Georg Cantor created set theory out of his desire to put the theory of real numbers on a sound basis. Despite initial prejudice and criticism, set theory was well established as a branch of mathematics by the turn of the 20th century.

The union of A and B

What are sets?

A set may be regarded as a collection of objects. This is informal but gives us the main idea. The objects themselves are called ‘elements’ or ‘members’ of the set. If we write a set A which has a member a, we may write a ∈ A, as did Cantor. An example is A = {1, 2, 3, 4, 5} and we can write 1 ∈ A for membership, and 6 ∈ A for non-membership.

Sets can be combined in two important ways. If A and B are two sets then the set consisting of elements which are members of A or B (or both) is called the ‘union’ of the two sets. Mathematicians write this as A ∪ B. It can also be described by a Venn diagram, named after the Victorian logician the Rev. John Venn. Euler used diagrams like these even earlier.

The set A ∩ B consists of elements which are members of A and B and is called the ‘intersection’ of the two sets.

The intersection of A and B

If A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 10, 21}, the union is A ∪ B = {1, 2, 3, 4, 5, 7, 10, 21} and the intersection is A ∩ B = {1, 3, 5}. If we regard a set A as part of a universal set E, we can define the complement set ¬A as consisting of those elements in E which are not in A.

The complement of A

The operations ⋂ and ⋃ on sets are analogous to × and + in algebra. Together with the complement operation ¬, there is an ‘algebra of sets’. The Indian-born British mathematician Augustus De Morgan, formulated laws to show how all three operations work together. In our modern notation, De Morgan’s laws are:

¬(A ∪ B) = (¬A) ∩(¬B)

and

¬(A ∩ B) = (¬A) ∪(¬B)

The paradoxes

There are no problems dealing with finite sets because we can list their elements, as in A = {1, 2, 3, 4, 5}, but in Cantor’s time, infinite sets were more challenging.

Cantor defined sets as the collection of elements with a specific property. Think of the set {11, 12, 13, 14, 15, . . .}, all the whole numbers bigger than 10. Because the set is infinite, we can’t write down all its elements, but we can still specify it because of the property that all its members have in common. Following Cantor’s lead, we can write the set as A = {x: x is a whole number > 10}, where the colon stands for ‘such that’.

In primitive set theory we could also have a set of abstract things, A = {x: x is an abstract thing}. In this case A is itself an abstract thing, so it is possible to have A ∈ A. But in allowing this relation, serious problems arise. The British philosopher Bertrand Russell hit upon the idea of a set S which contained all things which did not contain themselves. In symbols this is S = {x: x∉x}.

He then asked the question, ‘is S ∈ S?’ If the answer is ‘Yes’ then S must satisfy the defining sentence for S, and so S∉S. On the other hand if the answer is ‘No’ and S ∈ S, then S does not satisfy the defining relation of S = {x: x ∉ x } and so S ∈ S. Russell’s question ended with this statement, the basis of Russell’s paradox,

S ∈ S if and only if S ∉ S

It is similar to the ‘barber paradox’ where a village barber announces to the locals that he will only shave those who do not shave themselves. The question arises: should the barber shave himself? If he does not shave himself he should. If does shave himself he should not.

It is imperative to avoid such paradoxes, politely called ‘antinomies’. For mathematicians it is simply not permissible to have systems that generate contradictions. Russell created a theory of types and only allowed a ∈ A if a were of a lower type than A, so avoiding expressions such as S ∈ S.

Another way to avoid these antinomies was to formalize the theory of sets. In this approach we don’t worry about the nature of sets themselves, but list formal axioms that specify rules for treating them. The Greeks tried something similar with a problem of their own – they didn’t have to explain what straight lines were, but only how they should be dealt with.

In the case of set theory, this was the origin of the Zermelo–Fraenkel axioms for set theory which prevented the appearance of sets in their system that were too ‘big’. This effectively debarred such dangerous creatures as the set of all sets from appearing.

Gödel’s theorem

Austrian Mathematician Kurt Gödel dealt a knockout punch to those who wanted to escape from the paradoxes into formal axiomatic systems. In 1931, Gödel proved that even for the simplest of formal systems there were statements whose truth or falsity could not be deduced from within these systems. Informally, there were statements which the axioms of the system could not reach. They were undecidable statements. For this reason Gödel’s theorem is paraphrased as ‘the incompleteness theorem’. This result applied to the Zermelo–Fraenkel system as well as to other systems.

Cardinal numbers The number of elements of a finite set is easy to count, for example A = {1, 2, 3, 4, 5} has 5 elements or we say its ‘cardinality’ is 5 and write card(A) = 5. Loosely speaking, the cardinality measures the ‘size’ of a set.

According to Cantor’s theory of sets, the set of fractions Q and the real numbers R are very different. The set Q can be put in a list but the set R cannot . Although both sets are infinite, the set R has a higher order of infinity than Q. Mathematicians denote card(Q) by) , the Hebrew ‘aleph nought’ and card(R) = c. So this means <c.

The continuum hypothesis

Brought to light by Cantor in 1878, the continuum hypothesis says that the next level of infinity after the infinity of Q is the infinity of the real numbers c. Put another way, the continuum hypothesis asserted there was no set whose cardinality lay strictly between and c. Cantor struggled with it and though he believed it to be true he could not prove it. To disprove it would amount to finding a subset X of R with < card(X) < c but he could not do this either.

The problem was so important that German mathematician David Hilbert placed it at the head of his famous list of 23 outstanding problems for the next century, presented to the International Mathematical Congress in Paris in 1900.

Gödel emphatically believed the hypothesis to be false, but he did not prove it. He did prove (in 1938) that the hypothesis was compatible with the Zermelo–Fraenkel axioms for set theory. A quarter of a century later, Paul Cohen startled Gödel and the logicians by proving that the continuum hypothesis could not be deduced from the Zermelo–Fraenkel axioms. This is equivalent to showing the axioms and the negation of the hypothesis is consistent. Combined with Gödel’s 1938 result, Cohen had shown that the continuum hypothesis was independent of the rest of the axioms for set theory.

This state of affairs is similar in nature to the way the parallel postulate in geometry  is independent of Euclid’s other axioms. That discovery resulted in a flowering of the non-Euclidean geometries which, amongst other things, made possible the advancement of relativity theory by Einstein. In a similar way, the continuum hypothesis can be accepted or rejected without disturbing the other axioms for set theory. After Cohen’s pioneering result a whole new field was created which attracted generations of mathematicians who adopted the techniques he used in proving the independence of the continuum hypothesis.

the condensed idea

Many treated as one