L'axiome de normalité pour les espaces totalement ordonnés

1990 ◽  
Vol 55 (1) ◽  
pp. 277-283 ◽  
Author(s):  
Labib Haddad ◽  
Marianne Morillon
Keyword(s):  
Ordered Set ◽  
Axiom Of Choice ◽  
Order Topology ◽  
Cohen Model ◽  
The Axiom Of Choice ◽  
Linearly Ordered Set

AbstractWe show that the following property (LN) holds in the basic Cohen model as sketched by Jech: The order topology of any linearly ordered set is normal. This proves the independence of the axiom of choice from LN in ZF, and thus settles a question raised by G. Birkhoff (1940) which was partly answered by van Douwen (1985).

On a theorem of Fleischer

1986 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
G. Mehta
Keyword(s):  
Ordered Set ◽  
Separation Theorem ◽  
Order Topology ◽  
Topological Spaces ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractFleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

scholarly journals Quantum Invariance

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Information ◽  
Coherent State ◽  
Set Theory ◽  
Ordered Set ◽  
Axiom Of Choice ◽  
Statistical Ensemble ◽  
Planck Constant ◽  
Physical Processes ◽  
Statistical Representation ◽  
The Axiom Of Choice

Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement.A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to a well-ordered set after measurement and thus requires the axiom of choice.Quantum invariance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information.Quantum invariance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity.

Support structures for the axiom of choice

1971 ◽  
Vol 36 (1) ◽  
pp. 28-38 ◽  
Author(s):  
David Pincus
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Embedding Theorems ◽  
Support Structures ◽  
Support Structure ◽  
Cohen Model ◽  
The Axiom Of Choice

The notion of “support” was introduced by Mostowski in [4] in order to prove that a certain universe satisfied the ordering principle but not the axiom of choice. The notion was refined in [3] and in [1] it was shown to be satisfied in a certain Cohen model of full ZF set theory. This paper is an axiomatic study of universes whose undefined relations are ∈ and a “support structure”, T.In §2 the general theory is introduced and the universes of [4] and [1] are characterized. §3 examines a more complicated universe which will be used in [5] to show that in many cases a consistency in full ZF set theory may be proven directly by the methods of [4]. The embedding theorems of §4 are crucial to this application.

The dense linear ordering principle

1997 ◽  
Vol 62 (2) ◽  
pp. 438-456 ◽  
Author(s):  
David Pincus
Keyword(s):  
Prime Ideal ◽  
Set Theory ◽  
Ordered Set ◽  
Axiom Of Choice ◽  
Linear Ordering ◽  
Independence Result ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Prime Ideal Theorem ◽  
The Way

AbstractLet DO denote the principle: Every infinite set has a dense linear ordering. DO is compared to other ordering principles such as O, the Linear Ordering principle, KW, the Kinna-Wagner Principle, and PI, the Prime Ideal Theorem, in ZF, Zermelo-Fraenkel set theory without AC, the Axiom of Choice.The main result is:Theorem. AC ⇒ KW ⇒ DO ⇒ O, and none of the implications is reversible in ZF + PI.The first and third implications and their irreversibilities were known. The middle one is new. Along the way other results of interest are established. O, while not quite implying DO, does imply that every set differs finitely from a densely ordered set. The independence result for ZF is reduced to one for Fraenkel-Mostowski models by showing that DO falls into two of the known classes of statements automatically transferable from Fraenkel-Mostowski to ZF models. Finally, the proof of PI in the Fraenkel-Mostowski model leads naturally to versions of the Ramsey and Ehrenfeucht-Mostowski theorems involving sets that are both ordered and colored.

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
The Axiom Of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

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.

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