"Continuity belongs to the realm of consciousness" (Franklin Merrell-Wolff).
"The continuum is an inextinguishable source of numbers, but it consists not of points but of infinitesimal parts" (Charles Sanders Peirce).
"Throughout the ages, the problems of the continuum and of infinity were two horns of a single dilemma" (Tobias Dantzig).
The Problem
The continuum problem is essentially the problem of the cardinality of the real numbers. Hilbert, at the famous International Congress of Mathematicians in Paris in 1900, posed 23 problems that remained to be solved. The first of these was the continuum problem, which he considered to be the most important mathematical problem. Today, this problem remains unsolved.
Cantor's Theory
The theorem on the points of the continuum
According to this theorem, the points of a continuous manifold of n dimensions (n > 1) can be made to correspond biunivocally to the points of a continuous manifold of one dimension. Therefore, both varieties have the same cardinality. Cantor, in a letter to Dedekind on June 29, 1877, asked for his opinion and said "If I do not see it, I do not believe it".
A manifold is a geometric object that generalizes the intuitive notion of curve (1-manifold) and surface (2-manifold) to any dimension.
For example, the points of a square can be made to correspond biunivocally to the points of a segment. The demonstration in this case is very simple. If we have a square of side 1 with the lower left vertex at the origin of coordinates, an interior point P has coordinates (0. a1a2a3..., 0. b1b2b3...). To this point P we make correspond the point Q of the base segment of the square 0. a1b1a2b2a3b3. ...And conversely, to every point Q of the unit base segment 0. a1b1a2b2a3b3. .. can be made to correspond to the point P of coordinates (0. a1a2a3..., 0. b1b2b3...).
Dedekind intuited that the application of a square on a segment was discontinuous. In 1910, Brower proved Dedekind's conjecture. In 1878, Cantor proved that the sets Rn and R have the same cardinality, that is, every Euclidean space of dimension n has the same cardinality as the space of dimension 1 (the real line).
The continuum hypothesis
Cantor introduced the concept of cardinal number to compare the size of infinite sets, proving in 1874 that the cardinal of the set of natural numbers is strictly smaller than that of the real numbers. Cantor wondered whether there exist infinite sets whose cardinality is strictly included between those of both sets.
The continuum hypothesis (CH) states that there is no set of infinite cardinality that is strictly included between that of the integers and that of the real numbers, i.e., between ℵ< sub>0 and ℵ1 and that c (the cardinality of the continuum) is ℵ 1, i.e., c = 2^ℵ0 = ℵ1,. Cantor believed in this hypothesis, but could neither prove nor disprove it.
In 1940, Gödel proved that CH was consistent with ZF (Zermelo-Fraenkel axiomatic set theory) and with ZFC (ZF including the axiom of choice).
Paul Cohen showed in 1963 that neither the continuum hypothesis (CH) nor the axiom of choice (AE) can be proved from the standard axioms of set theory, the Zermelo-Fraenkel axioms (ZF). That is, these two axioms (CH and AE) can neither be proved nor disproved from the ZF axioms. Both are undecidable.
The axiom of choice in set theory states, "In a collection (finite or infinite) of nonempty sets there exists a set that contains one element, and only one, from each set in the collection."
This axiom is very controversial and not all mathematicians accept it. If it is accepted it is because it simplifies the mathematical proofs. In the case of a finite collection, the axiom is self-evident. If the collection is infinite, it is impossible to develop an algorithm capable of constructing a choice set because the algorithm would never terminate. From MENTAL's point of view, the choice set could only be described, not constructed, by being infinite.
Anyway, CH may or may not be included as an axiom. Doing so or not leads to two different versions of the real straight line.
The generalized continuum hypothesis (CHG) states that there are no intermediate infinities between ℵn and ℵn+1. It has been shown that CHG is also independent of ZF theory and implies the axiom of choice.
Critique of Cantor's Theory and Solution to the Continuum Problem
On the concept of "continuum"
There is much confusion about the concept of "continuum". In the light of the principle of descending causality and the two modes of consciousness, everything becomes clearer:
With the concept of continuous (opposed to discrete) we encounter a situation similar to the concept of infinite (opposed to finite). The continuous and the infinite belong to the synthetic mode of consciousness, of the right hemisphere (RH) of the brain. The discrete and the finite belong to the analytic mode of consciousness, of the left hemisphere (LH) of the brain.
LH mode
RH mode
Finite
Infinity
Discreet
Continuous
The concept of continuum comes from geometry, specifically the straight line. Its dual concept is arithmetic, which is discrete in nature.
A straight line is not made up of points nor does it "contain" points because a straight line has extension and the point does not. The continuum contains segments. Points are (discrete) manifestations of the continuum. The point, not having extension, has no parts, in the same way that zero is indivisible. A point is a direction of the real line, not a content.
The point is inexpressible, it belongs to a transcendent realm of non-space and non-time. Nevertheless, it can be represented as a number or sequence of numbers. A point is represented in space nD by nordered numbers (sequence). It is a geometric archetype (the point) represented by arithmetic archetypes (the numbers). The point as an archetype is an intermediary element between the continuous and the discrete.
It makes no sense to speak of a "real line" because the real numbers are not contained in the continuous. They are only manifestations of the continuum. The real numbers do not constitute the continuum, they are not part of the continuum, like the points.
The continuum contains itself. A segment has holographic structure. Each part is like the whole or similar to the whole. At the qualitative level, all segments are the same segment, not so quantitatively.
Since the point has no extension, all segments are qualitatively identical, including the infinitely large and the infinitely small. The continuum is made up of parts. Each part has the quality (the continuous) of the whole.
The manifested points of the straight line can be made to correspond to the real numbers, thus obtaining the real line. The real line has infinite numbers and is of infinite density and each of its parts also has these properties.
The continuous is a quality, like the infinite The quantitative is the superficial. The continuous cannot be captured from the discrete. The qualitative cannot be captured from the quantitative. The profound cannot be captured from the superficial. The intuitive cannot be captured from the rational.
The continuum cannot be rationalized. From the superficial one cannot access the profound. The points and the real numbers are manifestations of the continuum. From points and real numbers the continuum cannot be accessed.
According to Brower's continuity theorem, the continuum cannot be divided, it is indecomposable. Formally: every decidable subset of R is either empty or includes all elements of R.
On the problem of the cardinality of the continuum
Naively posed, the continuum problem is: how many points does a line segment contain? This is a false problem for the following reasons:
It makes no sense to talk about the cardinality of the continuum because cardinality only applies to the discrete. The continuous has no cardinality because it is a quality.
A segment, however small, has extension. A point has no extension. Therefore, it does not make sense to say that the line consists of infinitely many points, but it does make sense to say that it is manifested in infinitely many points. To each point corresponds a real number.
A segment, being continuous, belongs to a deeper level of reality than points. Points are manifestations of the segment. To ask how many points there are in a segment is to apply a superficial approach to the deep, which is doomed to failure. The approach must be the opposite. The principle of descending causality must be applied, in this case from the continuous to the discrete, from the segment to the points.
The continuous and the discrete correspond to the two poles of consciousness, to the deep and the superficial, respectively. From the continuous, the profound, emanates or manifests the discrete, the superficial. The real line (the continuous) contains or manifests the numbers (discrete), but in itself the real line is not numerable, neither finite nor infinite. The discrete is the analytic; the continuous is the synthetic.
The continuous manifests itself as points and infinitesimals. Points have no extension (or content), they are just directions, and these directions are expressible by coordinates in a Cartesian system. Infinitesimals have imaginary extension and are not expressible. The continuum is not constituted by points because then it would have no extension. Neither is it constituted by infinitesimals because its extension would be imaginary. The continuum is inexpressible.
On the equipotence of all segments
If we have two segments of different lengths in the plane, a biunivocal correspondence can be established between the points of one and the other segment. If the lengths of the segments A and B are a and b, respectively, to the point fa of A is matched by the point fb of B, where f is a factor between 0 and 1. Ultimately all segments can be put in biunivocal correspondence with the segment [0−1].
But this does not mean that all segments have the same cardinality for several reasons: 1) points have no extension; 2) segments have the quality of the continuum and have no cardinality; 3) their manifestations (points) have the common property of being infinite, but being infinite is a quality, not a quantity.
Conclusions
There is no continuum problem. There is no sense in the concept of "continuum cardinality".
To paraphrase Wittgenstein, "The continuum hypothesis is not an unsolved problem, but rather a pseudo-problem."
Points and real numbers are manifestations of the continuum.
The cardinality of points and real numbers is infinite, in a qualitative sense.
Surprisingly, it is still admitted today that Cantor was right in distinguishing different classes of infinities.
Addenda
The Banach-Tarski paradox (1 = 2)
This paradox states that it is possible to take a solid sphere, cut it into a finite number of pieces, restructure them using only rigid translational and rotational motions, and reassemble them into two copies identical to the original sphere, thus doubling the original volume. Surprisingly, nothing more than 5 pieces are needed to achieve this result. Moreover, one of these pieces consists of a single point, which is the center of the sphere. This paradox is explained by the following:
This is not a physical sphere, but a mathematical sphere, which is the set of points that compose a 3D sphere in R3. The mathematical sphere is continuous, contains infinite points, is of infinite density and is infinitely divisible (as occurs in a segment of R). A physical sphere, on the other hand, is not infinitely divisible because it consists of a finite number of atoms.
Each of the pieces has a strange and complex shape, so the volume is not measurable (it is not well defined and is incompatible with the notion of volume) because it consists of isolated points distributed over the entire volume of the sphere. Not all 3D objects have volume.
Use is made of the axiom of choice, In the paradox case, each of the infinitely complex pieces is constructed from choice sets.
The two spheres still have the same infinite number of points and the same infinite density as the original sphere. But in a physical sphere, the two resulting spheres would have half the density.
An alternative version of this theorem states that it is possible to take a sphere the size of a pea, cut it into a finite number of pieces, and reassemble it to form a new sphere the size of the Sun.
Bibliography
Dauben, Joseph Warren. George Cantor. His Mathematics and Philosophy of the Infinite. Princeton University Press, 1990.
Cohen, Paul J. Set Theory and the Continuum Hypothesis. Dover Books on Mathematics, 2008.
Gödel, Kurt. Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press, 1940.
Jech, Thomas J. The Axiom of Choice. Dover Books on Mathematics, 2008.
Kanamori, Akihiro. The Mathematical Development of Set Theory from Cantor to Cohen. Elsevier, 2012. Disponible online.
Moore, Gregory H. Zermelo’s Axiom of Choice: Its Origins, Development, and influence. Dover Books on Mathematics, 2013.
Smullyan, Raymond M.; Fitting, Melvin. Set Theory and the Continuum Problem. Dover Books on Mathematics, 2010.
