Editing Absolute infinite (section)

== The Burali-Forti paradox ==
{{main|Burali-Forti paradox}}
The idea that the collection of all ordinal numbers cannot logically exist seems [[paradoxical]] to many. This is related to the Burali-Forti's paradox which implies that there can be no greatest [[ordinal number]]. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.

More generally, as noted by [[A. W. Moore (philosopher)|A. W. Moore]], there can be no end to the process of [[set (mathematics)|set]] formation, and thus no such thing as the ''totality of all sets'', or the ''set hierarchy''. Any such totality would itself have to be a set, thus lying somewhere within the [[cumulative hierarchy|hierarchy]] and thus failing to contain every set.

A standard solution to this problem is found in [[Zermelo set theory]], which does not allow the unrestricted formation of sets from arbitrary properties. Rather, we may form the set of all objects that have a given property ''and lie in some given set'' (Zermelo's [[Axiom schema of specification|Axiom of Separation]]). This allows for the formation of sets based on properties, in a limited sense, while (hopefully) preserving the consistency of the theory.

While this solves the logical problem, one could argue that the philosophical problem remains. It seems natural that a set of individuals ought to exist, so long as the individuals exist. Indeed, [[naive set theory]] might be said to be based on this notion. Although Zermelo's fix allows a [[Class (set theory)|class]] to describe arbitrary (possibly "large") entities, these predicates of the [[metalanguage]] may have no formal existence (i.e., as a set) within the theory. For example, the class of all sets would be a [[proper class]]. This is philosophically unsatisfying to some and has motivated additional work in [[set theory]] and other methods of formalizing the foundations of mathematics such as [[New Foundations#How NF.28U.29 avoids the set-theoretic paradoxes|New Foundations]] by [[Willard Van Orman Quine]].