Concept
Axiom schema of replacement
Overview
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image 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 of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection 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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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In respect to any complex eternal object, _S(D1, D2, ... Dn)_, the eternal objects _D1, ... Dn_, whose individual essences are constitutive of the individual essence of _S(D1, ... Dn)_, are called the components of _S(D1, ... Dn)_. It is obvious that the grade of complexity to be ascribed to _S(D1, ... Dn)_ is to be taken as one above the highest grade of complexity to be found among its components.
Alfred North Whitehead, Science and the modern world
States what Axiom schema of replacement is.
These two are sufficient; for the corresponding propositions respecting unequals may be proved from these by a _reductio ad absurdum_. These axioms, and likewise the so-called definitions, are, as has already been said, results of induction; true of all objects whatever, and, as it may seem, exactly true, without the hypothetical assumption of unqualified truth where an approximation to it is all that exists. The…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Axiom schema of replacement is.
This complete quantity of the extent in relation to such a condition is called universality (universalitas). To this corresponds totality (universitas) of conditions in the synthesis of intuitions. The transcendental conception of reason is therefore nothing else than the conception of the totality of the conditions of a given conditioned. Now as the unconditioned alone renders possible totality of conditions, and…
Immanuel Kant, The Critique of Pure Reason
States what Axiom schema of replacement is.
Related ideas
- First-order logic
- Journal of Symbolic Logic
- Georg Cantor
- Skolem's paradox
- If and only if
- Zermelo–Fraenkel set theory
- John von Neumann
- Law of excluded middle
- Ernst Zermelo
- Jena
- Axiom of infinity
- Set (mathematics)
- Set theory
- Zermelo set theory
- Second-order arithmetic
- Association for Symbolic Logic
- Type theory
- Well-formed formula
- Axiom of extensionality
- Abraham Fraenkel
- Gödel's incompleteness theorems
- Cardinal number
- Cardinality
- Consistency
Place in the tradition
Part of Logicism.
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