Problems in Set Theory, Mathematical Logic and the Theory of Algorithms

Substantiation of the structure theory of design of technological machines and devices

“…The search for regularities in the structure of designs of machines and devices can be rationally carried out using set theory, the calculation of predicates of the first order, the theory of groups [3][4][5][6]. This makes it possible to create objective quantitative criteria for an a priori assess ment of the manufacturability of assembly, maintainability, unification, functional saturation of designs, contributing to their improvement.…”

“…When constructing a theory, it is necessary to mini mize its initial assumptions, that is, to strive for a smaller number of axioms and basic concepts [5]. All parts of the theory must satisfy the requirements below.…”

Substatiation of quantitative criteria of structural parts and units manufacturability evaluation

“…where K -any assembly unit. Let's carry out the proof by induction [18][19][20][21][22]. Take the structure of the assembly, consisting, for example, of the base, sheet metal part and fastening.…”

How to release Frege's system from Russell's antinomy

“…either as axiom ([10] 27), or theorem ([15] 141). Any derivation of ( * ) in ZF requires necessarily both statements, so that we adopted directly (E 1 ) ( [17] 109). As in Frege's Grundgesetze, see section (2), ( * ) follows easily from (E 2 ), whereas the derivation of ( * ) from (E 1 ) is bound to the methodological concerns of ZF.…”