Geodesic
A~property of recursively enumerable sets containing ``hardly deducible'' formulas
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 200-207

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There are considered recursively enumerable sets of formulas of the predicate calculus with the following property: for any sufficiently large $n$ there exists a formula in such a set, whose complexity of establishing of deducibility is maximal among all deducible formulas having the length $\leq n$. The article contains a low bound for some characteristic of density of such sets. 
@article{ZNSL_1971_20_a18,
     author = {A. O. Slisenko},
     title = {A~property of recursively enumerable sets containing ``hardly deducible'' formulas},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {200--207},
     publisher = {mathdoc},
     volume = {20},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a18/}
}
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A. O. Slisenko. A~property of recursively enumerable sets containing ``hardly deducible'' formulas. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 200-207. http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a18/