A note on Russell’s paradox is that the problem with the Russell set isn’t that it’s nonconstructive, but rather the problem is that we allowed too much freedom in asserting for every property, there is a set of things that satisfy the property, and the conventional way it’s solved is by instead dropping the axiom of unrestricted comprehension, and adding the axiom of specification as well as a couple of other axioms to ensure that we have the sets we need.
Even without Russell’s paradox, you can still prove things nonconstructively.
A note on Russell’s paradox is that the problem with the Russell set isn’t that it’s nonconstructive, but rather the problem is that we allowed too much freedom in asserting for every property, there is a set of things that satisfy the property, and the conventional way it’s solved is by instead dropping the axiom of unrestricted comprehension, and adding the axiom of specification as well as a couple of other axioms to ensure that we have the sets we need.
Even without Russell’s paradox, you can still prove things nonconstructively.