In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, the axiom schema of replacement is a
schema
The word schema comes from the Greek word ('), which means ''shape'', or more generally, ''plan''. The plural is ('). In English, both ''schemas'' and ''schemata'' are used as plural forms.
Schema may refer to:
Science and technology
* SCHEMA ...
of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s in
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
(ZF) that asserts that the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of any
set under any definable
mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.
The axiom schema is motivated by the idea that whether a
class is a set depends only on the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the class, not on the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of its elements. Thus, if one class is "small enough" to be a set, and there is a
surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
from that class to a second class, the axiom states that the second class is also a set. However, because
ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining
formulas.
Statement
Suppose
is a definable binary
relation (which may be a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
) such that for every set
there is a unique set
such that
holds. There is a corresponding definable function
, where
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
. Consider the (possibly proper) class
defined such that for every set
,
if and only if there is an
with
.
is called the image of
under
, and denoted