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Title: An investigation of logical systems incorporating Hilbert's ε-operator
Author: Leisenring, A. C.
ISNI:       0000 0004 2677 1241
Awarding Body: Royal Holloway University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1966
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Hilbert's ε-operator is a powerful logical instrument which can be used to simplify-the formulation of logical calculi and mathematical theories and to facilitate metamathematical investigations of the predicate calculus. The purpose of this thesis is to investigate the nature of the c-operator and to demonstrate the useful role it can play in metamathematica and in the formulation of mathematical theories. In chapter I a semantic interpretation is given to first-order languages containing the ε-symbol (ε-languages). By defining the notion of a logical closure, we establish an abstract model-theoretic property of E-languages which provides new proofs of the compactness theorems for e-languages and c-free languages. These proofs are model-theoretic in the sense that they do not depend on a particular choice of axioms and rules of inference for the languages concerned. Furthermore, this abstract property is used to establish the completeness of the ε-calculus which is defined in this chapter. In chapter II a new system of natural deduction is defined, and with the help of the ε-symbol the equivalence of this system and the predicate calculus is established. This result, which is analogous to Gentzen's Hauptsatz, provides new proofs of Hilbert's E-theorems and Herbrand's theorem. Also the ε-symbol is used to explain the conditions which must be imposed on the rule of existential instantiation in systems of natural deduction. In chapter IQ a proof is given of the eliminability of the c-symbol for E-calculi which contain the axiom schema Vx(A<->B) -> εxA = εxB. This result, which constitutes a strengthened form of Hilbert's second E-theorem, is then used to clarify the role which the ε-symbol plays in axiomatic set theory, particularly in connection with the axiom of choice.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available