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Axiom schema of specification - Wikipedia, the free encyclopedia
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification , axiom schema of separation , subset axiom scheme or axiom s...
en.wikipedia.org/wiki/Axiom_schema_of_specification |
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Axiom schema of predicative separation - Wikipedia, the free encyclopedia
In axiomatic set theory, the axiom schema of predicative separation , or of restricted , or Δ 0 separation, is a schema of axioms which is a restriction of the usual axiom schema of separatio...
en.wikipedia.org/wiki/Axiom_schema_of_predicative_separ... en.wikipedia.org/wiki/Axiom_schema_of_predicative_separation |
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Axiom schema of replacement - Wikipedia, the free encyclopedia
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is nece...
en.wikipedia.org/wiki/Axiom_schema_of_replacement |
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The Axiom Schema of Separation is an axiom schema of Zermelo-Fraenkel set theory. ... The Axiom Schema of Separation implies that $\phi$ may depend on more than one parameter $p$ . ... Another consequence of the Axiom Schema of Separation is that a subclass of any set is a set.
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Aug 5, 2009 ... In set theory, the axiom schema of replacement is a schema of axioms in .... instance of the axiom schema of separation is the empty set. ...
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Axiom schema of replacement. Axiom schema of restricted comprehension, Axiom schema of separation. Axiom schema of specification ...
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schema of separation. The pre-critical Axiom of Abstraction permits us to form ... Now, symbolically, Zermelo's Axiom schema of separation is ...
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But in this case, the set ''B'' required for the axiom of separation is the Empty Set , so the axiom of separation follows from the axiom of replacement together with the Axiom Of Empty Set . For this reason, the axiom schema of separation is often left out of modern lists of the Zermelo-Fraenkel axioms.
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This set S may contain more than just the natural numbers, forming a subset of it, but we may apply the axiom schema of specificationSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification or axiom schema of separation or axiom schema...
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[edit] Axiom schema of separation ... In the discussion of the axiom of separation, it seems awkward to me that the predicate is described as ranging over the members of A; a naiive reader might miss that, or think that it's some special kind of predicate.
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