Editing Axiom schema of specification (section)

== Unrestricted comprehension<!--'Unrestricted comprehension' and 'Axiom schema of unrestricted comprehension' redirect here--> ==
{{Unreferenced section|date=June 2024}}
{{also|Basic Law V}}
The '''axiom schema of unrestricted comprehension'''<!--boldface per WP:R#PLA--> reads:

<math display="block">\forall w_1,\ldots,w_n \, \exists B \, \forall x \, ( x \in B \Leftrightarrow \varphi(x, w_1, \ldots, w_n) )</math>

that is:
{{block indent|There exists a set {{mvar|B}} whose members are precisely those objects that satisfy the predicate {{mvar|φ}}.}}

This set {{mvar|B}} is again unique, and is usually denoted as {{math|{{{var|x}} : {{var|φ}}({{var|x}}, {{mvar|w}}{{sub|1}}, ..., {{var|w}}{{sub|{{mvar|b}}}})}.}}

In an unsorted material set theory, the axiom or rule of '''full''' or '''unrestricted comprehension''' says that for any property ''P'', there exists a set {''x'' | ''P''(''x'')} of all objects satisfying ''P.''<ref>{{Cite web |title=axiom of full comprehension in nLab |url=https://ncatlab.org/nlab/show/axiom+of+full+comprehension?t&utm_source=perplexity#definition |access-date=2024-11-07 |website=ncatlab.org |language=en}}</ref>

This axiom schema was tacitly used in the early days of [[naive set theory]], before a strict axiomatization was adopted. However, it was later discovered to lead directly to [[Russell's paradox]], by taking {{math|{{var|φ}}({{var|x}})}} to be {{math|¬({{var|x}}&nbsp;∈&nbsp;{{var|x}})}} (i.e., the property that set {{mvar|x}} is not a member of itself). Therefore, no useful [[axiomatization]] of set theory can use unrestricted comprehension. Passing from [[classical logic]] to [[intuitionistic logic]] does not help, as the proof of Russell's paradox is intuitionistically valid.

Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo–Fraenkel axioms (but not the [[axiom of extensionality]], the [[axiom of regularity]], or the [[axiom of choice]]) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification – each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i.e. it is a special case of the axiom schema of comprehension.

It is also possible to prevent the schema from being inconsistent by restricting which formulae it can be applied to, such as only [[Stratification (mathematics)|stratified]] formulae in [[New Foundations]] (see below) or only positive formulae (formulae with only conjunction, disjunction, quantification and atomic formulae) in [[positive set theory]]. Positive formulae, however, typically are unable to express certain things that most theories can; for instance, there is no [[Complement (set theory)|complement]] or relative complement in positive set theory.