The lattice of bi-numerations of arithmetic. II.
Marie Hájková · Czech digital mathematics library · 1971
This paper is a direct continuation of our 16].The knowledge of [6] is presupposed.Similarly as in [6J, in the whole paper A - denotes a fixed axiomatic theory with the following properties: (1) A is a primitive recursive set, (2) JL is consistent, (3) ?c Jl ( P is the Peano's arithmetic).Numbering of definitions and theorems in this paper begins with 3.1; references like 2*24 or 1.18 refer to definitions and theorems from [61.III.Reducibility; a non-describability theorem We shall now study the problem of reducibility of elements of C'&t/rv.]-We recall the definition: 3*1 • Definition.An element r of a lattice M = =• is irreducible if, for each x , /^ 6 At , x u ^ .=# implies x -* z, or /y, as 3*2.Theorem.Let A .be reflexive, let f f ft €. € Btyns and suppose / **c fi> .Then there is a AMS, Primary 02D99 Ref.2.2.664 Secondary --281 - vn> such that cc' Coup A Cjm r ) -> V y (y) , (2) bt-A (^ CqrVfl A Cgn^) -* Vf (y) .Proof of Lemma 3.3* Let cT & 3JJTV satisfy the conditions (# ).It suffices to put EocJ-* C'yJ r\ I cTl -282 and y(y.) ** T/cfj.(0 & 4, ty>) .Conversely, let *y (/y,) and ot e 3-Orv satisfy the conditions (1) and (2).Put <r(x)»*oc(x)v?fin ( f' ) (x) A V (nr(<*, )A Br,4 C0# 4, ^J).