What results are known about the construction of models for a theory $T$ of the logic BL$\forall$ for languages of higher cardinality? The construction for the countable case relies on

1) The fact that you can form sentences of arbitrary finite length.

2) The fact that there is an undecidable statement in $T_{n}\supset{T}$ that helps you to determine an undecidable statement for $T_{n+1}$.

This leads to a Henkin theory $T'=\bigcup_{n=0}^{\infty}T_{n}$. But this procedure will fail at limit ordinals.

This is somewhat related to: Compactness and completeness in Gödel logic but I'm looking for a more broader set of results (or if they exist) and some references.


I'm definitely not an expert, but you could be interested in A Henkin-style proof of completeness for first-order algebraizable logics by Cintula and Noguera.


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