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Classification of Sets [set theory]
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[math] classes = predicate scopes
aka. sets in the wider, naive sense
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«A class [in mathematics] is a collection of sets (or sometimes other mathematical
objects) that can be unambiguously defined by a property that all its members share»
[wiki].
In light of the question "property = predicate?"
[^],
mathematical classes to me seem to be more precisely
the collections of entities specified by a predicate over entities,
ie., the extensions/scopes of unary predicates.
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| | sets in the strict, mathematical sense
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The most commonly used classes are sets.
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Instances:
N, Z, Q, R, C
(sets of numbers)
 | extension-of/
scope-of
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In the ZFC context, classes exist only in the metalanguage as
equivalence classes of logical formulas.
If a paradox of naive set theory applies to a class,
this is proof that it is a proper class.
Eg. the class of all sets not containing themselves (Russell's paradox);
the class of all ordinal numbers[^] (Burali-Forti paradox).
«Several objects in mathematics are too big for sets
and need to be described with classes,
for instance large categories or the class-field of surreal numbers.»
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