∃G (G ↔ ~Provable(G))

“there exists a proposition that is materially equivalent to a statement of its own unprovability.”

If G was Provable this contradicts its assertion: G is not Provable.

If ~G was Provable this contradicts its assertion: G is Provable

Therefore G is neither Provable nor Refutable and does not exist.

2019-07-16 update

If G was false then it would be ~Provable (F ↔ T) is false

If G was true then it would be ~~Provable (T ↔ F) is false

∴ ~∃G (G ↔ ~Provable(G))

Copyright 2018 Pete Olcott