Abstract
The first method is based on the familiar method of lowering thinnings downwards and is a further development of the lemmas on “weeding” of [1]. The second method is based on the use of sufficiently wide classes of sequents, for which derivability in the intuitionistic predicate calculus coincides with derivability in the classical predicate calculus and the familiar property of disjunction is true. By this method one can get, e.g., a syntactic proof of the following assertion. If the positive formula A is derivable in the theory of groups under additional assumptions of the form
then A is also derivable in the theory of groups without these assumptions. As the third method there is proposed a syntactically formulated test for the conservativeness of extensions of intuitionistic axiomatic theories. With the help of this test one can get, for example, a syntactic proof of the hereditary undecidability of the intuitionistic theory of equality, with additional axioms which are the formula
, all formulas of the form
and all negations of formulas derivable in the classical predicate calculus.
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 163–175, 1979.
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Orevkov, V.P. Three ways of recognizing inessential formulas in sequents. J Math Sci 20, 2351–2357 (1982). https://doi.org/10.1007/BF01629445
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DOI: https://doi.org/10.1007/BF01629445