## Constructive Order Completeness (2004)

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Baroni04constructiveorder,

author = {Marian Alexandru Baroni},

title = {Constructive Order Completeness},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1

### Citations

442 | Foundations of Constructive Analysis
- Bishop
- 1967
(Show Context)
Citation Context ...ind completeness of R. The main goal of this paper is to provide a constructive definition of order completeness for arbitrary partially ordered sets. Our setting is Bishop’s constructive mathematics =-=[4, 5]-=-, mathematics developed with intuitionistic logic, 1 a logic based on the strict interpretation of the existence as computability. One advantage of working in this manner is that proofs and 1 We also ... |

278 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...interpretations. On the one hand, Bishop’s constructive mathematics is consistent with the traditional mathematics. On the other hand, the results can be interpreted recursively or intuitionistically =-=[3, 8, 16]-=-. If we are working constructively, the first problem is to obtain appropriate substitutes of the classical definitions. The classical theory of partially ordered sets is based on the negative concept... |

167 |
Infinite Dimensional Analysis
- Aliprantis, Border
- 1999
(Show Context)
Citation Context ...ess and suprema and the one with lower locatedness and infima. We will also give a definition of incompleteness which is not merely the negation of completeness and we will prove constructively that C=-=[0, 1]-=-, a standard classical example of an Archimedean Riesz space that is not order complete, satisfies this definition. In Section 6 we will prove that a Cartesian product of n order complete sets is orde... |

59 |
Constructive Analysis, Grundlehren der Math. Wissenschaften 279
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...ind completeness of R. The main goal of this paper is to provide a constructive definition of order completeness for arbitrary partially ordered sets. Our setting is Bishop’s constructive mathematics =-=[4, 5]-=-, mathematics developed with intuitionistic logic, 1 a logic based on the strict interpretation of the existence as computability. One advantage of working in this manner is that proofs and 1 We also ... |

22 |
Constructive mathematics: a foundation for computable analysis
- Bridges
- 1999
(Show Context)
Citation Context ...with x<z<y. An example is the standard strict order relation < on R, as described in [4]. For an axiomatic definition of the real number line as a constructive ordered field, the reader is refered to =-=[6]-=- or [7]. A detailed investigation of linear orders in lattices can be found in [9]. The binary relation � on X is called an excess relation if it satisfies the following axioms: E1 ¬(x � x), E2 x � y ... |

17 |
Introduction to operator theory in Riesz space
- Zaanen
- 1997
(Show Context)
Citation Context ...pace endowed with the pointwise ordering, is the standard example of an Archimedean Riesz space that is not Dedekind complete. (For background information about Riesz spaces, the reader is refered to =-=[17]-=-. Constructive definitions of ordered vector spaces and Riesz spaces, can be found in [2].) To prove that C[0, 1] is Dedekind incomplete let us consider, as in the classical proof [1], the sequence (f... |

13 |
Constructivism in Mathematics: An Introduction (two volumes
- Troelstra, Dalen
- 1988
(Show Context)
Citation Context ...interpretations. On the one hand, Bishop’s constructive mathematics is consistent with the traditional mathematics. On the other hand, the results can be interpreted recursively or intuitionistically =-=[3, 8, 16]-=-. If we are working constructively, the first problem is to obtain appropriate substitutes of the classical definitions. The classical theory of partially ordered sets is based on the negative concept... |

7 |
Markov’s principle, church’s thesis and lindelöf’s theorem
- Ishihara
- 1993
(Show Context)
Citation Context .... Although this principle is accepted in the recursive constructive mathematics developed by A.A. Markov, it is rejected in Bishop’s constructivism. For further information on Markov’s principle, see =-=[10]-=-. To end this section, let us consider an example. Let X be a set of real–valued functions defined on a nonempty set S, andlet� be the relation on X defined by f � g if there exists x in S such that g... |

5 |
A constructive uniform continuity theorem
- Ishihara, Schuster
- 2002
(Show Context)
Citation Context ... counterpart [5, 12]: a nonempty subset of R that is bounded above has a supremum if and only if it satisfies a certain condition of (upper) order locatedness. As pointed out by Ishihara and Schuster =-=[11]-=-, this equivalence expresses constructively the order completeness of the real number line. Furthermore, the definitions of upper and lower locatedness were extended by Palmgren [13] to the case of a ... |

3 |
Linear Order in Lattices: A Constructive Study
- Greenleaf
- 1978
(Show Context)
Citation Context ...in [4]. For an axiomatic definition of the real number line as a constructive ordered field, the reader is refered to [6] or [7]. A detailed investigation of linear orders in lattices can be found in =-=[9]-=-. The binary relation � on X is called an excess relation if it satisfies the following axioms: E1 ¬(x � x), E2 x � y ⇒∀z ∈ X (x � z ∨ z � y). We say that x exceeds y whenever x � y. Clearly, each lin... |

3 |
Constructive Continuity, Memoirs of the American
- Mandelkern
- 1983
(Show Context)
Citation Context ...f R [4]. Although this supremum is classically equivalent to the least upper bound, we cannot expect to prove constructively that its existence is guaranteed by the existence of the least upper bound =-=[12]-=-. Having described the general framework, let us examine the notion of order completeness. When working constructively, we have to get over a main difficulty: the least–upper–bound principle is no lon... |

2 |
On the order dual of a Riesz space
- Baroni
- 2003
(Show Context)
Citation Context ...elation obtained by the negation of an excess relation. To develop a constructive theory, the classical supremum; that is, the least upper bound, is too weak a notion. We will use a stronger supremum =-=[2]-=-, a generalization of the usual constructive supremum of a subset of R [4]. Although this supremum is classically equivalent to the least upper bound, we cannot expect to prove constructively that its... |

2 | Constructive completions of ordered sets, groups and fields
- Palmgren
- 2003
(Show Context)
Citation Context ...hara and Schuster [11], this equivalence expresses constructively the order completeness of the real number line. Furthermore, the definitions of upper and lower locatedness were extended by Palmgren =-=[13]-=- to the case of a dense linear order. According to [13], a set X endowed with a dense linear order is order complete if each nonempty subset of X that is bounded above and upper located has a least up... |

1 |
Positive lattices, in: Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum
- Plato
- 2001
(Show Context)
Citation Context ...classical definitions. The classical theory of partially ordered sets is based on the negative concept of partial order. Unlike the classical case, an affirmative concept, von Plato’s excess relation =-=[14]-=-, will be used as a primary relation. Throughout this paper a partially ordered set will be a set endowed with a partial order relation obtained by the negation of an excess relation. To develop a con... |

1 |
Authors’ Note Funds from the National Institute Of Child Health and Development Grant HD-01994 to Haskins Laboratories supported the research reported in the present paper. Reprint requests should be sent to the first or third author at Haskins Laboratori
- unknown authors
- 1986
(Show Context)
Citation Context ...working in this manner is that proofs and 1 We also assume the principle of dependent choice [8], which is widely accepted in constructive mathematics. For constructivism without dependent choice see =-=[15]-=-. 1sresults have more interpretations. On the one hand, Bishop’s constructive mathematics is consistent with the traditional mathematics. On the other hand, the results can be interpreted recursively ... |