Since ancient times, the role of Mathematics for rational enquiry of the truth in the foundation of all scientific thoughts has always been acknowledged. Dawn of \({19}^{th}\) century was marked by the beginning of systematic search for the foundations of Mathematics and after going through a series of difficulties in the early \({20}^{th}\) century, the mathematical discoveries started stabilizing resulting in a large, intelligible body of mathematical knowledge which is still providing impetus to active research fields.
The wheel in the late nineteenth century turned in a new direction when a mathematical theory of sets was crafted and advanced by the renowned mathematician, Georg Cantor (1845-1918). The seed of this theory was sown when Cantor proved a key theorem in real analysis, through which a method for creating real number sets that elaborated on a countless iteration of the limit operation, was introduced. The novelty of this proof steered him into an insight of sets of real numbers and to the abstraction of his set theory. The conception of ideas given by Cantor has now permeated mathematics, offering an adaptable means for exploring even deeper concepts of infinity and sets of infinite measure.
The credit for investigating the following question goes to Cantor:
Can the concept of “size” be extended to infinite sets?
The uncountability of the set of real numbers led Cantor to establish the following result: the existence of “infinitely many different infinites”, which had an obvious mathematical and philosophical implication.
In 1874, Cantor established that the set of natural numbers and the set of algebraic numbers were in one-to-one correspondence. At the same time, he established that no such one-to-one correspondence existed between sets of natural and real numbers. Cantor tried to explore the possibility of the existence of any infinite sets of real numbers that corresponded in one-to-one manner with the set of natural numbers, but not with the real numbers set. In 1878, the Continuum Hypothesis was proclaimed by Cantor after introducing uncountable sets, which stated that: “every infinite set of real numbers matched in size with either the set of natural numbers or the entire set of real numbers and no set of intermediate size existed”. Since this result led to the conclusion that there were more than one level of infinity, it faced opposition by many mathematicians of that time. Some denied its existence based on their understanding that infinity was a non-legitimate mathematical abstraction. On the other hand, Christian theologians held the view that his work was contrary to the uniqueness of the absolute infinity in the nature of God.
Throughout most of his career, Cantor grappled, without any success, to find a resolution to the Continuum Hypothesis. However, the problem continued to remain as one of the most prominent and baffling problems of the 20th century. In 1940, mathematician Kurt Gödel showed that it couldn’t be disproved within the usual axioms of set theory. In the 1960s, mathematician Paul Cohen showed that the continuum hypothesis can’t be proved by set theory. This won Cohen the Fields Medal, the highest honor in mathematics.
Eventually, Cantor’s contribution to mathematical investigations was acknowledged. David Hilbert, a prominent mathematician of the 20th century, described Cantor’s work as “the finest creation of mathematical genius and one of the unbeatable achievements of purely scholarly human activity”.
Hilbert, who had expertise in posing mathematical questions, published 23 open questions in the year 1900. Many have been solved but some still remain unanswered. But all endeavors to seek a solution to these problems have resulted in some very deep mathematics. Riemann hypothesis is one such example. The act lies in asking a good question.
In 1946, he presented 10 of these 23 problems at the first major international gathering of mathematics after the World War II, ‘The Princeton University Bicentennial Conference on Problems of Mathematics’. The problem which topped the list still remains unreturned, and it is the famous ‘Continuum Hypothesis’.
After Cantor’s demise, most of the researchers in the field of set theory declared that the Continuum Hypothesis was unresolvable. From Cantor until 1940, Continuum remained the focus of the advancement in Set Theory. Although both, the set of natural numbers and that of real numbers, are infinite, there are more real numbers than the natural numbers; this commenced the journey to the exploration of different sizes of infinity. However, Cantor’s ideas gained momentum and set theory gained significance in the unearthing and creation of new mathematical results, especially in fields like the theory of functions and measure theory. The efficacy of this tool regarding traditional mathematics was realized by the mathematical community and received acceptance due to corresponding change in the attitudes. The theory of abstract sets propounded by Cantor would revolutionize the mathematical progress in due course.
There are two general approaches to set theory “Naïve set theory” credited to Cantor and “Axiomatic set theory” which is due to Zermelo-Fraenkel (ZFC). The most important difference in the two approaches is that the naive theory doesn’t have much by way of axioms. As stated by Cantor:
“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.”
Cantor did not define the concept of ‘set’ in his Naïve set theory. The nature of elements of the sets was generally ignored. While the set was considered as a collection of objects, it was also presumed that any object can be a member of a set. “Naïve set theory” in the sense of naïve theory is a nonformalized theory wherein sets and operations on sets are styled using natural language. The terms and, or, if . . . then, not, for some, for every, are the same as in ordinary mathematics.
Paradoxes are witnessed, without restriction, upon assuming that any property may be used to form a set, a common example being Russell’s paradox; there is no set consisting of “all sets that do not contain themselves”.
More set theories emerged from the questions raised on the unambiguity and consistency of what exactly comprised and what did not comprise a set. One such widely accepted theory is Zermelo- Fraenkel set theory.
An axiomatic structure was advocated by Ernst Zermelo and Abraham Fraenkel so as to frame a theory of sets restricting paradoxes such as Russell’s paradox in the early twentieth century.
ZFC is the truncation of Zermelo-Fraenkel set theory with the debatable axiom of choice counted in, where C denotes “choice”, and ZF stands for the axioms of Zermelo-Fraenkel set theory with the axiom of choice omitted. Following axioms comprise ZFC set theory:
Another “special” and initially controversial axiom ‘Axiom of choice’ is usually stated separately.
Axiom of choice: For every set \(A\) of non-empty sets there is a function \(f\), which associates to every set \(A\) in \(A\), an element \(a \in A\).
Sets have abundant significance in mathematics. In modern formal treatments, sets are used to define almost all mathematical objects like numbers, functions, relations, etc. Naïve set theory is sufficient for many reasons, as well as serves as a starting point for more formal treatments. If a naïve set theory correctly identifies the sets permissible to be studied, it is not essentially inconsistent.
Similarly, an axiomatic set theory is neither essentially stable nor essentially free of paradoxes as is evident from Gödel’s incompleteness theorems.
The choice between an axiomatic approach and other approaches largely depends on convenience. As far as mathematics in everyday is concerned, informal use of axiomatic set theory may be the best choice. Depending on notation, this informal usage of axiomatic set theory can precisely have the form of naïve set theory, which is noticeably simpler to read, write and grasp.
P.R. Halmos lists these properties as axioms in his book “Naïve Set Theory” as follows:
The humble notion of a set is generally introduced casually and regarded as self-evident. Very deep and magnificent, yet fundamental and humble, the concepts of Set Theory pervade all branches of mathematics. Remarkably, all usual mathematical objects can be represented as sets. For example, within set theory, one can create the natural numbers as well as the real numbers. The formal language of pure set theory allows one to formalize all mathematical notions and arguments. All algebraic structures, functional spaces, vector spaces, and topological spaces can be regarded as sets in the universe of sets. Therefore, while mathematical theorems can be regarded as statements about sets, they can also be proven from ZFC, which, in turn, are the axioms of set theory. One can, hence, infer that mathematics is rooted in set theory.
Since all of conventional mathematics can be developed within set theory, one can view certain results in set theory as being part of Metamathematics, the specialized branch within mathematics, which encompasses available mathematical tools to explore the nature and potential of mathematics.
Both aspects of set theory, namely, the mathematical science of the infinite, and the foundation of mathematics, are of philosophical importance.
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