Question 1. (10 marks) Consider the following argument:
Let U be the set of all sets. Define a partial ordering on U by inclusion: A ? B
iff A ? B for A, B ? U. Consider a chain C of U under this partial ordering:
C : A1 ? A2 ? A3 ? · · · . Define B = ?i?1Ai
. Clearly, B ? U and it is an upper
bound of the chain C. Hence, Zorn’s Lemma implies that U has a maximal
element, say M.
The argument is clearly wrong since M is not a maximal element: M ? {M, {M}} ? U.
Identify which step in the argument is wrong and why.