Axiom schema of replacement | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-01-13

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Axiom_schema_of_replacement


00:00:50 1 Statement
00:06:34 2 Axiom schema of collection
00:12:32 3 Example applications
00:12:44 4 History
00:15:19 5 Impact
00:19:12 6 Relation to the axiom schema of separation
00:21:07 7 References



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SUMMARY
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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.