scholarly journals The Principle of Mathematical Induction Applied to the Generalized Model for the Economic Design of X-Control Charts

2012 ◽  
Vol 02 (04) ◽  
pp. 236-240 ◽  
Author(s):  
Mohamed E. Seliaman ◽  
Salih O. Duffuaa
Keyword(s):  
Control Charts ◽  
Mathematical Induction ◽  
Economic Design ◽  
Generalized Model ◽  
Principle Of Mathematical Induction

Weaknesses of the Economic Design of Control Charts

Technometrics ◽  
1986 ◽  
Vol 28 (4) ◽  
pp. 408 ◽  
Author(s):  
William H. Woodall ◽  
Thomas J. Lorenzen ◽  
Lonnie C. Vance
Keyword(s):  
Control Charts ◽  
Economic Design

Economic design of EWMA control charts using regions of maximum and minimum ARL

2005 ◽  
Vol 37 (11) ◽  
pp. 1011-1021 ◽  
Author(s):  
J. Carlos García-Díaz ◽  
Francisco Aparisi
Keyword(s):  
Control Charts ◽  
Economic Design

Arithmetic, philosophical issues in

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Natural Number ◽  
Decision Procedure ◽  
Mathematical Induction ◽  
Universal Quantifier ◽  
Number Concept ◽  
Special Character ◽  
The Status ◽  
Proof By Induction ◽  
Philosophy Of Arithmetic ◽  
Principle Of Mathematical Induction

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

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