Machine proof of Equivalence between Axiom of Choice and Product Theorem of Standard Family Sets

2020 ◽  
Author(s):  
Hongwei Lv ◽  
Jia Liu ◽  
Yaoshun Fu ◽  
Wensheng Yu
Keyword(s):  
Axiom Of Choice ◽  
Product Theorem ◽  
Standard Family

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.

On the Relationship between Alonso's Theory of Movement and Wilson's Family of Spatial-Interaction Models

1981 ◽  
Vol 13 (2) ◽  
pp. 217-224 ◽  
Author(s):  
J Ledent
Keyword(s):  
Input Data ◽  
Spatial Interaction ◽  
System Of Equations ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Standard Family ◽  
Original Contribution ◽  
Standard Models ◽  
The Relationship

This paper compares the system of equations underlying Alonso's theory of movement with that of Wilson's standard family of spatial-interaction models. It is shown that the Alonso model is equivalent to one of Wilson's four standard models depending on the assumption at the outset about which of the total outflows and/or inflows are known. This result turns out to supersede earlier findings—inconsistent only in appearance—which were derived independently by Wilson and Ledent. In addition to this, an original contribution of this paper—obtained as a byproduct of the process leading to the aforementioned result—is to provide an exact methodology permitting one to solve the Alonso model for each possible choice of the input data.

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.

Data Acquisition from Light-Duty Vehicles Using OBD and CAN

10.4271/r-458 ◽  
2018 ◽  
Author(s):  
Eric Walter ◽  
◽  
Richard Walter ◽  
Keyword(s):  
Data Acquisition ◽  
Accident Reconstruction ◽  
Intermittent Faults ◽  
Light Duty ◽  
Control Units ◽  
Standard Family ◽  
Engineering Parameters ◽  
Light Duty Vehicles ◽  
Share Information ◽  
Vehicle Network

Modern vehicles have multiple electronic control units (ECU) to control various subsystems such as the engine, brakes, steering, air conditioning, and infotainment. These ECUs are networked together to share information directly with each other. This in-vehicle network provides a data opportunity for improved maintenance, fleet management, warranty and legal issues, reliability, and accident reconstruction. Data Acquisition from Light-Duty Vehicles Using OBD and CAN is a guide for the reader on how to acquire and correctly interpret data from the in-vehicle network of light-duty (LD) vehicles. The reader will learn how to determine what data is available on the vehicle's network, acquire messages and convert them to scaled engineering parameters, apply more than 25 applicable standards, and understand 15 important test modes. Topics featured in this book include: • Calculated fuel economy • Duty cycle analysis • Capturing intermittent faults Written by two specialists in this field, Richard P. Walter and Eric P. Walter of HEM Data, the book provides a unique roadmap for the data acquisition user. The authors give a clear and concise description of the CAN protocol plus a review of all 19 parts of the SAE International J1939 standard family. Data Acquisition from Light-Duty Vehicles Using OBD and CAN is a must-have reference for product engineers, service technicians fleet managers and all interested in acquiring data effectively from the light-duty vehicles.

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