A Spector-Gandy theorem for cPCd() classes

1992 ◽  
Vol 57 (2) ◽  
pp. 478-500
Author(s):  
Shaughan Lavine
Keyword(s):  
Usual Sense ◽  
Admissible Set ◽  
Similarity Type ◽  
Definition Of ◽  
Key Theorem ◽  
Countable Structure

AbstractLet be an admissible structure. A cPCd() class is the class of all models of a sentence of the form , where is an -r.e. set of relation symbols and Φ is an -r.e. set of formulas of ℒ∞,ω that are in . The main theorem is a generalization of the following: Let be a pure countable resolvable admissible structure such that is not Σ-elementarily embedded in HYP(). Then a class K of countable structures whose universes are sets of urelements is a cPCd() class if and only if for some Σ formula σ (with parameters from ), is in K if and only if is a countable structure with universe a set of urelements and σ, where , the smallest admissible set above relative to , is a generalization of HYP to structures with similarity type Σ over that is defined in this article. Here we just note that when Lα is admissible, HYPLα() is Lβ() for the least β ≥ α such that Lβ() is admissible, and so, in particular, that is just HYP() in the usual sense when has a finite similarity type.The definition of is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem.

Uncountable models and infinitary elementary extensions

1973 ◽  
Vol 38 (3) ◽  
pp. 460-470 ◽  
Author(s):  
John Gregory
Keyword(s):  
Countable Subset ◽  
Elementary Extension ◽  
Admissible Set ◽  
Natural Numbers ◽  
Elementary Equivalence ◽  
Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

Some local definability results on countable topological structures

1983 ◽  
Vol 48 (3) ◽  
pp. 683-692 ◽  
Author(s):  
Holger Eisenmenger
Keyword(s):  
Formal Language ◽  
Second Order ◽  
Usual Sense ◽  
Countable Base ◽  
Relation Symbol ◽  
Topological Structures ◽  
Similarity Type ◽  
Individual Variables

L denotes a fixed finitary similarity type, B, respectively P a new relation symbol, an L-structure in the usual sense, and (, σ) a topological L-structure, where σ is a topology on A. (, σ) is countable if is countable and σ has a countable base. The formal language for our study of topological structures is . is the least fragment of the (monadic) second-order, infinitary language closed under negation (⇁), countable disjunction (∨), countable conjunction (∧), quantification over individual variables (∃ν, ∀ν), and quantification over set variables in the form ∃V(t ∈ V → φ) [respectively ∃V(t ∈ V → φ] where t is an L-term and each free occurrence of V in φ is negative [respectively positive]. We abbreviate ∃V(t ∈ V ∧ φ) and ∀V(t ∈ V → φ) by ∃V ∈ ν φ respectively ∀V ∈ ν φ. (For detailed information on we reIer to [1].)i, j, … m, n range over ω. a, x, etc. denote finite tuples; a ∈ A means that all members of a are in A. IdA denotes the identity on A, Perm(A) the set of all permutations of A, and Aut() (respectively Aut(, σ)) the set of all automorphisms of (respectively (σ)). Let F ⊆ Perm(A), B ⊆ Am(m ≥ 1), and μ be a system of subsets of A. B (respectively μ) is called invariant under F if for all ƒ ∈ F, ƒ(B) = B (respectively ƒ(μ) = μ). denotes the least system of subsets of A which contains μ and which is closed under arbitrary union, .For the rest of this paragraph let A be a countable nonempty set.

scholarly journals TRACTABLE FALSIFIABILITY

2015 ◽  
Vol 31 (2) ◽  
pp. 259-274 ◽  
Author(s):  
Ronen Gradwohl ◽  
Eran Shmaya
Keyword(s):  
Computational Complexity ◽  
Short Proof ◽  
Usual Sense ◽  
Economic Theories ◽  
Additional Requirement ◽  
Definition Of

Abstract:We propose to strengthen Popper’s notion of falsifiability by adding the requirement that when an observation is inconsistent with a theory, there must be a ‘short proof’ of this inconsistency. We model the concept of a short proof using tools from computational complexity, and provide some examples of economic theories that are falsifiable in the usual sense but not with this additional requirement. We consider several variants of the definition of ‘short proof’ and several assumptions about the difficulty of computation, and study their different implications on the falsifiability of theories.

Σ1 compactness for next admissible sets

1974 ◽  
Vol 39 (1) ◽  
pp. 105-116 ◽  
Author(s):  
Judy Green
Keyword(s):  
Compactness Theorem ◽  
Admissible Set ◽  
Compact Sets ◽  
Admissible Sets ◽  
New Class ◽  
Negative Results ◽  
Power Set ◽  
Consistency Property ◽  
Definition Of

Let σ be any sequence B0, B1 …, Bn, … of transitive sets closed under pairs with for each n. In this paper we show that the smallest admissible set Aσ with σ ∈ Aσ is Σ1 compact. Thus we have an entirely new class of explicitly describable uncountable Σ1 compact sets.The search for uncountable Σ1 compact languages goes back to Hanf's negative results on compact cardinals [7]. Barwise first showed that all countable admissible sets were Σ1 compact [1] and then went on to give a characterization of the Σ1 compact sets in terms of strict reflection [2]. While his characterization has been of interest in understanding the Σ1 compactness phenomenon it has led to the identification of only one class of uncountable Σ1 compact sets. In particular, Barwise showed [2], using the above notation, that if ⋃nBn is power set admissible it satisfies the strict reflection principle and hence is Σ1 compact. (This result was obtained independently by Karp using algebraic methods [9].)In proving our compactness theorem we follow Makkai's approach to the Barwise Compactness Theorem [12] and use a modified version of Smullyan's abstract consistency property [14]. A direct generalization of Makkai's method to the cofinality ω case yields a proof of the Barwise-Karp result mentioned above [6]. In order to obtain our new result we depart from the usual definition of language and use instead the indexed languages of Karp [9] in which a conjunction is considered to operate on a function whose range is a set of formulas rather than on a set of formulas itself.

scholarly journals TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES

2004 ◽  
Vol 16 (05) ◽  
pp. 583-602 ◽  
Author(s):  
DEBASHISH GOSWAMI
Keyword(s):  
Spectral Data ◽  
Noncommutative Geometry ◽  
Positive Operator ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Spectral Triple ◽  
Usual Sense ◽  
Spectral Triples ◽  
Definition Of ◽  
Made In

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

scholarly journals The Concept of Virtue in Religious Philosophy of Hermann Cohen

2019 ◽  
Vol 23 (4) ◽  
pp. 398-412
Author(s):  
Z. A. Sokuler
Keyword(s):  
Usual Sense ◽  
Religious Philosophy ◽  
Moral Ideal ◽  
The Social ◽  
Attributes Of God ◽  
The Common ◽  
Religion Of Reason ◽  
Definition Of ◽  
The Mind ◽  
Hermann Cohen

The concept of virtue was of great interest and importance for H. Cohen. In the interpretation of this concept in his latest work “Religion of reason from the sources of Judaism” the most important concepts of this work were brought in the focus: the specificity of definition of what is the religion of reason; understanding of the uniqueness of God; correlation; messianism. For Cohen, a single system of virtues presupposes a single and unique ethics and correlates with the idea of the unity of humanity. The last concept, in his opinion, maturated in the fold of monotheism. Humanity is one, because all people are creations of the unique God. “Religion of reason” treats of the common universal virtues. In the religion of reason, the idea of God and morality are inextricably linked. Cohen rejects metaphysical speculation about the nature of God, about the attributes of God inherent in himself. The religion of the mind speaks of God only in correlation with man. God is a moral ideal and reveals himself to man by giving him moral commandments. Morality connects man and God, and this connection is revealed in detail by Cohen in the theme of virtues. Understanding God as Truth is important for the disclosure of this topic. The corresponding virtue for a person is faithfulness to truth, or truthfulness. In addition to truthfulness in the usual sense, for Cohen, faithfulness to truth requires correct worship of God. The correlation culminates in the idea of messianism, which is interpreted by Cohen as an endless movement of a whole humanity to the social justice.

scholarly journals Electronic Documents in Archives’ Information Exchange System

2021 ◽  
pp. 520-531
Author(s):  
Irina V. Sabennikova ◽  
Keyword(s):  
Information Exchange ◽  
New Technologies ◽  
Personal Information ◽  
Social Activity ◽  
Usual Sense ◽  
Cultural Perceptions ◽  
Electronic Documents ◽  
Record Keeping ◽  
Electronic Copy ◽  
Definition Of

The functioning of various types of documents in the information space is determined by new technologies: social networks, blogs, forums, Internet exhibitions, electronic periodicals and non-periodical editions; they directly affect the formation of ethical, political, socio-cultural perceptions of modern users. There is a tendency to diversify these technologies as applicable to various types of electronic documents and for purposes of their use. Nowadays, the archives face a number of important tasks concerning study, analysis, admission for storage, and further use of documents. Among the issues requiring consideration and clarification in relation to electronic documents there is definition of the original, distinction between the original and electronic copy, the original and falsified document, definition of its material carrier, broadcasting of information, as well as some legal issues, primarily, questions of ownership, copyright, etc. The accumulation of information in the virtual sphere proceeds in various forms: there are photographs, videos from digital cameras, all kinds of electronic documents; it necessitates a clarification of some provisions concerning acquisition, storage, and usage of the documents and the information they contain. The article uses a research metaphor of “digital cultural layer” to express the changes in the archival sphere, where there emerges a new layer (complex) of electronic documents that has been created and is functioning in the virtual sphere. Electronic digital document doesn’t fit the usual paradigm, according to which a document always has a visualized material medium. The notion of the “original” becomes a subject for clarification. Acceptance of electronic documents for storage leads to adjustment of such traditional notions as “storage item” and “record-keeping item.” Identification of electronic document also requires clarification, since it does not lend itself to “identification” in the usual sense and we can only speak of conditional identification. The issues of acquisition, storage, and usage of electronic documents are considered in the article on the example of personal collections. As a significant part of citizens participate in various forms of social activity on the Internet (blogs, forums, electronic correspondence, etc.), thus expanding the social base of creators and holders of personal information, the near future will see a replenishment of personal provenance collections by electronic documents of personal origin.

Some Obstacles to Duality in Topological Algebra

1982 ◽  
Vol 34 (1) ◽  
pp. 80-90 ◽  
Author(s):  
Paul Bankston
Keyword(s):  
Full Subcategory ◽  
Topological Algebra ◽  
Usual Sense ◽  
Similarity Type ◽  
First Order ◽  
Equivalence Of Categories ◽  
Order Language

0. Introduction. Functors form an equivalence of categories (see [8,]) if Γ(Φ(A)) ≅ A and Φ (Γ(B)) ≅ B naturally for all objects A from and B from . Letting denote the opposite of we say that and are dual if there is an equivalence between and .Let τ be a similarity type of finitary operation symbols. We let Lτ denote the first order language (with equality) using nonlogical symbols from τ, and consider the class of all algebras of type τ as a category by declaring the morphisms to be all homomorphisms in the usual sense (i.e., those functions preserving the atomic sentences of Lτ). If is a class in (i.e., and is closed under isomorphism), we view as a full subcategory of , and we define the order of to be the number of symbols occurring in τ.

Rowbottom cardinals and Jonsson cardinals are almost the same

1973 ◽  
Vol 38 (3) ◽  
pp. 423-427 ◽  
Author(s):  
E. M. Kleinberg
Keyword(s):  
Large Cardinal ◽  
Order Logic ◽  
Elementary Substructure ◽  
Similarity Type ◽  
First Order ◽  
Reflection Principles ◽  
Uncountable Cardinals ◽  
The Moment ◽  
Definition Of ◽  
Jonsson Cardinals

Each of the various “large cardinal” axioms currently studied in set theory owes its inspiration to concrete phenomena in various fields. For example, the statement of the well-known compactness theorem for first-order logic can be generalized in various ways to infinitary languages to yield definitions of compact cardinals, and the reflection principles provable in ZF, when modified in the appropriate way, yields indescribable cardinals.In this paper we concern ourselves with two kinds of large cardinals which are probably the two best known of those whose origins lie in model theory. They are the Rowbottom cardinals and the Jonsson cardinals.Let us be more specific. A cardinal κ is said to be a Jonsson cardinal if every structure of cardinality κ has a proper elementary substructure of cardinality κ. (It is routine to see that only uncountable cardinals can be Jonsson. Erdös and Hajnal have shown [2] that for n < ω no ℵn is Jonsson. (In fact, they showed that if κ is not Jonsson then neither is the successor cardinal of κ and that, assuming GCH, no successor cardinal can be Jonsson.) Keisler and Rowbottom first showed that the existence of a Jonsson cardinal contradicts V = L.) The definition of a Rowbottom cardinal is only slightly more intricate. We assume for the moment that our similarity type has a designated one-place relation.

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