scholarly journals Ample thoughts

2013 ◽  
Vol 78 (2) ◽  
pp. 489-510 ◽  
Author(s):  
Daniel Palacín ◽  
Frank O. Wagner
Keyword(s):  
General Notion ◽  
Base Property ◽  
Simple Theories ◽  
Canonical Base ◽  
Weakened Version ◽  
Special Case

AbstractNon-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

Relativized realizability in intuitionistic arithmetic of all finite types

1978 ◽  
Vol 43 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Extension Theorem ◽  
General Notion ◽  
Conservative Extension ◽  
First Order ◽  
New Information ◽  
Reflection Theorem ◽  
Order Arithmetic ◽  
Special Case ◽  
Dependent Choice ◽  
Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

Uniqueness, definability and interpolation

1988 ◽  
Vol 53 (2) ◽  
pp. 554-570 ◽  
Author(s):  
Kosta Došen ◽  
Peter Schroeder-Heister
Keyword(s):  
Proof Theory ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
General Notion ◽  
Structural Part ◽  
First Order ◽  
First Case ◽  
Additional Constant ◽  
Special Case

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, Tα a set of sentences, α* an additional constant of the same syntactical category as α and Tα, a copy of Tα with α* instead of α, then for implicit definability of α in Tα one has, in the case of predicate constants, to derive α(x1,…,xn) ↔ α*(x1,…,xn) from Tα ∪ Tα*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata Sα and derivability from instances of Sα ∪ Sα* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

Certain predicates defined by induction schemata

1953 ◽  
Vol 18 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Formal System ◽  
General Notion ◽  
Constant Number ◽  
Natural Numbers ◽  
System L ◽  
Truth Definition ◽  
Consistency Proofs ◽  
Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

scholarly journals CONSTRUCTING TYPES IN DIFFERENTIALLY CLOSED FIELDS THAT ARE ANALYSABLE IN THE CONSTANTS

2018 ◽  
Vol 83 (04) ◽  
pp. 1413-1433
Author(s):  
RUIZHANG JIN
Keyword(s):  
Minimal Length ◽  
Canonical Analysis ◽  
Base Property ◽  
Canonical Base ◽  
Closed Fields ◽  
Image Position ◽  
Positive Integers

AbstractAnalysability of finiteU-rank types are explored both in general and in the theory${\rm{DC}}{{\rm{F}}_0}$. The well-known fact that the equation$\delta \left( {{\rm{log}}\,\delta x} \right) = 0$is analysable in but not almost internal to the constants is generalized to show that$\underbrace {{\rm{log}}\,\delta \cdots {\rm{log}}\,\delta }_nx = 0$is not analysable in the constants in$\left( {n - 1} \right)$-steps. The notion of acanonical analysisis introduced–-namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of reductions and coreductions in theories with the canonical base property, it is constructed, for any sequence of positive integers$\left( {{n_1}, \ldots ,{n_\ell }} \right)$, a type in${\rm{DC}}{{\rm{F}}_0}$that admits a canonical analysis with the property that theith step hasU-rank${n_i}$.

scholarly journals On definable Galois groups and the strong canonical base property

2017 ◽  
Vol 17 (01) ◽  
pp. 1750002
Author(s):  
Daniel Palacín ◽  
Anand Pillay
Keyword(s):  
Finite Rank ◽  
Galois Groups ◽  
Strong Form ◽  
Current Paper ◽  
Base Property ◽  
Canonical Base ◽  
Definition Of

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank context than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid.

God and Rationality

1974 ◽  
Vol 4 (2) ◽  
pp. 283-292
Author(s):  
Robert C. Solomon
Keyword(s):  
Soren Kierkegaard ◽  
General Notion ◽  
Alvin Plantinga ◽  
Philosophical Language ◽  
Belief In God ◽  
Natural Theologian ◽  
Relevant Notion ◽  
Special Case

Is belief in God rational? Over a century ago, Hegel (following Kant) and Søren Kierkegaard established one set of parameters for discussing that question, but in a language that appears opaque to many philosophers today. Very recently, Alvin Plantinga, James Ross, and George Mavrodes have been debating similar issues in a modern analytic idiom. In this essay, I want to use this modern philosophical language in an attempt to clarify certain issues surrounding the relevant notion of “rationality” and related notions essential to the natural theologian, and in so doing attempt to make presentable the dispute between Hegel and Kierkegaard.For our purposes here, I take “rationality” to be predicated of an epistemological concept of belief, even if, as I believe, any such notion would have to be a special case and a logical derivative of a more general notion of “rationality” as primarily practical.

On types of CB-rank 1 in simple theories

2008 ◽  
Vol 7 (4) ◽  
pp. 895-899 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Recent Result ◽  
Simple Theory ◽  
Countable Model ◽  
Stable Case ◽  
Simple Theories ◽  
Complete Type ◽  
Special Case

AbstractWe prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor–Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable case, and is a special case of a recent result of Tanovic.

Plus ultra

2015 ◽  
Vol 15 (02) ◽  
pp. 1550008
Author(s):  
Frank O. Wagner
Keyword(s):  
Equivalence Relations ◽  
Base Property ◽  
Canonical Base

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.

scholarly journals On the canonical base property

2013 ◽  
Vol 19 (4) ◽  
pp. 865-877 ◽  
Author(s):  
Ehud Hrushovski ◽  
Daniel Palacín ◽  
Anand Pillay
Keyword(s):  
Base Property ◽  
Canonical Base

scholarly journals A Physical Theory of Information

2020 ◽  
Author(s):  
Manouchehr Amiri
Keyword(s):  
Quantum Mechanics ◽  
Binary Data ◽  
Black Hole Thermodynamics ◽  
Data Matrix ◽  
General Notion ◽  
Mass Energy ◽  
Energy Equivalence ◽  
Physical Information ◽  
Fundamental Equations ◽  
Special Case

In this article, I develop a general notion of physical bit information which is compatible with the basics of quantum mechanics and incorporates the Shannon entropy as a special case. This notion of physical information leads to Binary data matrix model (BDM) which predicts the basic results of quantum mechanics, general relativity, and black hole thermodynamics. the compatibility of the model with holographic, information conservation and Landauers principles is investigated. After deriving the Bit Information principle as a consequence of BDM, the fundamental equations of Planck, De Broglie, Beckenstein and mass-energy equivalence are derived.

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