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### Page 459 of Mathematical Reviews Vol. , Issue 84b [page]

1984 Mathematical Reviews
If FS,(7) and US,(7T) are recursively enumerable and T is p-inseparable then PC (a version of predicate calculus) is p-reducible to T. The results of part I are valid for p-inseparability.  ...  Otherwise, by the minimality of B, 0( B) and B have the same partial degree, hence there is an r.e. enumeration operator taking @(B) to B.  ...

### Page 578 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews
It is well known that effective inseparability of pairs of sets is an effective version of recursive inseparability of pairs of sets.  ...  A recursively enumerable set A is called a diagonal if there is a partial recursive function v such that {Ax.v(i,x)}icw is the set of all partial recursive functions of one variable and A = {ne w: u(n,  ...

### Page 1701 of Mathematical Reviews Vol. , Issue 87d [page]

1987 Mathematical Reviews
Hence the corollary: There is a minimal pair of recursively inseparable r.e. sets. Bai Ying Shen (Nanjing) Ishmukhametov, Sh. T. Differences of recursively enumerable sets. (Russian) Izv. Vyssh.  ...  Then there are two r.e. sets Ag and A, such that Ag UA, C B, AgN A; = ©, and Ap, A; are recursively inseparable and form a minimal pair.  ...

### Page 700 of Mathematical Reviews Vol. 23A, Issue 6A [page]

1962 Mathematical Reviews
Let w1, we, --- be the standard Post-Kleene enumeration of all recursively enumerable (r.e.) sets.  ...  A pair (a, 8) of sets of natural numbers is called effectively inseparable (E.I.) when there is a recursive function 4(z,y) such that if «4, wj are, respectively, disjoint r.e. supersets of a, 8, then  ...

### Page 633 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews
Theorem 2: For any high r.e. set C and nonrecursive r.e. set D, there exists a pair of recursively inseparable r.e. sets A, B such that A is high and B is low with C <7 A and @ <7 B<7D.  ...  Theorem 3: There exist r.e. sets A, B, C, D such that (1) A, B are recursively inseparable sets and form a minimal pair; (2) A <7 C, B <1 D; (3) A| B, A| D, C| B, C | D; (4) A, B are low; (5) C, D are  ...

### Page 6154 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews
The proof proceeds in two steps: They first construct an w-sequence of nonbounding r.e. degrees that pairwise form minimal pairs, combining the Lachlan- Yates construction of a minimal pair and Lachlan  ...  (NZ-VCTR) Recursively enumerable m- and tt-degrees. II. The distribution of singular degrees. Arch. Math. Logic 27 (1988), no. 2, 135-147.  ...

### Some properties of pseudo-complements of recursively enumerable sets

Robert A. Di Paola
1966 Transactions of the American Mathematical Society
recursively enumerable set.  ...  Thus, we are led to consider those sets of integers whose members are probably (in a system T) in the complement of a given recursively enumerable set S, or, as we shall call them, the pseudo-complements  ...  PSEUDO-COMPLEMENTS OF RECURSIVELY ENUMERABLE SETS the same re set. This can be expressed as the decision problem of a set, and the 1-degree of unsolvability of this set is computed.  ...

### Some Properties of Pseudo-Complements of Recursively Enumerable Sets

Robert A. Di Paola
1966 Transactions of the American Mathematical Society
recursively enumerable set.  ...  Thus, we are led to consider those sets of integers whose members are probably (in a system T) in the complement of a given recursively enumerable set S, or, as we shall call them, the pseudo-complements  ...  PSEUDO-COMPLEMENTS OF RECURSIVELY ENUMERABLE SETS the same re set. This can be expressed as the decision problem of a set, and the 1-degree of unsolvability of this set is computed.  ...

### Finding the limit of incompleteness I [article]

Yong Cheng
2020 arXiv   pre-print
We prove that for each recursively inseparable pair 〈 A,B〉, we can construct a r.e. theory U_〈 A,B〉 such that U_〈 A,B〉 is weaker than R w.r.t. interpretation and G1 holds for U_〈 A,B〉.  ...  As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which G1 holds.  ...  If A is recursively enumerable and not recursive, there is a recursively inseparable pair B, C such that A, B and C have the same Turing degree.  ...

### Page 6086 of Mathematical Reviews Vol. , Issue 87k [page]

1987 Mathematical Reviews
Consequently, there are r.e. degrees a containing no A of great- est or least w-degree. The next result of the paper concerns pairs of intervals of r.e. T- and w-degrees.  ...  (SGP-SING); Stob, M. (1-CLVN) Structural interactions of the recursively enumerable T- and w-degrees. Special issue: second Southeast Asian logic conference (Bangkok, 1984). Ann. Pure Appl.  ...

### Page 1768 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews
Marius Zimand (1-RCT-C) 93d:03047 03D25 03D30 Lerman, Manuel (1-CT) Minimal degrees and recursively inseparable pairs of recursively enumerable sets. Z. Math. Logik Grundlag.  ...  As Solovay observed, the degrees of such models are those that separate r.e. effectively inseparable sets. A.  ...

### Strong reducibilities

Piergiorgio Odifreddi
1981 Bulletin of the American Mathematical Society
If for some set B> A and B are recursively inseparable, we say that A is part of an r.i. pair. In most cases such as Theorems 2.2 and 2.4 A and B will be r.e. E A with ƒ recursive.  ...  If A is retraceable and immune, the m-degrees of A and A form a minimal pair. PROOF.  ...

### Page 4202 of Mathematical Reviews Vol. , Issue 80K [page]

1980 Mathematical Reviews
Finally, the author defines a notion of effectively negatively defined properties of r.e. sets and gives as examples the complete creative sets and the effectively inseparable pairs of sets.  ...  This paper is a survey of some ideas and results from three areas of the study of the class P of recursively enumerable sets.  ...

### Recursion-Theoretic Ranking and Compression [article]

Lane A. Hemaspaandra, Daniel Rubery
2017 arXiv   pre-print
We show that the rankable and compressible sets broadly populate the 1-truth-table degrees, and we prove that every nonempty coRE cylinder is recursively compressible.  ...  The present paper defines and studies the recursion-theoretic versions of compression and ranking functions, and in particular studies the question of which sets have, or lack, such functions.  ...  Let A and B be a pair of disjoint, r.e. sets that are recursively inseparable. It is well-known that such pairs exist, e.g., {i | M i (i) halts and outputs 1} and {i | M i (i) halts and outputs 2}.  ...

### The Recursively Enumerable Degrees [chapter]

Richard A. Shore
1999 Studies in Logic and the Foundations of Mathematics
(Recursive inseparability means that there is no recursive set separating the set of theorems of the theory of partial orderings from set of sentences of the language of partial orderings that are false  ...  than that of the recursive sets, 0, and 0 , the degree of K?  ...
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