Editing New Foundations (section)

=== Large sets ===
NF (and NFU + ''Infinity'' + ''Choice'', described below and known consistent) allow the construction of two kinds of sets that [[ZFC]] and its proper extensions disallow because they are "too large" (some set theories admit these entities under the heading of [[proper class]]es):
* ''The [[universal set]] V''. Because <math>x=x</math> is a [[stratified formula]], the [[universal set]] ''V'' = {''x'' | ''x=x''} exists by ''Comprehension''. An immediate consequence is that all sets have [[complement (set theory)|complements]], and the entire set-theoretic universe under NF has a [[Boolean algebra (structure)|Boolean]] structure.
* ''[[cardinal number|Cardinal]] and [[ordinal number|ordinal]] numbers''. In NF (and TST), the set of all sets having ''n'' elements (the [[circular reasoning|circularity]] here is only apparent) exists. Hence [[Frege]]'s definition of the cardinal numbers works in NF and NFU: a cardinal number is an [[equivalence class]] of sets under the [[Relation (mathematics)|relation]] of [[equinumerosity]]: the sets ''A'' and ''B'' are equinumerous if there exists a [[bijection]] between them, in which case we write <math>A \sim B</math>. Likewise, an ordinal number is an [[equivalence class]] of [[well-ordering|well-ordered]] sets.