NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

2019 ◽  
pp. 1-35
Author(s):  
WILFRIED SIEG ◽  
PATRICK WALSH
Keyword(s):  
Proof Theory ◽  
Mathematical Proof ◽  
Formal Proofs ◽  
Bernstein Theorem ◽  
Search System ◽  
Inference Mechanism ◽  
Mathematical Proofs ◽  
Computational Work ◽  
Definitional Extension

Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

The Cantor–Bernstein theorem: how many proofs?

2019 ◽  
Vol 377 (2140) ◽  
pp. 20180031 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Argument Structure ◽  
Proof Theory ◽  
Careful Analysis ◽  
Methodological Framework ◽  
Theme Issue ◽  
Bernstein Theorem ◽  
Computational Tools ◽  
Chain Theory ◽  
The Many

Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo , respectively. This paper is a case study in analysing proofs of a single theorem within a given methodological framework, here Zermelo–Fraenkel set theory (ZF). It uses tools from proof theory, but focuses on heuristic ideas that shape proofs and on logical strategies that help to construct them. It is rooted in a perspective on Beweistheorie that predates its close connection and almost exclusive attention to the goals of Hilbert's finitist consistency programme. This earlier perspective can be brought to life (only) with the support of powerful computational tools. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

The Development of the Idea of Mathematical Proof: A 5-Year Case Study

1996 ◽  
Vol 27 (2) ◽  
pp. 194 ◽  
Author(s):  
Carolyn A. Maher ◽  
Amy M. Martino
Keyword(s):  
Mathematical Proof

Trusting Computers Through Trusting Humans

2008 ◽  
pp. 3460-3474
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This article considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

Trusting Computers Through Trusting Humans

2011 ◽  
pp. 384-398
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This chapter considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

Graphical rules of steady-state reaction systems

1981 ◽  
Vol 59 (4) ◽  
pp. 737-755 ◽  
Author(s):  
Chou Kuo-Chen ◽  
Sture Forsen
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Mathematical Proof ◽  
Complex Reaction ◽  
Consecutive Reaction ◽  
Reaction Systems ◽  
First Order ◽  
Mathematical Proofs ◽  
Pseudo First Order ◽  
Manual Calculation

Four rules to deal with first-order or pseudo-first-order steady-state reaction systems are presented.By means of Rule 1, we can immediately write down the apparent rate constants of consecutive reaction systems. This rule is actually the same as the "Rule of Thumb" proposed by Gilbert, but here its mathematical proof is given.Rule 2 and Rule 3 may serve to derive the apparent rate constants of various complex reaction systems. In comparison with the general algebraic methods, these two rules can simplify laborious calculations that would otherwise be tedious and liable to errors.Rule 4 presents a new schematic method to calculate the concentrations of the reactants. The new method, in simplifying the calculation of complex problems, is extraordinarily efficacious in comparison with the existing schematic methods. For complex mechanisms which are too complicated to be treated with the general manual calculation method, the practical calculations show that we can easily write down the desired results by means of Rule 4.In addition, Rules 2, 3, and 4 include corresponding check formulae, by use of which we can avoid missing subgraphs to be counted. Their advantages will be manifested particularly in dealing with complex mechanisms.The mathematical proofs of these rules are given in the Appendices.

The Role of Programmable Calculators and Computers in Mathematical Proofs

1978 ◽  
Vol 71 (9) ◽  
pp. 745-750
Author(s):  
Stephen L. Snover ◽  
Mark A. Spikell
Keyword(s):  
Mathematical Proof ◽  
Mathematical Proofs

Data generated by computing devices may he used as an essential part of a mathematical proof.

Process Fault Diagnosis Based on Process Model Knowledge: Part II—Case Study Experiments

1991 ◽  
Vol 113 (4) ◽  
pp. 627-633 ◽  
Author(s):  
R. Isermann ◽  
B. Freyermuth
Keyword(s):  
Fault Diagnosis ◽  
Knowledge Acquisition ◽  
Knowledge Base ◽  
Process Models ◽  
Normal Operation ◽  
Problem Solution ◽  
Inference Mechanism ◽  
Process Knowledge ◽  
Fault Trees

A computer assisted fault diagnosis system (CAFD) is considered which allows the early detection and localization of process faults during normal operation or on request. It is based on an on-line engineering expert system and consists of an analytical problem solution, a process knowledge base, a knowledge acquisition component and an inference mechanism. The analytic problem solution uses a process parameter estimation, and the detection of process coefficient changes, which are symptoms of process faults. The process knowledge base is comprised of analytical knowledge in the form of process models and heuristic knowledge in the form of fault trees and fault statistics. In the phase of knowledge acquisition the process specific knowledge like theoretical process models, the normal behavior and fault trees is compiled. The inference mechanism performs the fault diagnosis, based on the observed symptoms, the fault trees, fault probabilities and the process history. This is described in Part I. In Part II, case study experiments with a d.c. motor, centrifugal pump, a heat exchanger and an industrial robot show practical results of the model based fault diagnosis.

scholarly journals Generating certified code from formal proofs: a case study in homological algebra

2009 ◽  
Vol 22 (2) ◽  
pp. 193-213 ◽  
Author(s):  
Jesús Aransay ◽  
Clemens Ballarin ◽  
Julio Rubio
Keyword(s):  
Homological Algebra ◽  
Formal Proofs

scholarly journals Language Constraints in Constructing Arguments for Mathematical Proofs

2021 ◽  
Vol 2123 (1) ◽  
pp. 012046
Author(s):  
I Minggi ◽  
Alimuddin ◽  
Sabri
Keyword(s):  
Mathematical Proof ◽  
Learning Trajectory ◽  
Mathematical Proofs ◽  
Mathematical Arguments ◽  
Step Procedure

Abstract A learning trajectory for constructing mathematical proof has been developed. The trajectory is to provide the students with a step-by-step procedure in constructing arguments for proving mathematical statements. However, in proving activities, the students were found to encounter difficulties in completing a deductive axiomatic argument constituting an accepted mathematical proof. An investigation has been conducted to explore the problems the students experienced in constructing proofs. It was found that they faced language constraints in constructing mathematical arguments. They encountered challenges in how to correctly express the mathematical statements in their constructed proofs.

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