We prove that there exists a computable—and hence continuous,-function F(v,x) defined α a rectangle R of the plane such that the differential equation x′=Fx,v has no computable solution of any neighborhood within R. As an immediate corollary, we obtain from the form of the above differential equation a computable transformation with no computable fixed point.
Several problems in recursion theory on admissible ordinals (α-recursion theory) and recursion theory of inadmissible ordinals (β-recursion theory) are studied. Fruitful interactions between both theories are stressed. In the first part of the admissible collapse is used in order to characterize for some inadmissible β the structure of all β-recursively enumerable degrees as an accumulation of structures of -recursively enumerable degrees for many admissible structures . Thus problems about the β-recursively enumerable degrees can be solved by considering “locally” the analogous problem in an admissible (where results of α-recursion theory apply). In the second part β-recursion theory is used as a tool in infinite injury priority constructions for some particularly interesting α (e.g. ω1CK). New effects can be observed since some structure of the inadmissible world above O′ is projected into the α-recursively enumerable degrees by inverting the jump. The gained understanding of the jump of α-recursively enumerable degrees makes it possible to solve some open problems.
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