2

Exercise 1:

Suppose we use the above rule to order the four definitions given
earlier.What are their position numbers?

Exercise 2:

Think of another characteristic of whole numbers and make up a
definition for it. Then assign it a position number in the list of five
definitions and reorder the other definitions as necessary.

Now, we note the following. For a given position number, either it possesses the
characteristic given in the definition to which it corresponds or it does not. For
example, suppose 17 has been assigned to the definition of divisible by five. Then
17 does nothave the characteristic described. On the other hand, if 14 has been
assigned to the definition divisible by seven, then 14 doeshave this characteristic.

Definition:

A position number that does notpossess the characteristic described
in the definition to which it corresponds is called Richardian.

In the above examples, 17 is Richardian and 14 is not Richardian.

Exercise 3:

From Exercise 2, you have assigned five position numbers. Which
ones are Richardian and which ones are not?

Exercise 4:

Consider the characteristic of being Richardian. Calculate its
position number in the list of six definitions—five from Exercise 2
and the definition of Richardian above. We will call this number n
so that we can refer to it.

Exercise 5:

Now, you should explain why there is a paradox inherent in the
following question: Is the number nRichardian? This paradox is
called the Richard paradox.

Exercise 6:

Propose a way out of the Richard paradox based on a theory-of-types
kind of argument. That is, is being Richardian an arithmetic property
or something else?

3.

Here is a paradox of Bertrand Russell involving sets. Some sets (taken as a whole)

are not members of themselves. For example, a set (i.e. collection) of butterflies is

not a butterfly. Other sets aremembers of themselves. For example, the collection of

all things that are not butterflies is a thing that is not itself a butterfly, and is hence a

member of itself. Now, consider the set R, where

R is the set of all sets that are not members of themselves.