Some elementary degree-theoretic reasons why structures need similarity types

1986 ◽  
Vol 51 (3) ◽  
pp. 732-747
Author(s):  
T. G. McLaughlin
Keyword(s):  
Natural Numbers ◽  
Similarity Type ◽  
First Order ◽  
Order Language ◽  
Disjoint Sets ◽  
And Function ◽  
The Way

By a “partly numerical structure” (p.n.s.) we shall here mean a quadruple , where M is a set, ω = the natural numbers, ω ⊆ M, and are disjoint sets, is a set of relations (of various positive integral arities) on M, and is a set of functions (of various positive integral arities) with arguments and values in M. Thus, in calculated disharmony with common practice, we do not (except as noted below, in connection with naming the elements of ω) fix a similarity type as part of our notion of a “structure”. Suppose a finitary first-order language (with identity) has been specified, with constant symbols , n ∈ ω, and with exactly enough relation and function symbols of each arity to enable us to interpret in . We wish to consider the variation in the degree (relative to a fixed Gödel-numbering of ) of the complete -theory of as we vary the way in which elements of ∪ are assigned as interpretations to the relation and function symbols of . We shall in fact, therefore, be concerned exclusively with p.n.s.'s for which ∪ is countable. More: we assume to be such that we can effectively tell, uniformly in n > 0, exactly how many n-ary relations has and exactly how many n-ary functions has.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Theories without countable models

1972 ◽  
Vol 37 (3) ◽  
pp. 562-568
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Countable Model ◽  
Compactness Theorem ◽  
Complete Theory ◽  
Natural Numbers ◽  
Axiom Schema ◽  
First Order ◽  
Order Language ◽  
The Standard Model ◽  
Skolem Theorem

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

scholarly journals הכל הבל in Ecclesiastes 1:2 and 12:8 – Descriptive metaphysics of properties as comparative-philosophical supplement

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Jaco Gericke
Keyword(s):  
Second Order ◽  
Property Theory ◽  
First Order ◽  
Conceptual Clarification ◽  
Order Language ◽  
The World ◽  
Original Contribution ◽  
The Way

In this article, a supplementary yet original contribution is made to the ongoing attempts at refining ways of comparative-philosophical conceptual clarification of Qohelet’s claim that הבל הכל in 1:2 (and 12:8). Adopting and adapting the latest analytic metaphysical concerns and categories for descriptive purposes only, a distinction is made between הבל as property of הכל and the properties of הבל in relation to הכל. Involving both correlation and contrast, the second-order language framework is hereby extended to a level of advanced nuance and specificity for restating the meaning of the book’s first-order language on its own terms, even if not in them.Contribution: By considering logical, ontological, mereological and typological aspects of property theory in dialogue with appearances of הכל and of הבל in Ecclesiastes 1:2 and 12:8 and in-between, a new way is presented in the quest to explain why things in the world of the text are the way they are, or why they are at all.

Categoricity regained

1976 ◽  
Vol 41 (3) ◽  
pp. 639-643 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Infinite Length ◽  
Modern Logic ◽  
Mathematical Structures ◽  
Similarity Type ◽  
First Order ◽  
Nonstandard Model ◽  
Partial Isomorphism ◽  
Order Language ◽  
Order Arithmetic ◽  
Universe Of Sets

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by(i) enlarging our language so as to allow expressions of infinite length, and(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that(i) if f ∈ P then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and(ii) if f ∈ P, a ∈ A, and b ∈ B then there exist g,h ∈ P, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .

“Something”

2018 ◽  
Author(s):  
Mark Sainsbury
Keyword(s):  
First Order ◽  
Positive Account ◽  
Order Language ◽  
The Way

“Something” does not work in English the way “∃” works in a first-order language: it cannot always be associated with a domain of objects, and it can quantify into some positions that cannot be occupied by names or nominative phrases (as in “You are something that I am not—kind”). The best positive account of how it works, detailed and defended in this chapter, is substitutional. Since not all substitutes introduce objects, not all uses of “something” are ontologically committing. One merit of the view is that it explains the consistency of believing that there are things that don’t exist while not believing that there are nonexistent things.

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.

scholarly journals Current advances in nanomaterials affecting morphology, structure, and function of erythrocytes

RSC Advances ◽  
2021 ◽  
Vol 11 (12) ◽  
pp. 6958-6971
Author(s):  
Yaxian Tian ◽  
Zhaoju Tian ◽  
Yanrong Dong ◽  
Xiaohui Wang ◽  
Linsheng Zhan
Keyword(s):  
Structure And Function ◽  
Erythrocyte Membranes ◽  
And Function ◽  
The Way

This review focuses on the way how nanoparticles affect the structure and function of erythrocyte membranes, and is expected to pave the way for development of new nanodrugs.

PAL – Propositional Algorithmic Logic

1981 ◽  
Vol 4 (3) ◽  
pp. 675-760
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Data Structures ◽  
Completeness Theorem ◽  
Natural Numbers ◽  
First Order ◽  
Algorithmic Logic ◽  
Axiomatic Systems ◽  
Nondeterministic Program ◽  
Basic Sets

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

2. Speaking of God in the Post-Human Era: An Examination of Radical Life Extension in Relation to Key Theological Concepts and Narratives

2018 ◽  
Author(s):  
Kaitlyn Barton
Keyword(s):  
Human Life ◽  
Life Extension ◽  
The Core ◽  
Psychological Suffering ◽  
The Cross ◽  
And Function ◽  
Radical Life Extension ◽  
The Way ◽  
Changing World

Rapid advancements in radical life extension technologies contribute to humanity’s ever-changing world. The normalization of radical life extension technologies would signify that the present era in which biology and evolution act as dictators of human life and health would come to an end, thereby ushering in the age of the post-human. The purpose of this paper is to engage in a theological analysis of how and to what degree the ways in which humanity speaks about God could be changed or influenced if radical life extension becomes normative within society. . It is likely that this powerful technology would have a significant impact on many facets of culture, including the way in which humanity engages with religion, in particular Christianity. To accomplish this, the technology that could potentially support radical life extension, namely nanotechnology and cybernetic immortality, will be explained in terms of their relevance and function. Subsequently, the affects of radical life extension for human life will be addressed. Specifically, the implications of the partial or full eradication of human biological and psychological suffering and death through the use of cybernetic immortality and nanotechnology and will be considered. From there, the core theological concepts and narratives will be analyzed in the context of the potential actualization of radical life extension technology. A focus will be placed on the ethic of loving thy neighbour, Christ’s suffering on the cross, the hope of salvation and the Christian hope of entrance into heaven after death. 

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