Urelement - Urelements in Set Theory

Urelements in Set Theory

The Zermelo set theory of 1908 included urelements. It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements. (For an exception, see Suppes.) Axiomatizations of set theory that do invoke urelements include Kripke–Platek set theory with urelements, and the variant of Von Neumann–Bernays–Gödel set theory described by Mendelson. In type theory, an object of type 0 can be called an urelement; hence the name "atom."

Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen proved the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem. Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.

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