scholarly journals A Simple Character String Proof of the "True but Unprovable'' Version of Gödel's First Incompleteness Theorem

2014 ◽  
Vol 151 ◽  
pp. 355-369
Author(s):  
Antti Valmari
Keyword(s):  
Incompleteness Theorem ◽  
Character String ◽  
Simple Character

A Simple Character

2018 ◽  
Author(s):  
Wylie Breckenridge
Keyword(s):  
Visual Experience ◽  
Simple Character

The author starts by considering a simple character, c, and a visual experience that has that character—your visual experience of a certain patch. A ‘look’ sentence is chosen that we might use to say that you have a visual experience of the patch with character c: ‘The patch looks grey to you’. The author argues that we use ‘grey’ to refer to a certain way of looking, w, and that by ‘The patch looks grey to you’ we mean that you have a visual experience of the patch that is occurring in way w. The author concludes that this is what it is for you to have a visual experience of the patch with character c, and, more generally, that to have a visual experience with character c is to have a visual experience that is occurring in way w.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Character string pattern matching in Algol 68

1977 ◽  
Vol 12 (6) ◽  
pp. 144-152 ◽  
Author(s):  
R. J.W. Housden ◽  
N. Kotarski
Keyword(s):  
Pattern Matching ◽  
Character String ◽  
String Pattern ◽  
Algol 68

Incomplete information in a relational database

1980 ◽  
Vol 3 (3) ◽  
pp. 363-377
Author(s):  
John Grant
Keyword(s):  
Incomplete Information ◽  
Relational Database ◽  
Functional Dependency ◽  
Relational Algebra ◽  
Database Model ◽  
Character String

In this paper we investigate the inclusion of incomplete information in the relational database model. This is done by allowing nonatomic entries, i.e. sets, as elements in the database. A nonatomic entry is interpreted as a set of possible elements, one of which is the correct one. We deal primarily with numerical entries where an allowed set is an interval, and character string entries. We discuss the various operations of the relational algebra as well as the notion of functional dependency for the database model.

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