Assignment 9. The Paradoxes of Naive Set Theory

For submission

1 (a) Give an example (not already in the text) of a set that is not a member of itself.

(b) Give an example (not already in the text) of a set that is a member of itself.

2 What problem for set theory results from the set of all sets that are not members of themselves?

3 (a) What is the vicious-circle principle?

(b) How did Russell use it to escape the problem of 2?

For discussion

Not for submission

A. Is the naive notion of set really in trouble? Sets are used in many parts of quantitative science and even mathematics by researchers who pay no attention to foundational issues and likely do not even know of them. How can this be?

B. Find something prohibited by the vicious circle principle.

C. A routine construction of the numbers 0, 1, 2, 3, ... is shown in the text. They are conjured up as sets from nothing other than sets. Recall 0 is {}, 1 is { {} }, 2 is { {} , { {} } }, ... and so on. Can these really be the numbers? If not, what is missing from them?

D. Consider the two resolutions of the Russell paradox: a hierarchical construction like the theory of types and an axiomatic approach such as the ZFC system. What are the good and the bad of each? Which do you prefer?

E. These resolutions of the Russell paradox do some violence to our intuitive notions. Is there a less destructive solution?

F. Does the Banach-Tarski paradox provide sufficient basis for excluding the axiom of choice from set theory?