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powerset of the domain, to actually prove that this is a function would require some possibly tricky uses of the ZFC axioms), however I would much rather change 'naive set theory' to just 'set theory' because by this article's reasoning we should say 'naive set theory' whenever we aren't rigorously applying ZFC axioms, for example with images, preimages, indexed families, set operations, etc. etc.
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352:. The preimage is a separate thing, a generalization of the fiber to whole subsets of the codomain, instead of single elements at a time. We use the same notation for it, but definitions are the place where clarity is most valued. We don't need a notational allusion to a related concept. We need a simple function
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Is there really any need to say that fibers are a notion of naive set theory and not just set theory? I understand that we are not formally proving that the fiber of an element is a set using the ZFC axioms (and similarly if we want to consider the 'fiber function' mapping from the codomain to the
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as the definition for the fiber. I think this conflates the concept of "fiber" with "preimage". We are trying to express, in symbols, "the fiber of a function over a value in its codomain is the set of all inputs in the domain that map to that output", and that translates to
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The lead implies that there are two topics here. I tried to split the article but that wasn't appreciated :-( Sorry, but anyway, can the lead be re-written or expanded in a way that explains this as a single topic?
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Knowledge. If you would like to participate, please visit the project page, where you can join
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generalizes the concept of a fiber to whole subsets of the codomain
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750:{\displaystyle f^{-1}:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}
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400:{\displaystyle f^{-1}:Y\to {\mathcal {P}}(X)}
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