Some properties of the upper semilattice of computable families of computably enumerable sets

2021 ◽  
Vol 60 (2) ◽  
pp. 195-209
Author(s):  
M. Kh. Faizrakhmanov
Keyword(s):  
Computably Enumerable Sets ◽  
Upper Semilattice ◽  
Computably Enumerable

scholarly journals Definable properties of the computably enumerable sets

1998 ◽  
Vol 94 (1-3) ◽  
pp. 97-125 ◽  
Author(s):  
Leo Harrington ◽  
Robert I. Soare
Keyword(s):  
Computably Enumerable Sets ◽  
Computably Enumerable

Signal Recovery Beyond Linear Estimates

2019 ◽  
pp. 386-446
Author(s):  
John Stillwell
Keyword(s):  
Peano Arithmetic ◽  
Computable Analysis ◽  
Signal Recovery ◽  
Church’S Thesis ◽  
Equivalent Definition ◽  
Computably Enumerable Sets ◽  
Natural Concept ◽  
Definition Of ◽  
Computably Enumerable

This chapter explains why Σ‎0 1 formulas of Peano arithmetic (PA) capture all computably enumerable sets, as claimed by Alonzo Church's thesis from the previous chapter. This allows us to capture “computable analysis” in the language of PA, since computable sets and functions are definable in terms of computable enumerability. To justify the claim that Σ‎0 1 = “computably enumerable,” this chapter makes a thorough analysis of the concept of computation. It takes a precise, but intuitively natural, concept of computation and translates it into the language of PA. The chapter demonstrates that the translation is indeed Σ‎0 1, but with a slightly different (though equivalent) definition of Σ‎0 1.

Definable incompleteness and Friedberg splittings

2002 ◽  
Vol 67 (2) ◽  
pp. 679-696
Author(s):  
Russell Miller
Keyword(s):  
Partial Order ◽  
Computably Enumerable Sets ◽  
Computably Enumerable

AbstractWe define a property R(A0, A1) in the partial order of computably enumerable sets under inclusion, and prove that R implies that A0 is noncomputable and incomplete. Moreover, the property is nonvacuous. and the A0 and A1 which we build satisfying R form a Friedberg splitting of their union A, with A1 prompt and A promptly simple. We conclude that A0 and A1 lie in distinct orbits under automorphisms of , yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit.

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