Urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set (has no elements), but that may be an element of a set. It is also referred to as an atom or individual. Ur-elements are also not identical with the empty set.
Theory
There are several different but essentially equivalent ways to treat urelements in a
One way is to work in a first-order theory with two sorts, sets and urelements, with a ∈ b 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.
Another way is to work in a
This situation is analogous to the treatments of theories of sets and
Urelements in set theory
The
Adding urelements to the system
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}.[7]
Quine atoms cannot exist in systems of set theory that include the
Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.[9]
Quine atoms are the only sets called reflexive sets by Peter Aczel,[8] 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.[10]
References
- ^ Dexter Chua et al.: ZFA: Zermelo–Fraenkel set theory with atoms, on: ncatlab.org: nLab, revised on July 16, 2016.
- ISBN 0486466248.
- ISBN 0486616304. Retrieved 17 September 2012.
- ISBN 978-0412808302. Retrieved 17 September 2012.
- S2CID 46960777.
- ^ Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant.
- ISBN 978-0-521-53361-4.
- ^ MR 0940014, retrieved 2016-10-17.
- ISBN 1575860090.
- ISBN 1575860090.