Basic Set Theory by Exercises: The Class of Sets.
--
Set theory is a very fundamental and interesting part of mathematics. Typically its basics are being taught in the first semester of a course in mathematics. Everybody who uses mathematics in some form should have a basic knowledge of this subject. In this series of articles I want to provide some nice exercises together with their solutions, that I discovered at various places. It is like everywhere, learning by doing makes you a master, but not holding some certificate or title.
Set theory can lead quicker than any other mathematical discipline to ‘weird’ logical constructs. To mention one very apparent, let us ask if there exists a set that contains all sets.
Problem 1:
Show that there is no set that contains all sets.
Hint: Proof by contradiction and consider the following construct.
Solution:
You might notice that this solution contained a part that talks about ‘believing’, something that we are not used to when doing mathematics. And indeed, the consistency of set theory cannot be proofed nor refuted unless it is not true. We come back to this in a latter post since it requires some more techniques to understand.
In conjunction with this exercise it is worthwhile to mention that there is an informal definition of so called classes that allows for a class to hold all sets. It’s not hard to understand and a good reference is this.
Problem 2:
This problem looks more difficult than it actually is. Don’t let yourself getting confused by the notation! Just try to imagine the textual contents and how it matches with the given sentences from predicate logic.
Solution:
Beside of the above mentioned intention, the axiom of regularity has many more far reaching and deep consequences in axiomatic set theory.
Thanks for reading!
