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Noncomputability

When we want to specify the elements of a set that contains only a few elements, the most direct and obvious way is to exhaustively list all the elements in the set. However, when a set contains a large number of an infinite number of elements, exhaustively listing all elements in the set becomes impractical or impossible. For example, we may have

P = {x|x is a high school student in Illinios}

where P is a finite set with a large number of elements. We may have,

Q = {x|x is a perfect square}

where Q is a countably infinite set of integers. Also, we may have,

R = {x| {a, b} ⊆ x}

Note that R is a set of sets such that every element in R has the set {a, b} as a subset.

We want to show that there is a possible pitfall when we specify the elements of a set by specifying the properties that uniquely characterize these elements.

Consider the set

S = {x|x ∉ x}

It seems that we have followed the “recipe” and have defined a set S such that a set x is an element of S if x ∉ x. Thus for example, {a, b} is an element of S because {a, b} ∉ {a, b}. {{a}} is also an element of S because {{a}} ∉ {{a}}. However, suppose someone wants to know whether S is an element of S. In other words, she wants to know whether S ϵ S. Following the specification, we say that for S to be an element of S it must be the case that S ∉ S, which is a self contradictory statement. Let us turn around and assume that S is not an element of S; that is S ∉ S. Then, according to the specification, S should be an element of S. That is, if S ∉ S then S ϵ S- again, a self-contradictory statement. We hasten to point out that what we have said is not just a pun and have by no means attempted to confuse the reader with entangled and complicated syntax. Rather, contrary to our intuition, it is not always the case that we can precisely specify the elements of a set by specifying the properties of the elements in the set. Such an observation was first made by B. Russell in 1911, and is referred to as Russell’s appendix.