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.

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.

Review: Yehuda Rav (Mathematical Reviews 2000a:03057)

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