You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers.

Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?

The problem with such sleight of hand counterargument is that you haven't even defined what "a thing" is nor "all things" are in this world. And such discussions will just come back to set theory, ZFC, axiom of choice and real numbers.

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