Source: Notre Dame J. Formal Logic
Volume 36, Number 3
We present a formalization of first-order predicate calculus with
equality which, unlike traditional systems with axiom schemata or substitution
rules, is finitely axiomatized in the sense that each step in a formal proof
admits only finitely many choices. This formalization is primarily based on
the inference rule of condensed detachment of Meredith. The usual
primitive notions of free variable and proper substitution are absent, making
it easy to verify proofs in a machine-oriented application. Completeness
results are presented. The example of Zermelo-Fraenkel set theory is shown to
be finitely axiomatized under the formalization. The relationship with
resolution-based theorem provers is briefly discussed. A closely related
axiomatization of traditional predicate calculus is shown to be complete in a
strong metamathematical sense.
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