Is the theory of real closed fields expanded by restricted analytic functions decidable? I have been doing a lot of reading on the subject, but I can't quite find a straight answer on this one. The theory seems to enjoy many good properties, o-minimality, completeness, quantifier elimination. But I can't seem to find anything about it being decidable or undecidable. I think it needs to be recursively axiomatized in order for it to be decidable, but I am not quite sure. Does that mean that real closed fields augmented with a specified list of restricted analytic functions is decidable?
2026-04-05 00:05:07.1775347507
Is the theory of real closed fields expanded by restricted analytic functions decidable?
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As Levon Haykazyan pointed out in a comment, since there are uncountably many analytic functions, the language is uncountable. In order to discuss whether a theory is decidable, the appropriate definitions require that the language in question be presented computably: that is, the function and relation symbols are given by natural numbers, and the arity function (telling how many arguments a symbol takes) must be a computable function. Since this presentation requires us to associate symbols with natural numbers, it doesn't make sense for uncountable languages.