Recursive categoricity and persistence

1986 ◽  
Vol 51 (2) ◽  
pp. 430-434 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
First Order ◽  
Existential Sentence ◽  
Order Language ◽  
Recursive Structures ◽  
Definition Of

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci∣i<ω} be distinct new constants, and let {θi∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θi is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

scholarly journals Semantics for Hyperclassical Logic and the Problem of Negation in the Formal Language

2018 ◽  
Vol 16 (3) ◽  
pp. 5-15
Author(s):  
V. V. Tselishchev
Keyword(s):  
Theoretical Model ◽  
Formal Language ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Order Language ◽  
Winning Strategies ◽  
Game Theoretic ◽  
Definition Of ◽  
Game Theoretic Semantics

The application of game-theoretic semantics for first-order logic is based on a certain kind of semantic assumptions, directly related to the asymmetry of the definition of truth and lies as the winning strategies of the Verifier (Abelard) and the Counterfeiter (Eloise). This asymmetry becomes apparent when applying GTS to IFL. The legitimacy of applying GTS when it is transferred to IFL is based on the adequacy of GTS for FOL. But this circumstance is not a reason to believe that one can hope for the same adequacy in the case of IFL. Then the question arises if GTS is a natural semantics for IFL. Apparently, the intuitive understanding of negation in natural language can be explicated in formal languages in various ways, and the result of an incomplete grasp of the concept in these languages can be considered a certain kind of anomalies, in view of the apparent simplicity of the explicated concept. Comparison of the theoretical-model and game theoretic semantics in application to two kinds of language – the first-order language and friendly-independent logic – allows to discover the causes of the anomaly and outline ways to overcome it.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

Logic of reduced power structures

1983 ◽  
Vol 48 (1) ◽  
pp. 53-59
Author(s):  
G.C. Nelson
Keyword(s):  
Boolean Algebra ◽  
Complete Theory ◽  
Power Structures ◽  
Index Set ◽  
First Order ◽  
Order Language ◽  
Power Set ◽  
Definition Of ◽  
Proper Filter ◽  
Horn Sentence

We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by I/D the reduced power of modulo D as defined in [2, pp. 167–169]. The first-order language associated with I/D will always be the same language as associated with . We denote the 2-element Boolean algebra 〈{0, 1}, ⋂, ⋃, c, 0, 1〉 by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures I/D and 2I/D given in [2, pp. 341–345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that φ is a reduced product sentence iff φ is provably equivalent to a Horn sentence [2, Theorem 6.2.5/ (iff φ is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T(any set of sentences), ⊨ T denotes that is a model of T.In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation 〈n, p, q〉 of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each 〈n, p, q〉 there exist an index set I and filter D such that 2I/D ⊨ 〈n, p, q〉 [3] or [2, Lemma 6.3.21].

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.

Truth Definitions, Skolem Functions and Axiomatic Set Theory

1998 ◽  
Vol 4 (3) ◽  
pp. 303-337 ◽  
Author(s):  
Jaakko Hintikka
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Skolem Functions ◽  
Order Language ◽  
Definition Of ◽  
Axiomatic Set Theory

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals Solving puzzles in deformed JT gravity: phase transitions and non-perturbative effects

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Clifford V. Johnson ◽  
Felipe Rosso
Keyword(s):  
Phase Structure ◽  
Scalar Potential ◽  
String Equation ◽  
Two Dimensional ◽  
Leading Order ◽  
Classical Analysis ◽  
First Order ◽  
First Order Phase Transition ◽  
Definition Of ◽  
First Order Phase

Abstract Recent work has shown that certain deformations of the scalar potential in Jackiw-Teitelboim gravity can be written as double-scaled matrix models. However, some of the deformations exhibit an apparent breakdown of unitarity in the form of a negative spectral density at disc order. We show here that the source of the problem is the presence of a multi-valued solution of the leading order matrix model string equation. While for a class of deformations we fix the problem by identifying a first order phase transition, for others we show that the theory is both perturbatively and non-perturbatively inconsistent. Aspects of the phase structure of the deformations are mapped out, using methods known to supply a non-perturbative definition of undeformed JT gravity. Some features are in qualitative agreement with a semi-classical analysis of the phase structure of two-dimensional black holes in these deformed theories.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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