 | | 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.
- The metaphor of potential infinity is a wheel that, starting from rest, turns indefinitely, without ever stopping. The natural infinity is the infinity generated by adding 1 indefinitely to a variable initialized to 0 and is the cardinality of the set of natural numbers. Potential infinity is an active, quantitative and operative infinity, characterized by an endless repetitive process, with no final result.
- The metaphor of actual infinity −and the one that best illustrates this concept− is that of a regular polygon inscribed in a circle. One starts with a 3-sided polygon, then 4, etc. up to an infinite-sided polygon, which is identified with the circumference, which is the final result of the process. The current infinity is a passive, qualitative and descriptive infinity, which corresponds to an infinite process that is considered already realized, completed.
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 ):
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 numbers | 1 | 2 | 3 | 4 | 5 | ...
|
| Even natural numbers | 2 | 4 | 6 | 8 | 10 | ...
|
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:
C = {a, b}
P(C) = {}, {a}, {b}, {a, b} }
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....
- 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.
- 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:
- The infinite is a quality. The finite is a quantity. It is not that there is only one infinity or that all infinities are equal. It is that infinity is not a quantity but a quality. It makes no sense to speak of one infinity greater than another because "greater" is a quantitative comparison.
- An infinite set can be numerable or non-numerable. That a set is not numerable does not imply that it is a higher-order infinite. This does not make sense just as it does not make sense to say of its dual element (the zero) "a zero of lower order".
- It makes no sense to speak of the cardinality of the real numbers. Nor does the continuum hypothesis make sense.
- Infinity is something undifferentiated, so it has a certain analogy with zero.
- Many properties of the synthetic mode of consciousness can be applied to the infinite: potential, abstract, imaginary, open, undifferentiated, recursive, and so on. And the corresponding dual properties apply to the finite (mode of analytic consciousness): actual, concrete, real, closed, differentiated, nonrecursive, etc.
- In the real numbers there is no recursive pattern (as in the natural numbers) that allows us to conceptualize their cardinality.
- If the cardinality of the real numbers refers to a number of elements and is not finite, this simply means that it is qualitatively infinite.
- In convergent series the potential and actual infinities converge. For example, the series
1/2 + 1/4 +1/8 + ... = 1
0.9 + 0.09 + 0.009 + ... = 1
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:
- The horizontal infinity.
It refers to the irrational numbers in the list, numbers that have in general infinite decimal places. Among them there are expressible irrational numbers (having a descriptive pattern) and inexpressible irrational numbers. The latter are impossible to know in order to change the corresponding decimal place. We can only do this with the finite or expressible (describable) irrational numbers.
- The vertical infinity.
It refers to the infinite process of generating an irrational number that is not in the list. This number is incomputable, the process never ends because we must go through the infinite numbers in the list, so it is impossible to construct such a new number.
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:
⟨( ∞+n = ∞ )⟩
⟨( 0+n = n )⟩
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:
( ∞+∞ = ∞ )
( ∞*∞ = ∞ )
( ∞^∞ = ∞ )
⟨( n^∞ = ∞ )⟩
⟨( ∞^n = ∞ )⟩
( ∞*0 = 0 ) // so (1+1+...)*0 = 0+0+.. = 0
⟨( ∞ > n )⟩ // ∞ is greater than any natural number (by its definition)
(∞+1 = ∞) // ∞ is the successor of itself
(2 4 ...)# = ∞ // the cardinality of the set of even numbers is ∞
(N# = ∞) // the cardinality of the set of natural numbers is ∞
(Q# = ∞) // the cardinality of the set of rational numbers is ∞
(Z# = ∞) // the cardinality of the set of integers is ∞
(R# = ∞) // the cardinality of the set of real numbers is ∞
(In R# we only consider the expressible real numbers: the rationals and the irrationals that have a pattern.)
Remarks
- The justification of the expression
(∞+∞ = ∞) is well illustrated by Hilbert's hotel metaphor. It is a hotel with infinitely many rooms, all occupied. When infinitely many new guests arrive, they are accommodated without any problems. To do this, each guest in room n moves to room 2n, leaving the odd ones free for new guests.
- In the definition of infinity, potential substitution is used. It is an inaccessible, unreachable entity. We can move towards it as much as we want, but without the possibility of reaching it. But, unlike what happens with irrational numbers, we cannot approach it. In fact, we are just as far away, no matter how far we advance. Infinity potentially exists because we can imagine and describe it.
- Infinity is also actual (or closed) because, as a symbol, it is a mathematical object that can be operated on, as we have seen in the properties. We say that infinity is an entity that has been "reified", reified, turned into something practical and manageable.
- According to the definition of infinity, the expression is fractal: infinity contains itself infinitely many times. For example, the set of natural numbers {1, 2, 3, 4, 4, 5, ...} includes as a subset the multiples of 2 {2, 4, 6, ...}, this in turn the multiples of 4 {4, 8, 12, ...}, this in turn the multiples of 8, etc., and so on to infinity. All these subsets have the same cardinality: infinite.
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):
a/bi> = a/(bi>+1) + a/(bi>+1)2 + a/(bi>+1)3 + ...
In the case of b=1, we have the series
a = a/2 + a/22 + a/23 + ...
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.
- "If you can show that there are numbers greater than infinity, your head spins. This may be the main reason they were invented."
- There is no such thing as actual infinity. It only makes sense to consider infinity as potential. It makes no sense to speak of an "infinite totality". It makes no sense to speak of infinite numbers or infinite sets. Infinity is incomplete, an endless process that is never completed. Cantor reifies mathematical objects that do not exist. "There is no such thing as 'all numbers.'" ' It is nonsense to speak of the number of all objects."
- Cantor mixes intensities with extensions. Infinity is not a quantity. Therefore, it is not a mathematical entity. The term "infinity" should be avoided in mathematics.
- The diagonal argument is incorrect because the irrational numbers in the numbered theoretical list of irrational numbers cannot be expressed because they contain infinite decimal places.
- For all the above reasons, the continuum hypothesis is not "unsolved problem" (as Hilbert said), but rather a pseudo-problem.
This criticism of Cantor's theory is part of Wittgenstein's conception of mathematics:
- Mathematics is algorithmic in nature.
- We cannot describe mathematics, we can only make it.
- The only mathematical reality is in the constructive process, not in the result.
- Mathematical truth is created, not discovered.
Other criticisms
Wittgenstein was not alone in criticizing Cantor's theory. We highlight the following:
- Kronecker considered Cantor's ideas about transfinite numbers absurd. He even claimed that irrational numbers did not exist, so he opposed their use. Kronecker was a finist. Finitism defends that a mathematical object only exists if it can be constructed in a finite number of steps. Kronecker famously said "God created the natural numbers, everything else is the work of man".
- Poincaré said that "There is no actual infinity". His criticism intensified when in 1905 Cantor's ideas were accepted. He said that transfinite number theory was a "disease" from which mathematics would eventually be cured.
- Brower said that Cantor's theory was "a pathological incident in the history of mathematics of which future generations will be horrified."
Brower admitted the non-numerability of the real numbers, so he rejected admitting sets that were not numerable. He also rejected the principle of the excluded third for infinite sets.
- Hermann Weyl, like Poincaré and Brower, rejected the idea of actual infinity: a set containing all the natural numbers and operable with that set.
- Mathematician Richard Hamming −the creator of the Hamming code, detector and corrector of errors in data− replied to Hilbert's "No one will expel us from the paradise Cantor created for us" with "I see no reason to enter it."
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):
- 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 ...
- 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).
- Ordering the expressions by their codes, it follows that the set Ω of all possible expressions of the language is infinite numerable.
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