Set Theory - mathSight

The text below makes use of logic symbols.

Set theory provides a foundation of mathematics. Developed by George Cantor in the later 19th Century.It is claimed that everything in mathematics can be described as sets and relationships between sets.

George Cantor 1895 : “A set is a gathering together into a whole of definite, distinct objects of our perception or thought.”

A number can be described as a set of sets, a function as an set of ordered pairs. Spaces, such as a vector space, are sets with additional order on them. Algebra is formed from sets that have operations on the elements. Sets connect mathematics with logic.

We intuitively call a set a collection of objects that has something in common. If the set comprises of elements ${x}$ then the statement of ‘commonality’ could be written ${P(x)}$ . We then form a set by stating:

$\displaystyle \begin{array}{rcl} A = \{x:P(x)\} \end{array}$

The elements of a set can be elements themselves. A famous paradox arises from this position. Say ${P(x) = x \notin x}$ . We could try to find the set:

$\displaystyle \begin{array}{rcl} A = \{x : x \notin x\} \end{array}$

This says that the set A is the set of all elements ${x}$ that are not elements of themselves. ${A \notin A \implies A \in A}$ which is a contradiction. ${A \in A \implies A \notin A}$ which is also a contradiction. Hence the contradiction ${A \in A \iff A \notin A}$ arises.

One solution to this paradox is to not admit ${A}$ as a set. In a sense, it is too large to be a set. Instead the notion of a class is introduced. All sets are classes but not all classes are sets. A proper class is is a class that cannot be an element of another class.

At first sight this appears to be a patch on the problem. This is not the case. It states that not every definable collection can be treated as an object. In a grand sense, it is the difference between an object that is all of mathematics (a proper class) and objects within mathematics (classes and sets).

Another solution to this ‘too large a set’ problem is to restrict the objects that can be selected to form a set from pre-existing sets. This allows a safe way of building a set. If ${A}$ is a set then:

$\displaystyle \begin{array}{rcl} B = \{x \in A : P(x)\} \end{array}$

Both concepts affectively avoid sets that are too large.

We also need to be careful how ${P(x)}$ is phrased. Imagine we want the smallest positive integer not describable in fewer than twelve words.This statement is 11 words long. As there are an infinite number of positive integers but a finite possible permutation of 12 words then the smallest integer exists. However, the overall sentence itself is a description of an integer, under 12 words long and it is not the number of the last sentence. This is a paradox. We therefore have to be very careful how we describe the selection statement ${P(x)}$ . The solution is that ${P(x)}$ should be constructed from first order logic.

There is sufficient motivation in the above to study further. What is more grand than the foundation of mathematics.

If you need a recap on logic symbols please see this page.