Abstract
The \(\varepsilon \)-elimination method of Hilbert’s \(\varepsilon \)-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s \(\varepsilon \)-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first \(\varepsilon \)-theorem, even if the formalisation incorporates so-called \(\varepsilon \)-equality axioms.
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Notes
More precisely on Page 21 of [3] it is stated that “If equality is present, however, the maximal rank of critical formulas will also play a role.”.
Kindly see https://en.wikipedia.org/wiki/Drinker_paradox.
A term is fully indicated if every occurrence of the term is obtained by a replacement [27, Definition 1.6].
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Miyamoto, K., Moser, G. Herbrand complexity and the epsilon calculus with equality. Arch. Math. Logic 63, 89–118 (2024). https://doi.org/10.1007/s00153-023-00877-3
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DOI: https://doi.org/10.1007/s00153-023-00877-3