A Rigorous Procedure for Generating a Well-ordered Set of Reals without use of Axiom of Choice/Well-ordering Theorem

2021 ◽  
pp. 1-5
Author(s):  
Karan Doshi
Keyword(s):  
Ordered Set ◽  
Axiom Of Choice ◽  
Rigorous Procedure

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.

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).

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.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

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 Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Homomorphisms and congruences on ωα-bisimple semigroups

1973 ◽  
Vol 15 (4) ◽  
pp. 441-460 ◽  
Author(s):  
J. W. Hogan
Keyword(s):  
Partial Order ◽  
Ordered Set ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

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