# Set-theoretic Definability of Constructions - Progress Report on a Disgracefully Old Problem

## Wilfrid Hodges

Queen Mary, London

In 1976 I asked for a proof that there is no formula of the language of ZFC which, provably in every model of ZFC, defines for each field an algebraic closure of that field. After many failed attempts Shelah and I put together a proof in 2001. The field-theoretic part (basically Galois theory) appeared in the Proceedings of the Oviedo DLMPS. The set-theoretic part (due to Shelah) is not yet in definitive form; it proves the result but something cleaner and stronger should also be true.

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