Note on generalizing theorems in algebraically closed fields

Baaz, Matthias, and Richard Zach. 1998. “Note on Generalizing Theorems in Algebraically Closed Fields.” Archive for Mathematical Logic 37 (5–6): 297–307. https://doi.org/10.1007/s001530050100.

The generalization properties of algebraically closed fields \(\mathit{ACF}_p\) of characteristic \(p > 0\) and \(\mathit{ACF}_0\) of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that \(\mathit{ACF}_p\) admits finite term bases, and \(\mathit{ACF}_0\) admits term bases with primality constraints. From these results the analogs of Kreisel’s Conjecture for these theories follow: If for some \(k\), \(A(1 + … + 1)\) (\(n\) 1’s) is provable in \(k\) steps, then \((\forall x)A(x)\) is provable.

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Labeled calculi and finite-valued logics

Baaz, Matthias, Christian G. Fermüller, Gernot Salzer, and Richard Zach. 1998. “Labeled Calculi and Finite-Valued Logics.” Studia Logica 61 (1): 7–33. https://doi.org/10.1023/A:1005022012721.

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite­-valued logic if the labels are interpreted as sets of truth values (sets­-as­-signs). Furthermore, it is shown that any finite­-valued logic can be given an axiomatization by such a labeled calculus using arbitrary “systems of signs,” i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.

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