scholarly journals Recent Advances in Σ-definability over Continuous Data Types

2003 ◽  
Vol 10 (23) ◽  
Author(s):  
Margarita Korovina
Keyword(s):  
Expressive Power ◽  
Continuous Data ◽  
Data Types ◽  
Real Numbers ◽  
Recursively Enumerable Set ◽  
The Real ◽  
Recursively Enumerable ◽  
Definable Relations ◽  
Computational Aspects ◽  
Equality Test

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas.

Further results on infinite valued predicate logic

1964 ◽  
Vol 29 (2) ◽  
pp. 69-78 ◽  
Author(s):  
L. P. Belluce
Keyword(s):  
Real Number ◽  
Present Author ◽  
Closed Interval ◽  
Abelian Groups ◽  
Predicate Logic ◽  
Real Numbers ◽  
The Real ◽  
Truth Values ◽  
Recursively Enumerable ◽  
Weakened Form

In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

A further consistent extension of basic logic

1950 ◽  
Vol 14 (4) ◽  
pp. 209-218 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Special Kind ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory ◽  
The Real ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Universal Quantification ◽  
Excluded Middle ◽  
Additional Assumptions

1.1. In two previous papers a consistent theory of real numbers has been outlined by the author, using a system K′. This latter system is an extension of a system K, which is “basic” in the sense that every finitary (recursively enumerable) subclass of its well-formed expressions is in a certain sense represented in it. The system L described below is a further extension of K. The system K′ lacks two important features possessed by L: a symbol for a special kind of implication (or “conditionality”) and a symbol for the modal concept “necessity.” The presence of the implication symbol, and the additional assumptions that go with it, make available in L various kinds of restricted universal quantification not available in K′, for example, universal quantification restricted to the real numbers of the author's theory of real numbers.1.2. If ‘~[a & ~a]’ is a theorem of L, then the proposition expressed by ‘a’ may be said to L-satisfy the principle of excluded middle. I t is always the case that ‘a’ L-satisfies the principle of excluded middle (or rather that the proposition expressed by ‘a’ does so) if and only if ‘a’ or ‘~a’ is a theorem of L. An example of a proposition that does not L-satisfy the principle of excluded middle is that expressed by ‘’, namely the proposition that asserts that the class of classes that are not members of themselves is a member of itself.

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

On expressible sets and p-adic numbers

2011 ◽  
Vol 54 (2) ◽  
pp. 411-422
Author(s):  
Jaroslav Hančl ◽  
Radhakrishnan Nair ◽  
Simona Pulcerova ◽  
Jan Šustek
Keyword(s):  
Haar Measure ◽  
Real Numbers ◽  
The Real ◽  
Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

Models of the Real Numbers

2020 ◽  
pp. 203-221
Author(s):  
Lorenz Halbeisen ◽  
Regula Krapf
Keyword(s):  
Real Numbers ◽  
The Real
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