Noninitial segments of the α-degrees

1973 ◽  
Vol 38 (3) ◽  
pp. 368-388 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Recursion Theory ◽  
Partial Ordering ◽  
First Order ◽  
Order Language ◽  
Suitable Notion ◽  
The Universe ◽  
Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

On the T-degrees of partial functions

1985 ◽  
Vol 50 (3) ◽  
pp. 580-588 ◽  
Author(s):  
Paolo Casalegno
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Order Theory ◽  
Partial Functions ◽  
First Order ◽  
First Order Theory ◽  
Degrees Of Unsolvability

AbstractLet 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

Almost strongly minimal theories. I

1972 ◽  
Vol 37 (3) ◽  
pp. 487-493 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Closure ◽  
Natural Extension ◽  
Stone Space ◽  
First Order ◽  
Order Language ◽  
Simple Class ◽  
The Universe ◽  
Natural Expansion ◽  
Minimal Formula

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

An axiomatic system for the first order language with an equi-cardinality quantifier

1966 ◽  
Vol 31 (4) ◽  
pp. 633-640 ◽  
Author(s):  
Mitsuru Yasuhara
Keyword(s):  
Rational Numbers ◽  
Axiomatic System ◽  
Universal Quantifier ◽  
Real Numbers ◽  
Relational System ◽  
First Order ◽  
Order Language ◽  
Relational Systems ◽  
Finite Domains ◽  
The Universe

The equi-cardinality quantifier1 to be used in this article, written as Qx, is characterised by the following semantical rule: A formula QxA(x) is true in a relational system exactly when the cardinality of the set consisting of these elements which make A(x) true is the same as that of the universe. For instance, QxN(x) is true in 〈Rt, N〉 but false in 〈Rl, N〉 where Rt, Rl, and N are the sets of rational numbers, real numbers, and natural numbers, respectively. We notice that in finite domains the equi-cardinality quantifier is the same as the universal quantifier. For this reason, all relational systems considered in the following are assumed infinite.

On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model

1975 ◽  
Vol 40 (2) ◽  
pp. 186-196 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Model Problem ◽  
Recursive Algorithm ◽  
Finite Model ◽  
Positive Direction ◽  
First Order ◽  
Sharp Form ◽  
Universal Sentence ◽  
Order Language ◽  
Finite Set ◽  
The Universe

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.

A Note on the Realization of Types

1980 ◽  
Vol 23 (1) ◽  
pp. 95-98
Author(s):  
Alan Adamson
Keyword(s):  
Complete Theory ◽  
First Order ◽  
Order Language ◽  
The Universe

Let L be a countable first-order language and T a fixed complete theory in L. If is a model of T, is an n-sequence of variables, and ā=〈a1,…, an〉 is an n-sequence of elements of M, the universe of , we let where ranges over formulas of L containing freely at most the variables υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

On initial segments of hyperdegrees

1970 ◽  
Vol 35 (2) ◽  
pp. 189-197 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Finite Distributive Lattice ◽  
Order Types

An initial segment of hyperdegrees is a set S of hyperdegrees such that whenever h ∈ S and k ≦ h then k ∈ S. The main results of this paper affirm the existence of initial segments having certain order types. In particular, if L is a finite distributive lattice then L is isomorphic to an initial segment of hyperdegrees [Theorem 1]; as a consequence the elementary theory of the ordering of hyperdegrees is recursively undecidable [Corollary 1].

A form of feasible interpolation for constant depth Frege systems

2010 ◽  
Vol 75 (2) ◽  
pp. 774-784 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Constant Depth ◽  
First Order ◽  
Order Language ◽  
Frege Systems ◽  
Image Position ◽  
The Universe ◽  
Dnf Formula

AbstractLet L be a first-order language and Φ and Ψ two L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n, m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.For any language L containing only constants and unary predicates we show that there is a constant CL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL . n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size 2log(S)O(1) and with bottom fan-in log(S)O(1).

scholarly journals The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

2005 ◽  
Vol 11 (3) ◽  
pp. 398-410
Author(s):  
Noam Greenberg
Keyword(s):  
Partially Ordered Sets ◽  
Recursion Theory ◽  
Ordered Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Order Language ◽  
Replacement Scheme ◽  
Admissible Ordinals ◽  
Models Of Arithmetic

AbstractWhen attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal α, the α-r.e. degrees and the classical r.e. degrees are not elementarily equivalent.

Stratification and cut-elimination

1991 ◽  
Vol 56 (1) ◽  
pp. 213-226 ◽  
Author(s):  
Marcel Crabbé
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Cut Elimination ◽  
Simple Type ◽  
First Order ◽  
Similar Reason ◽  
Separation Axioms ◽  
Simple Type Theory ◽  
Order Language ◽  
The Universe

In this paper, we show the normalization of proofs of NF (Quine's New Foundations; see [15]) minus extensionality. This system, called SF (Stratified Foundations) differs in many respects from the associated system of simple type theory. It is written in a first order language and not in a multi-sorted one, and the formulas need not be stratifiable, except in the instances of the comprehension scheme. There is a universal set, but, for a similar reason as in type theory, the paradoxical sets cannot be formed.It is not immediately apparent, however, that SF is essentially richer than type theory. But it follows from Specker's celebrated result (see [16] and [4]) that the stratifiable formula (extensionality → the universe is not well-orderable) is a theorem of SF.It is known (see [11]) that this set theory is consistent, though the consistency of NF is still an open problem.The connections between consistency and cut-elimination are rather loose. Cut-elimination generally implies consistency. But the converse is not true. In the case of set theory, for example, ZF-like systems, though consistent, cannot be freed of cuts because the separation axioms allow the formation of sets from unstratifiable formulas. There are nevertheless interesting partial results obtained when restrictions are imposed on the removable cuts (see [1] and [9]). The systems with stratifiable comprehension are the only known set-theoretic systems that enjoy full cut-elimination.

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