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A presemiological structure on a nonempty denumerable set S is a structure given by a set As of functions on set S and a composition os with respect to As, satisfying the following axioms: (I) S contains an element Φ8 or briefly Φ, called the root of S. (II) There exists a nonempty denumerable set As, called the successor set of 5, consisting of distinct functions σ: -5 -> 5, called successor functions of £, satisfying σ'(Φ) Φ Φ (σ* € As). (III) There exists a composition a os τ: S->S(#, τ ζ As), briefly denoted s σ ο τ, called the successor composition of /S, satisfying σ ο τ $ As(a, τ € As) and associativity, with values given by σ οτ(χ) = σ(τ(χγ) (σ, τ € Α8). (IV) Every nonroot χ ζ. S is representable, not necessarily uniquely, s

On the Goldbach conjecture and the consistency of general recursive arithemetic.