Recent Articles

Review of “The demoralization of women”

In this knol the author tries to use a tragic situation which actually exists, to support her personal conspiracy theory. The tragic situation is the participation of young women in prostitution. The conspiracy theory is that men deliberately demoralize women via prostitution and other illicit activities involving the participation of women. This seems to be […]

Bounds on the cardinality of partition

INTRODUCTION Two sets  and  are said to be disjoint if , i.e. they have no members in common. If  and  are sets, the meanings of  and  are identical. However,  means the same thing as , i.e. the set consisting of the sets  and .  means the same thing as , i.e. the set of members of  which are not members of .  means the collection of mappings of  into .  is shorthand for .  means […]

Order extensions of totally ordered sets

Definition: Let  where  is a totally ordered set. Call a nonempty subset  productive iff for any two members  and , we have either  or . If for a productive subset  there exist no  such that  for every , then call  a point-producing subset.  There is an induced relation  on the point-producing subsets:  iff there do not exist  and  such that . We say that two point-producing subsets  are equivalent iff  and , and in this case we write . Define the order […]

Well-defining real numbers

What would it mean to say that the real numbers are well-defined?

Some reputable mathematicians (and some amusing cranks) have asserted that the concept of real number is not well-defined. Much time can be wasted arguing this point without agreeing on what it would mean for real numbers to be well-defined. In this knol I will propose a definition of “well-defined real number.”

A general rating system for a two-player game

INTRODUCTION Given a set of players of a particular game, we consider the problem of assigning to each player a numerical rating, a real number intended to give information about that player’s skill. In this knol™, we show how the ratings of a population of players can be defined so as to estimate the probabilities that […]

Shapley Division of Expenses

Understanding Shapley's Approach to Fair Division of Expenses

Suppose a community is served by three airlines: A, B and C. The one available runway needs to be replaced, at a cost of 1 million dollars per 1000 feet. Airline A is a local airline with small planes requiring only 3000 feet for takeoff. B has more range and flies a few bigger planes […]

Shapley Division of Goods

Understanding Shapley's Approach to Fair Division of Goods

Suppose you have three sons, Aaron, Ben and Charles. Over the years you have accumulated a large library of rare books, and you intend to leave them to your sons when you die. There are 1000 books in the library, and you have promised Aaron (the eldest) 700 books, Ben 200 books, and Charles 100 […]

What is Internet Defamation?

Legally, defamation on a web site is materially no different than in other media

Western civilization has recognized defamation as a criminal offense for hundreds of years, and its definition has carried over into the Internet. Defamation may be defined simply as the act of making a false statement to a third party that damages the subject’s reputation. Defamation may be divided into the categories of Libel and Slander. […]

The First 561 Natural Number Pairs

As Cantor demonstrated, the natural number pairs can be placed in ono-to-one correspondence with the positive integers. The following is the beginning of such a correspondence, which of course is infinite in length. There is no duplication, and they are sorted in increasing order such that the first sort key is the sum of the two numbers, and the second sort key is the first number.

The First 529 Rational Numbers

As Cantor demonstrated, the positive rational numbers can be placed in a sequential list. The following is the beginning of such a list, which of course is infinite in length. All numbers are in lowest terms, there is no duplication, and they are sorted in increasing order such that the first sort key is the sum of numerator and denominator, and the second sort key is the numerator.