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Mathematics 5
Winter Term 2000
The World According to Mathematics

Dwight Lahr and Josh Laison

Friday Discussion: Week #4

Part 1: Linguistic Paradoxes (continued from last week)

1.

Liar's Paradox:
Explain why the following sentence is self-contradictory, neither true nor false:

This statement is false.

2.

The Richard Paradox:
Here is a logical paradox formulated by Jules Richard (a Frenchman) in 1903:

Suppose we want to make a list of the properties of the counting numbers, that is,
of the whole numbers. First, we would have to list the
characteristics—characteristics such as even, odd, multiple of 7, or perfect square.
Then we would have to write out the definitions of the characteristics. For
example, we might give the following definitions for the characteristics we have
just presented:

A whole number is evenif and only if it is divisible by two.

A whole number is oddif and only if it leaves a remainder of one when
divided by two.

A whole number is a multiple of sevenif and only if it is divisible by seven.

A whole number is a perfect squareif and only if it is the product of a whole
number with itself.

Next, we would have to decide a rule for ordering the definitions in the list. That
is, how would we decide which to list first, or which to list second, and so on?

A simple rule for ordering the definitions is based on the number of letters in a
definition. Thus, we count the number of letters in all the definitions and assign
position number 1 to the definition with the smallest number of letters. We assign
position number 2 to the definition with the next smallest number of letters, and
proceed in this way to assign positions to all of the definitions. If two or more
definitions have the same number of letters, we assign their positions on the basis
of the alphabetical order of the letters in each. Therefore, each definition will have
its own position number.