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DOI
10.4324/9780415249126-Y024-1
DOI: 10.4324/9780415249126-Y024-1
Version: v1,  Published online: 1998
Retrieved January 29, 2022, from https://www.rep.routledge.com/articles/thematic/paradoxes-of-set-and-property/v-1

Article Summary

Emerging around 1900, the paradoxes of set and property have greatly influenced logic and generated a vast literature. A distinction due to Ramsey in 1926 separates them into two categories: the logical paradoxes and the semantic paradoxes. The logical paradoxes use notions such as set or cardinal number, while the semantic paradoxes employ semantic concepts such as truth or definability. Both often involve self-reference.

The best known logical paradox is Russell’s paradox concerning the set S of all sets x such that x is not a member of x. Russell’s paradox asks: is S a member of itself? A moment’s reflection shows that S is a member of itself if and only if S is not a member of itself – a contradiction.

Russell found this paradox by analysing the paradox of the largest cardinal. The set U of all sets has the largest cardinal number, since every set is a subset of U. But there is a cardinal number greater than that of any given set M, namely the cardinal of the power set, or set of all subsets, of M. Thus the cardinal of the power set of U is greater than that of U, a contradiction. (The paradox of the largest ordinal, discussed below, is similar in structure.)

Among the semantic paradoxes, the best known is the liar paradox, found by the ancient Greeks. A man says that he is lying. Is what he says true or false? Again, either conclusion leads to its opposite. Although this paradox was debated in medieval Europe, its modern interest stems from Russell, who placed it in the context of a whole series of paradoxes, including his own.