Urelement - Theory

Theory

There are several different but essentially equivalent ways to treat urelements in a first-order theory.

One way is to work in a first-order theory with two sorts, sets and urelements, with ab only defined when b is a set. In this case, if U is an urelement, it makes no sense to say

,

although

,

is perfectly legitimate.

This should not be confused with the empty set where saying

is well-formed but false.

Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements.

This situation is analogous to the treatments of theories of sets and classes. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are minimal objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally.)

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