Logic Journal of the IGPL 13 (4):435-441 (2005)

Abstract
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 . We show that this set is even non-arithmetical and 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
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DOI 10.1093/jigpal/jzi033
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First-Order Gödel Logics.Richard Zach, Matthias Baaz & Norbert Preining - 2007 - Annals of Pure and Applied Logic 147 (1):23-47.
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On Witnessed Models in Fuzzy Logic III - Witnessed Gödel Logics.Petr Häjek - 2010 - Mathematical Logic Quarterly 56 (2):171-174.

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