On Splits of Computably Enumerable Sets

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.

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Definable incompleteness and Friedberg splittings

Journal of Symbolic Logic ◽

10.2178/jsl/1190150104 ◽

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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Computably enumerable sets and related issues

Algebra and Logic ◽

10.1007/s10469-012-9161-1 ◽

2012 ◽

Vol 50 (6) ◽

pp. 494-511

Author(s):

I. A. Lavrov

Keyword(s):

Computably Enumerable Sets ◽

Computably Enumerable

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Orbits of computably enumerable sets: low sets can avoid an upper cone

Annals of Pure and Applied Logic ◽

10.1016/s0168-0072(01)00119-1 ◽

2002 ◽

Vol 118 (1-2) ◽

pp. 61-85 ◽

Cited By ~ 1

Author(s):

Russell Miller

Keyword(s):

Computably Enumerable Sets ◽

Computably Enumerable

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Definable Encodings in the Computably Enumerable Sets

Bulletin of Symbolic Logic ◽

10.2307/421206 ◽

2000 ◽

Vol 6 (2) ◽

pp. 185-196 ◽

Cited By ~ 5

Author(s):

Peter A. Cholak ◽

Leo A. Harrington

Keyword(s):

Computability Theory ◽

Finite Sets ◽

Warm Up ◽

Computably Enumerable Sets ◽

Disjoint Sets ◽

New Form ◽

Historical Remarks ◽

Equivalent Class ◽

The Ideal ◽

Computably Enumerable

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an infinite non-computable subset). We use A* to denote the equivalent class of A under the ideal of finite sets.

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