Usuário(a):Leonardo Coelho/Testes

Urelementos na teoria dos conjuntos

A teoria de conjuntos de Zermelo de 1908 incluía urelementos e é, assim, agora chamada de ZFA (Zermelo–Fraenkel com átomos) ou ZFCA (i.e. ZFA com o axioma da escolha). Logo se percebeu que, no contexto desta e das demais teorias axiomáticas de conjuntos correlatas, os urelementos não eram necessários pois podiam ser facilmente modelados em uma teoria de conjuntos sem urelementos. Assim, as exposições mais corriqueiras das teorias de conjuntos axiomáticas canônicas como ZF e ZFC não mencionam urelementos. (Há uma exceção: ver Suppes.) Axiomatizações de teoria de conjuntos que lançam mão de urelementos incluem a `teoria de conjuntos de Kripke–Platek com urelementos` e a `teoria dos conjuntos de Von Neumann-Bernays-Gödel` descrita por Mendelson. Na teoria dos tipos, um objeto do tipo 0 pode ser chamado de urelemento: daí o nome "átomo".

Ao se adicionar urelementos ao sistema `Novas Fundações` (NF) para produzir NFU (Novas Fundações com urelementos) obtém-se consequências surpreendentes. Em especial, Jensen provou a consistência de NFU em relação à aritmética de Peano ao passo que a consistência de NF relativamente a qualquer outra coisa permance um problema em aberto

In particular, Jensen proved the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF.

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Urelements in set theory

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, pending verification of Holmes's proof of its consistency relative to ZF. 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. In finitist set theory, urelements are mapped to the lowest-level components of the target phenomenon, such as atomic constituents of a physical object or members of an organisation.

Quine atoms

An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. Quine atoms (named after Willard Van Orman Quine) are sets that only contain themselves, that is, sets that satisfy the formula x = {x}.

Quine atoms cannot exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist (or rather, this would mean ZF is inconsistent), but it is compatible with the existence of Quine atoms. Aczel's anti-foundation axiom implies there is a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's axiom of superuniversality, which implies that the distinct Quine atoms form a proper class.

Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.

Quine atoms are the only sets called reflexive sets by Peter Aczel, although other authors, e.g. Jon Barwise and Lawrence Moss use the latter term to denote the larger class of sets with the property x ∈ x.