A non-arithmetical Gödel logic
Petr Hájek · Logic Journal of IGPL · 2005
The logic in question is G↓ – Gödel predicate logic with the set of truth values being V↓ = {1/n | n = 1, 2, …} ∪ {0}. It is shown in [1] that the set of its tautologies is not recursively axiomatizable (is Π1-hard). We show that this set is even non-arithmetical and (before this) we prove the set of satisfiable formulas of G↓ to be non-arithmetical. In the last section we show that another important Gödel logic G↑ is arithmetical, more precisely, its set of tautologies is Π2-complete.