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FINITE UNDECIDABILITY IN NIP FIELDS
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 04 October 2023, pp. 1-24
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A field K in a ring language
$\mathcal {L}$ is finitely undecidable if 
$\mbox {Cons}(T)$ is undecidable for every nonempty finite 
$T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, Chapter 3].
