Truth In V for ∃*∀∀-Sentences is Decidable

2006 ◽  
Vol 71 (4) ◽  
pp. 1200-1222 ◽  
Author(s):  
D. Bellé ◽  
F. Parlamento
Keyword(s):  
Set Theory ◽  
Finite Sets ◽  
First Order ◽  
Free Variables ◽  
Order Language

AbstractLet V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and. following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in ∀ but not by finite sets ([5]). and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 419-426
Author(s):  
Manuel Lerman
Keyword(s):  
Lattice Theory ◽  
Regular Cardinal ◽  
Finite Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Regular Cardinals ◽  
Constructible Sets ◽  
Set Inclusion ◽  
Order Language ◽  
Recursively Enumerable Sets

Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*-finite if it is α-finite and has ordertype less than α* (the Σ1 projectum of α). An a-r.e. set S is simple if (the complement of S) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ1L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1L is, of course, such an α), there are simple α-r.e. sets with different 1-types.The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “S is a maximal α-r.e. set”, “S is an r-maximal α-r.e. set”, or “S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L, there are no maximal, r-maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

scholarly journals Much Ado About the Many

2021 ◽  
Vol 25 (1) ◽  
Author(s):  
Jonathan Mai
Keyword(s):  
Set Theory ◽  
Formal Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Plural Quantification ◽  
First Order ◽  
Order Language ◽  
The Many ◽  
Second Order Logic ◽  
Alternative Theories

English distinguishes between singular quantifiers like "a donkey" and plural quantifiers like "some donkeys". Pluralists hold that plural quantifiers range in an unusual, irreducibly plural, way over common objects, namely individuals from first-order domains and not over set-like objects. The favoured framework of pluralism is plural first-order logic, PFO, an interpreted first-order language that is capable of expressing plural quantification. Pluralists argue for their position by claiming that the standard formal theory based on PFO is both ontologically neutral and really logic. These properties are supposed to yield many important applications concerning second-order logic and set theory that alternative theories supposedly cannot deliver. I will show that there are serious reasons for rejecting at least the claim of ontological innocence. Doubt about innocence arises on account of the fact that, when properly spelled out, the PFO-semantics for plural quantifiers is committed to set-like objects. The correctness of my worries presupposes the principle that for every plurality there is a coextensive set. Pluralists might reply that this principle leads straight to paradox. However, as I will argue, the true culprit of the paradox is the assumption that every definite condition determines a plurality.

Almost combinatorial Skolem functions

1970 ◽  
Vol 35 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Finite Sets ◽  
First Order ◽  
Universal Sentence ◽  
Skolem Functions

Let be a version of class set theory admitting urelemente, and with AC (= axiom of choice) replaced by AC0 (= axiom of choice for sets of finite sets), ω = nonnegative integers, and Δ = Dedekind cardinals. Let be an arbitrarily quantified positive first order sentence in functors for + and ·. Let ƒ0, … , ƒκ - 1 be function variables and the universal sentence obtained from by replacing existential quantifiers by the ƒ1 as Skolem functions.

On an Ackermann-type set theory

1973 ◽  
Vol 38 (3) ◽  
pp. 410-412
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Universal Closure

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.

scholarly journals Undecidable properties of finite sets of equations

1976 ◽  
Vol 41 (3) ◽  
pp. 589-604 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Decision Problems ◽  
Equational Logic ◽  
Relation Algebras ◽  
Finite Sets ◽  
Operation Symbol ◽  
First Order ◽  
Finite Models ◽  
Order Language

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

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.

Natural models and Ackermann-type set theories

1975 ◽  
Vol 40 (2) ◽  
pp. 151-158 ◽  
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus

Our results concern the natural models of Ackermann-type set theories, but they can also be viewed as results about the definability of ordinals in certain sets.Ackermann's set theory A was introduced in [1] and it is now formulated in the first order predicate calculus with identity, using ∈ for membership and an individual constant V for the class of all sets. We use the letters ϕ, χ, θ, and χ to stand for formulae which do not contain V and capital Greek letters to stand for any formulae. Then, the axioms of A* are the universal closures ofwhere all the free variables are shown in A4 and z does not occur in the Θ of A2. A is the theory A* − A5.Most of our notation is standard (for instance, α, β, γ, δ, κ, λ, ξ are variables ranging over ordinals) and, in general, we follow the notation of [7]. When x ⊆ Rα, we use Df(Rα, x) for the set of those elements of Rα which are definable in 〈Rα, ∈〉, using a first order ∈-formula and parameters from x.We refer the reader to [7] for an outline of the results which are known about A, but we shall summarise those facts which are frequently used in this paper.

scholarly journals Boolean universes above Boolean models

1993 ◽  
Vol 58 (4) ◽  
pp. 1219-1250 ◽  
Author(s):  
Friedrich Wehrung
Keyword(s):  
Set Theory ◽  
Boolean Model ◽  
Positive Cone ◽  
Second Order ◽  
Boolean Models ◽  
First Order ◽  
Naive Set Theory ◽  
Order Language

AbstractWe establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.

On countable fractions from an elementary class

1994 ◽  
Vol 59 (4) ◽  
pp. 1410-1413
Author(s):  
C. J. Ash
Keyword(s):  
Model Theory ◽  
Relation Symbol ◽  
First Order ◽  
Elementary Class ◽  
Free Variables ◽  
Order Language ◽  
Skolem Theorem ◽  
Elementary Result ◽  
Countable Models

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

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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