3

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.

Question: Is R a member of itself, or not? Can you propose a way out of this paradox?