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Thanks. I dont understand why for all elements of a set is first order and for all functions is second order - since functions (lets say R->R) are elements of t

Thanks. I dont understand why for all elements of a set is first order and for all functions is second order - since functions (lets say R->R) are elements of the powerset of the cartesian product of RxR. So functions in math are usually part of first order logic, no?

yaantc 6y ago
As said by @dwohnitmok it depends on the domain you consider. If you have a proposition quantifying over objects and functions operating on said objects, it's second order. But if you have a proposition considering only said functions as objects, it is first order (with a different domain: not the base objects, but functions over those objects). And again: you can ignore such subtleties for most of mathematics.