We first make a few simple assumptions about sets. We allow them to be infinite, and we allow them to be elements of themselves.

Most of the sets we ordinarily deal with are not elements of themselves. For examples, the set of all integers is not itself an integer; the set of all pool tables is not a pool table; and so on. There are some sets, however, that can be thought of as elements of themselves. For example, consider the set I of all infinite sets. There are an unlimited number of such sets, so it appears that I contains itself.

Next, we consider the set V of sets that are not elements of themselves. In more formal language, V = {∀X: X ∉ X}.

We pose the question: Is V ∈ V? If so, then by definition, V ∉ V, a contradiction. So it must be the case that V ∉ V. But it then follows by the same definition that V ∈ V, another contradiction. This is called Russell's Paradox [1901], after its discoverer, the British mathematician and logician Bertrand Russell [1872-1970].

Note: Cesare Burali-Forti, an assistant to the Italian set theorist Giuseppe Peano [1858-1932], discovered a similar antinomy in 1897. He noticed that since the set of ordinals is well-ordered, it, too, must have an ordinal. However, this ordinal must be both an element of the set of all ordinals and yet greater than every such element.

Sources: Stanford Encyclopedia of Philosophy and Paul R Halmos, Naive Set Theory

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