Editing Axiom schema of replacement (section)

=== Collection ===
[[File:Codomain2 A B.SVG|thumb|Axiom schema of collection: the image <math>f[A]</math> of the domain set <math>A</math> under the definable class function <math>f</math> falls inside a set <math>B</math>.]]
The '''axiom schema of collection''' is closely related to and frequently confused with the axiom schema of replacement. 
Over the remainder of the ZF axioms, it is equivalent to the axiom schema of replacement. The axiom of collection is stronger than replacement in the absence of the [[power set axiom]]<ref>{{cite arXiv|eprint=1110.2430 |last1=Gitman |first1=Victoria |author2=Joel David Hamkins |last3=Johnstone |first3=Thomas A. |title=What is the theory ZFC without power set? |date=2011 |class=math.LO }}</ref> or its [[constructive set theory|constructive counterpart of ZF]] and is used in the framework of IZF, which lacks the [[law of excluded middle]], instead of Replacement which is weaker.<ref>{{cite journal | last1 = Friedman | first1 = Harvey M | last2 = Ščedrov | first2 = Andrej | title = The lack of definable witnesses and provably recursive functions in intuitionistic set theories | journal = Advances in Mathematics | volume = 57 | issue = 1 | pages = 1–13 | year = 1985 | issn = 0001-8708 | doi = 10.1016/0001-8708(85)90103-3
 | url = https://www.sciencedirect.com/science/article/pii/0001870885901033 }}</ref>

While replacement can be read to say that the image of a function is a set, collection speaks about images of relations and then merely says that some [[superset|superclass]] of the relation's image is a set. 
In other words, the resulting set <math>B</math> has no minimality requirement, i.e. this variant also lacks the uniqueness requirement on <math>\phi</math>. 
That is, the relation defined by <math>\phi</math> is not required to be a function—some <math>x\in A</math> may correspond to many <math>y</math>'s in <math>B</math>. In this case, the image set <math>B</math> whose existence is asserted must contain at least one such <math>y</math> for each <math>x</math> in the original set, with no guarantee that it will contain only one.

Suppose that the free variables of <math>\phi</math> are among <math>w_1,\dotsc,w_n,x,y</math>; but neither <math>A</math> nor <math>B</math> is free in <math>\phi</math>. Then the axiom schema is:
:<math>
\forall w_1,\ldots,w_n \,[(\forall x\, \exists\, y \phi(x, y, w_1, \ldots, w_n)) \Rightarrow \forall A\, \exists B\, \forall x \in A\, \exists y \in B\, \phi(x, y, w_1, \ldots, w_n)]
</math>

The axiom schema is sometimes stated without prior restrictions (apart from <math>B</math> not occurring free in <math>\phi</math>) on the predicate, <math>\phi</math>:
:<math>
\forall w_1,\ldots,w_n \, \forall A\, \exists B\,\forall x \in A\, [ \exists y \phi(x, y, w_1, \ldots, w_n) \Rightarrow \exists y \in B\,\phi(x, y, w_1, \ldots, w_n)]
</math>

In this case, there may be elements <math>x</math> in <math>A</math> that are not associated to any other sets by <math>\phi</math>. However, the axiom schema as stated requires that, if an element <math>x</math> of <math>A</math> is associated with at least one set <math>y</math>, then the image set <math>B</math> will contain at least one such <math>y</math>. The resulting axiom schema is also called the '''axiom schema of boundedness'''.