# On Imbedding Operators

• G. E. Mints
• V. P. Orevkov
Part of the Seminars in Mathematics book series (SM, volume 4)

## Abstract

Systems are understood to be logical or mathematical logic calculi based on predicate calculus (classical or constructive). An operator 9 revising a formula of the system S1 into a formula of the system S2, is called an imbedding operator from S1 into S2 (see [1]) if for each formula A of the system S1 the deducibility of the formula A in S1 is equivalent to the deducibility of φ(A)in S2 If every formula of the system S1 is a formula of the system S2 then the imbedding operator φ from S1 into S2 is called regular (see [1]) if for any formula A of the system S1 the formula A ⊃ φ(A) is deducible in S2 (when all the formulas deducible in S2 are true in a certain interpretation, there results from the regularity of the imbedding operator that it transforms a true formula into a true formula).

## Keywords

Predicate Calculus Recursive Function Axiom Schema Positive Entry Arithmetic Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Literature Cited

1. 1.
Shanin, N. A., “On some logical problems of arithmetic,” Trudy Steklov Mat. Inst. Akad. Nauk SSSR, Vol. 43, 1955.Google Scholar
2. 2.
Mal’tsev, A. I., Algorithms and Recursive Functions (in Russian), Moscow, 1965.Google Scholar
3. 3.
Harrop, R., “Concerning formulas of the types A → B V C, A→Ǝ x B (x) in intuitionistic formal systems,” J. Symb. Logic, 25(l):27–32 (1960).Google Scholar
4. 4.
Mints, G. E. and Orevkov, V. P., “Extension of a theorem of V. I. Glivenko and G. Kreisel to a class of formulas of predicate calculus,” Doklady Akad. Nauk SSSR, 152(3):553–554 (1963).Google Scholar