Article
On the scope of the validity domain of Venn-Euler diagrams. New version.
author: Carlos Oscar Rodríguez Leal
Abstract
In this work it is rigorously shown that the methods of Venn's diagrams are formal procedures for demonstrating the formulas of sets for certain subsets, determining exactly the family of sets that meet this condition. It should be mentioned that in this work only is considered classical set theory that
accepts the continuum hypothesis, so the domain of the validity of the Venn's diagrams shown here is only applicable to the sets considered under this assumption.
A previous paper regarding this topic that I develop in the past contained subtle errors, which are amended in this paper.
Introduction
In set theory, the validity of the use of Venn-Euler's diagrams as a formal proof of the formulas presented there is questioned. For many mathematicians and researchers the Venn's diagrams method represents an informal proof of set formulas, so according to they should only be used as a guide to see if said formulas are certain or not, and in case of passing the test of the diagrams, should be used the formal deductive-analytical conventional methods to demonstrate them. However, today there are many researchers who consider that Venn's diagrams do represent a formal proof of such formulas, this due to the evident and unobjectionable nature of their evidence.
However, this work rigorously demonstrates that Venn's diagram methods are formal demonstration procedures of set formulas but only for certain sets. And more to one, I know determine exactly the type of sets that fall within this category, that is, the family of sets that meet this condition.
It should be mentioned that in this work only is considered classical set theory that accepts the continuum hypothesis, so the domain of the validity of the Venn's diagrams shown here is only applicable to the sets considered under this assumption.
Finally, a previous paper regarding this topic that I develop in the past [1] contained subtle errors, which are amended in this paper.
Previous theorems
In this section I will mention certain theorems that I will need later. First I will establish the following theorem.
Theorem 1. Every open square of edge length $l$ has the cardinality of the continuum.
The proof of the previous theorem is immediate from the dedefinitions of cardinal numbers, which can be consulted in [8].
Teorem 2. An open bounded set in the normed space $R^2$ has cardinality $c$.
Demonstration. Given a bounded open set $B$ in the normed space $R^2$, it is obvious that $B$ can be contained within an open square large enough in size and consequently said open set it cannot have cardinality greater than $c$. Therefore it is enough to show that its cardinality is not less than $c$. To do this, let's take a point $p$ from $B$. Then, because $B$ is open, there is an open ball centered at $p$ and contained in $B$. But due to the equivalence of all the norms in $R^2$ [4], in particular we can consider the metric of the absolute value, so there is an open square centered on $p$ and contained in $B$. Therefore $B$ contains a subset of cardinality $c$ and consequently its cardinality cannot be less than $c$.
Now I will proceed to properly demonstrate the validity of the Venn's diagrams.
Precise determination of the region of validity of Venn-Euler's diagrams
This section establishes the properties that the sets must meet on which the procedures of diagrams can be applied.
We must first define our Venn's diagrams as bounded open sets in the normed space $R^2$, i.e. we should not consider the boundary line in our drawings. Also our initial sets with which we will begin to work must be simply connected [6], for so that our figures match the type of figures used in the Venn-Euler diagrams.
Thus, the geometric figurines of Venn's diagrams are sets of points of cardinality $c$ according to theorem 2, so that, given a universe set $U$ also in cardinality $c$, this can be put into correspondence one-to-one with the geometric points of a subset $\tilde{U}$ of a Venn diagram $\hat{U}$ (open, bounded, and simply connected) of an infinite number of ways, i.e. there are multiple possible bijective functions between the universe of discourse and some abstract region of the plane that we represent with an imperfect and imprecise drawing in its border points but whose abstraction that represents is ideal.
The same procedure is carried out in the geometric proofs of classical Euclidean geometry, where the demonstrations are made based on some ideal geometric figure, but that we have represented with a figure particular and imperfect, however it serves as a guide to be able to make all the constructions and deductions in our mind abstractly, where the figure is generic and ideal.
Therefore, if from that universal set we take a finite family $\tau$ of subsets (finite or infinite in size) with which we will have to work, then in turn these sets correspond to points within the Venn's diagram, from such that if we represent them with a family of open figures $\hat{\tau}$ (without the boundary line), simply connected, and contained within $\hat{U}$ , then by this choice many possibilities of bijectives functions are possible (which are an infinity of possibilities, see figure 1).
Now, in the case that our set $U$ is of cardinality $\aleph_0$ , such a set is still expressed in a Venn's diagram if we only consider a discrete number of points for $\tilde{U} \subset \hat{U}$. And the explanations given for the case $#U = c$ are still applicable to this case by analogy.
And as a last case, if $U$ is finite, the same argument applies, taking only a finite number of points for $\tilde{U}$ and considering only a family $\tau$ of finite subsets.
Consequently, the geometric-type abstract reasoning that we make with Venn's diagrams will be valid for sets of pairs $(U, \tau)$ of universes of finite cardinality, $\aleph_0$ or $c$ cardinality, together with their respective "finite" families of subsets $\tau$ (since in the proofs with diagrams we can only work with a finite family $\tau$, for obvious reasons). Or at least such proofs should be considered semi-formal and no less valid than those made in Euclidean geometry (for which it is also fair to say that there is a trend since the days of Descartes [2] to transform it into an analytical science, and more modernly into a purely axiomical science [3], but
whose graphical demonstration methods are still accepted today).
And for sets of cardinality greater than $c$, Venn's diagram methods no longer represent a formal proof of the proposed formulas, however, in practice it is not so common in many contexts to work with sets of cardinality greater than $c$.
Finally, I have to mention that in this work only the classical set theory that accepts the hypothesis of the continuum is considered, so the domain of validity of Venn's diagrams demonstrated here is also restricted to the sets considered under this logical assumption.
Conclusions
In this paper it was demonstrated the validity of Venn-Euler diagrams for the proofs in set theory. In turn, it was shown that said validity is restricted to sets whose cardinality is not greater than $c$ and where we have accepted the continuum hypothesis.
The importance of the above lies in the fact that it tells us in which cases we can consider valid and formal the proofs in set theory using Venn diagrams.
However, It should be noted that in a future paper that I am developing, I demonstrate logical errors of this paper, where Venn diagrams are valid even for sets of cardinality greater than $c$, as long as the family of $\tau$ subsets with which one works is finite.
References
[1] Carlos Rodríguez, "Sobre los alcances del dominio de validez de los diagramas de Venn-Euler", Collection of Articles on Mathematics, 2018.
[2] C. H. Lehmann, "Geometría Analítica", First edition, Limusa, Mexico, (1997).
[3] D. Hilbert, "Fundamentos de la Geometría", Second edition, Consejo Superior de Investigaciones Científicas, Spain, (1996).
[4] E. Kreyszig, "Introductory Functional Analysis with Applications", Second edicion, Wiley, United States of America, (1989).
[5] G. Villalobos y E. Gosteva, "Teoría de Conjuntos", First edition, amate editorial, Mexico, (2000).
[6] I. Bronshtein y K. Semendiaev, "Manual de matemáticas para ingenieros y estudiantes", Third reprint, Ediciones de Cultura Popular, Mexico, (1977).
[7] J. Wentworth y D. E. Smith, "Geometráa Plana y del Espacio", Twenty-fourth edition, Editorial Porrua, Mexico, (2003).
[8] L. Seymour, "Set Theory and Related Topics", Second edition, Schaum's outline series, McGraw-Hill, (1998).
