numerable, that is, it cannot be put in biunivocal correspondence with the natural numbers. The cardinality of the set of real numbers is greater than the cardinality of the natural numbers.

This is intuitively obvious, for there is no such thing as a number "next" to a given real number. Between two distinct real numbers, even if they are infinitely close, there are infinitely many real numbers.

But Cantor proved this formally by two methods using in both cases the real segment S = [0, 1), which includes the real numbers between 0 and 1 (0 is included and 1 is not). The points of this segment can be represented by infinite sequences of zeros and ones after the decimal point: from .0000... to .1111....
  1. The real numbers are formed by infinite sequences of zeros and ones, which correspond to the power set of the set of natural numbers, so their number is 2^ℵ0.

  2. The diagonalization method.
    It is proved by contradiction. The set of real numbers is assumed to be numerable and this assumption leads to a contradiction. Indeed, if one had an ordered set (sequence) of the real numbers (e.g., 0.10110..., 0.010011..., etc.), it would be possible to find a real number different from all of them simply by choosing for the first number a digit different from the first digit, a digit different from the second digit of the second number, etc.). Moreover, there are infinitely many real numbers that do not exist in this hypothetical sequence. For this, it is enough to order the sequence in another way. Since there are infinitely many ways of ordering them, there are infinitely many real numbers that are not contemplated in that sequence. Therefore, the cardinality of the real numbers is greater than that of the natural numbers.
The so-called "continuum hypothesis" states that there is no infinity between the cardinality of the set of natural numbers and the cardinality of the set of real numbers.

For Cantor, the set of points of a line segment is an actual infinity, i.e., they all exist already there, without the need to perform any process.

For Hilbert, actual infinity is to consider the totality of the natural numbers or the totality of the points of a segment as a complete entity. For Hilbert, potential infinity is not true infinity, since it is something that is not, but that becomes, that comes to be.


Solution to the Paradoxes of Infinity and Specification in MENTAL

The appearance of a paradox indicates that we have misunderstood something or that our concepts are incomplete or superficial. Therefore, it is necessary to rethink the concepts to look for something deeper that resolves the paradox. A paradox appears at a superficial level and is an opportunity to discover something profound that we were unaware of. At a deep level, paradoxes disappear. Niels Bohr said: "Formidable! We have stumbled upon a paradox. Now we really have hope for progress".


On the concept of "infinity"

Infinity belongs to the mode of synthetic consciousness. It is the opposite of "finite," which belongs to the analytic mode of consciousness:
On the diagonal argument

Cantor's diagonal argument to prove that the real numbers are not numerable is not correct, for it fails in two respects, both related to the concept of infinity:
On the cardinality of the power set of natural numbers

The cardinality n of a finite set C is always less than its power set P(C), since n < 2^n. This property can be generalized "for all n". But this property cannot be carried to infinity, since this relation is only satisfied for finite values of n. Therefore, it makes no sense to state that ∞ < 2 because 2 = 2·2·2·2·2·.... = ∞, and it is concluded that ∞ < ∞.

The same reasoning could be applied to the relation n < 2n, which makes sense only for finite values of n. It cannot be asserted that ∞ < 2*∞ because 2*∞ = ∞+∞ = ∞, and arriving at the same contradiction that ∞ < ∞.

The only thing that differentiates 2n and 2n is that in the latter the "speed" of advance towards infinity is greater. The derivatives are, respectively, 2 (constant) and n*2(n−1). In the latter case, the derivative is so much larger the larger n is.

Therefore, the power set of all natural numbers is infinite, that is, it has the quality of being infinite. And the principle that the whole is greater than its parts is fulfilled, for example, the set of even numbers is included in the set of natural numbers.


On the equiparability of infinite sets

An infinite set can be put in correspondence with one of its parts, but that does not imply that they have the same cardinality. Only finite sets can have the same cardinality. In different infinite sets there are always more elements to be matched in one of the sets than in the other.


On the non-numerability of irrational numbers

Finite real numbers (with a finite number of decimal places) are numerable, since they are rational numbers. But irrational numbers (involving infinite decimal places) are not numerable because, in general, their decimals have a random structure, i.e., they have no pattern.


Infinity in MENTAL

Potential infinity is defined descriptively and recursively. With an initial value of 0 and adding 1 indefinitely: It can also be defined like this: The definition of the set of natural numbers is of descriptive type (it does not imply the actual infinity: The cardinality of the set of natural numbers is infinite, but it must be interpreted not as a number but as a quality: (N# = ∞).

Infinity and zero are dual with respect to the sum: Infinity and zero have analogous behavior with respect to product and division:

⟨( ∞*n = ∞ )⟩   ⟨( 0*n = 0 )⟩
⟨( ∞÷n = ∞ )⟩   ⟨( 0÷n = 0 )⟩


There is an analogy between infinity and the speed of light (c): c+v = c. The speed of light is invariant with respect to all inertial systems. Infinity is invariant with respect to the sum. In this sense, infinity can be considered an imaginary number.

Other properties of infinity: (In R# we only consider the expressible real numbers: the rationals and the irrationals that have a pattern.)


Remarks
On transfinite ordinals

The cardinality of a set does not depend on the order in which its elements are specified. In sequences, when considering the order, one must additionally speak of "ordinal numbers", which are the numbers associated with the positions occupied by its components (first, second, etc.).

If we consider sets or sequences of finite type, numbers act as cardinal and ordinal. For example, the number 5 can refer to a set of 5 elements or to the fifth element of a sequence.

However, Cantor used the symbol ω to refer to the infinite ordinal, i.e., the "last" element of the sequence of natural numbers: (0, 1, 2, ..., ω). But, just as in the case of finite numbers, a double notation is not necessary (for cardinals and ordinals), with infinity it is not necessary either. Therefore, the two types of infinity (cardinal and ordinal) are both aspects of the same infinity. Therefore, it can be represented in MENTAL:
N={1...∞}, which is equivalent to N={1...}.
It is verified that In MENTAL we could also construct imaginary expressions with infinity, for example,

Addenda

Rational numbers as infinite series

Any rational number a/b can be expressed as an infinite series of distinct rational numbers forming a geometric progression of ratio 1/(b+1): In the case of b=1, we have the series
The reception of Cantor's ideas

Ever since Cantor enunciated his theory, it was questioned first by mathematicians and later by philosophers. Even its mathematical character was questioned: "I do not know whether what predominates in Cantor's theory is philosophy or theology, but I am sure that it is not mathematics" (Kronecker). Finally, Cantor's ideas were progressively accepted. Hilbert was one of the mathematicians who supported him the most: "No one will take us away from the paradise that Cantor created for us".

Cantor's ideas about infinity contributed to the development of schools of foundations of mathematics, with competing views among Platonists, formalists, logicists, intuitionists, and constructivists.

Cantor was the creator of modern mathematics, for by proving the non-numerability of the continuum of real numbers and developing the theory of transfinites, he created set theory, a theory that he hoped would be the true foundation of mathematics. Set theory is considered to have been born on that day in December 1873 when Cantor proved that the set of real numbers was not numerable.

At the end of his days, Cantor believed that the foundation of mathematics was metaphysical. "Without an ounce of metaphysics mathematics is inexplicable."

For Cantor, transfinite numbers are a way to God, the absolute infinite, which he denoted as Ω, but he said that this absolute infinite fell outside the mathematical field. Cantor considered mathematics, metaphysics and theology to be inextricably linked. For Cantor, mathematics is the language of the immanent reality of God.


Wittgenstein's critique of Cantor's theory

Wittgenstein harshly attacked Cantor's theory of infinite levels of actual infinities. He regarded this theory as a "cancer" of mathematics. This criticism of Cantor's theory is part of Wittgenstein's conception of mathematics:
Other criticisms

Wittgenstein was not alone in criticizing Cantor's theory. We highlight the following:
Ω is infinite numerable

In MENTAL, the symbol Ω represents all possible expressions of the language, and this set is infinite numerable. This is easily demonstrated by applying a gödelization process to the expressions (so called because it is the method used by Gödel in his famous incompleteness theorem for formal axiomatic systems):
  1. A successive natural number (starting at 1) is assigned to each individual symbol with which expressions can be formed, including white space:

    ( ) { } [ ] + * ^ v ← → = ↓ ... 0 1 ... 9 a ... z ... A ... Z ...

  2. Each language expression is assigned the Gödel number or code 2a·3b·5c·. .. being:

    2, 3, 5, 7, 11, 13, 13, 17, 19, 21, ... the sequence of prime numbers.

    a, b, c, d, e, ... the numbers associated to each of the symbols appearing in the expression.

    Each expression has a different code. And from a code you can reconstruct its associated expression (since the factorization of a number into primes is unique).

  3. Ordering the expressions by their codes, it follows that the set Ω of all possible expressions of the language is infinite numerable.

Bibliography