+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- # Zermelo--Fraenkel set theory with atoms * table of contents {: toc} ## Idea **ZFA** is a variant of the [[material set theory]] [[ZF]] which allows for objects, called _atoms_ or _[[urelements]]_ (hence the alternative name ZFU), which may be members of [[sets]], but are not made up of other elements. ZFA featured in early independence proofs, notably [[Fraenkel-Mostowski permutation models]], for example showing AC is independent of the rest of the axioms of ZFA. [[Zermelo|Zermelo's]] original 1908 axiomatisation of set theory included atoms, but they were soon discarded as a foundational approach as they could be modeled inside of atomless set theory. ## Definition There are two possible approaches to formulating ZFA. In both cases, we can further require that the [[axiom of choice]] is satisfied, and obtain ZFCA. ### Empty atoms In this approach, atoms are empty. We start by adding an additional unary predicate $A$, where we interpret $A(x)$ as saying "$x$ is an atom". We write $(\forall set x)$ to mean $\forall x, \neg A(x) \Rightarrow$, and similarly write $(\forall atom x)$ to mean $\forall x, A(x) \Rightarrow$. Then the [[axiom of extensionality]] says $$ (\forall set x) (\forall set y) (\forall z, z \in x \Leftrightarrow z \in y) \Rightarrow x = y, $$ and the [[axiom of empty set]] says $$ (\exists set x) (\forall y) (y \notin x). $$ We also add the axiom that says atoms are empty: $$ (\forall atom a) (\forall x) x \notin a. $$ Sometimes it is also convenient to assume that we have a set of atoms: $$ (\exists X) (\forall x) (A(x) \Leftrightarrow x \in X), $$ but in some cases, we might also like to consider models with a proper class of atoms. ### Reflexive/Quine atoms We can give up on the [[axiom of foundation]], and introduce the urelements as [[reflexive sets]], ie. sets $x$ such that $x = \{x\}$. In place of the axiom of foundation, we can have an axiom of _weak_ foundation, where we require the existence of a set A such that every element of A is reflexive, and the [[cumulative hierarchy]] built up from $A$ is the whole universe. In other words, if we define * $R(0) = A$, * $R(\alpha + 1) = P(R(\alpha))$ for any ordinal $\alpha$, * $R(\lambda) = \bigcup_{\gamma \lt \lambda} R(\gamma)$ for $\lambda$ a limit ordinal, then $V = \bigcup_\alpha R(\alpha)$. ## Models ### Fraenkel--Mostowski models By allowing atoms in our models, we lend ourselves to the method of [[Fraenkel-Mostowski models]], where we can obtain models in which the [[axiom of choice]] fails by imposing some symmetry on the atoms (so that we cannot uniformly pick an atom out of many). Such models are closely related to [[categories of G-sets]]. ## Related concepts * [[ZFC]] * [[Fraenkel-Mostowski models]] * [[permutation model]] * [[urelements]] [[!redirects ZFA]] [[!redirects ZFU]] [[!redirects ZFCA]] [[!redirects ZFAC]]