## Final Activity Report Summary - GOEDELREALS (Godel Logics and Descriptive Properties of the Reals)

This project discussed the mathematical connections between descriptive properties of the reals and semantic properties of Gödel logics. We studied these relations in several different ways. On the one hand we progressed significantly in providing an intensional definition of Gödel logics, using well-quasi-orderings together with continuous bi-embeddability. On the other hand we obtained for the first time an exact correspondence result between logics determined by linear Kripke frames and Gödel logics. Other work in the area treated calculi for Gödel logics and quantified propositional variants and fragments of Gödel logics.

The most important result was for sure the one relating well-quasi-orderings with Gödel logics. In this article we extended Laver's result settling Fraisse's conjecture on the well-quasi-ordering of scattered linear orderings with bi-embeddability to the class of all countable closed linear orderings with respect to continuous monotone embeddability. We showed that this class formed a well-quasi-ordering and derived from this that there were only Aleph-1 many equivalence classes with respect to this embeddability relation. As a consequence, we could finally settle the total number of different Gödel logics to Aleph-0, i.e. countable many, which was a rather surprising result.

The other main result treated logics determined by countable linear Kripke frames. We gave a construction on how to obtain from one of these frames a truth value set such that the respective Gödel logic and the logic determined by this frame coincided, and vice-versa. In this way we could show that the class of logics determined by countable linear Kripke frames and the class of Gödel logics coincided. Furthermore, using the above result, we saw that this class contained only countable many logics.

These two results gave together an already more intensional definition of Gödel logics in the sense that only one Gödel logic was defined by one semantical structure. Although we made enormous progress in this direction, we still did not have a sufficient classification.

Another result of the fellowship was a direct proof of the completeness result for the hyper-sequent calculus HGIF, extending the use of the calculus to a much larger group of Gödel logics than before. This calculus was an extension of the famous sequent calculus of Gentzen, the basis of most techniques in automatic proof search.

Furthermore, we treated quantified propositional variants of Gödel logics. In first-order logic we dealt with objects and quantification was done over these objects. In quantified propositional logics we only quantified over propositions, i.e. over propositional variables, and not over external objects. These logics were very strong as they could express arbitrary properties of the underlying truth-value set. For one of the most important of these quantified propositional Gödel logics, we could obtain a quantifier elimination procedure using an extension of the underlying language.

Another obtained result was the characterisation of axiomatisability of various fragments of Gödel logics. These fragments were often used for specific applications as they sometimes exhibited useful properties and the full power of the first order language was not necessary.

The most important result was for sure the one relating well-quasi-orderings with Gödel logics. In this article we extended Laver's result settling Fraisse's conjecture on the well-quasi-ordering of scattered linear orderings with bi-embeddability to the class of all countable closed linear orderings with respect to continuous monotone embeddability. We showed that this class formed a well-quasi-ordering and derived from this that there were only Aleph-1 many equivalence classes with respect to this embeddability relation. As a consequence, we could finally settle the total number of different Gödel logics to Aleph-0, i.e. countable many, which was a rather surprising result.

The other main result treated logics determined by countable linear Kripke frames. We gave a construction on how to obtain from one of these frames a truth value set such that the respective Gödel logic and the logic determined by this frame coincided, and vice-versa. In this way we could show that the class of logics determined by countable linear Kripke frames and the class of Gödel logics coincided. Furthermore, using the above result, we saw that this class contained only countable many logics.

These two results gave together an already more intensional definition of Gödel logics in the sense that only one Gödel logic was defined by one semantical structure. Although we made enormous progress in this direction, we still did not have a sufficient classification.

Another result of the fellowship was a direct proof of the completeness result for the hyper-sequent calculus HGIF, extending the use of the calculus to a much larger group of Gödel logics than before. This calculus was an extension of the famous sequent calculus of Gentzen, the basis of most techniques in automatic proof search.

Furthermore, we treated quantified propositional variants of Gödel logics. In first-order logic we dealt with objects and quantification was done over these objects. In quantified propositional logics we only quantified over propositions, i.e. over propositional variables, and not over external objects. These logics were very strong as they could express arbitrary properties of the underlying truth-value set. For one of the most important of these quantified propositional Gödel logics, we could obtain a quantifier elimination procedure using an extension of the underlying language.

Another obtained result was the characterisation of axiomatisability of various fragments of Gödel logics. These fragments were often used for specific applications as they sometimes exhibited useful properties and the full power of the first order language was not necessary.