"Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all real numbers <https://en.wikipedia.org/wiki/Real_number>, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀ *x* ∃*y* (*x* + *y* = 0) but one needs second-order logic to assert the least-upper-bound <https://en.wikipedia.org/wiki/Supremum> property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum <https://en.wikipedia.org/wiki/Supremum>." Wikipedia 'Second-Order Logic'.
I insist. That kind of sentence is wrong. In FOL + ZFC we can express the least-upper-bound property for real numbers. Maybe in FOL+ Real if Real is a system of axioms of real numbers we cannot. But at last we must stop giving this kind of example without specifying the axiomatic added to FOL. -- FL -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/9edf0428-a73a-43de-8fa9-87437494642b%40googlegroups.com.
