I figure that if you’re going to take the time to do and/or think about the exercises, I might as well provide some solutions so that you can have the confidence of knowing that you’re doing it right!
Exercise 1) Try to define some crazy sets, and try to determine if two sets are the same or not. For example, can we define the set of all ideas? What about the set of all thoughts? What makes two different thoughts distinct? (Because remember, elements in a set need to be distinct!) Is the set of all ideas equal to the set of all thoughts? It depends on how you define them! Don’t get too worked up about this stuff yet though—I’m just trying to show you that “mathematical thought” is indeed much broader than you could have ever imagined!
Solution: Well, this exercise doesn’t really have an exact “solution”, since you’re really just supposed to think of some examples yourself, but I’ll get the ball rolling. Some crazy sets that one could define are the following: the set of all toenails (crazy just because it’s gross), the set of all universes (each universe is a single element—is there more than one element? are there infinitely many elements?), and the set of all sets! The set of all sets is simply the set whose elements are themselves sets, and moreover it contains all of the sets. We’ll discuss this wacky set a whole lot more here and in lesson 5.
Now, the part of this exercise asking about the set of ideas vs. the set of thoughts is really there to show the importance of having rigor in one’s definitions. We need to define what we mean by “a thought” and “an idea” before we can ask whether or not these two sets are the same. Let me make the following definitions (although it should be clear that there is no right or wrong definition, nor any exact, mathematical definition). A thought is any reaction, feeling, emotion, opinion, observation, or any other mental process, either verbalized or not, by any sentient being anywhere. Now, this is of course a complex issue because now we’d have to define every term in that definition—namely “feeling”, “emotion”, “sentience”, etc.—but let’s just go with this. An idea, on the other hand, is any mental facility brought forth by any sentient being from the motivation to accomplish any kind of goal.
With these (highly inexact) definitions, it is clear that the set of all ideas and the set of all thoughts are not the same, because there are several thoughts that are not ideas. It is the case, however, that any idea is indeed a thought, simply because an idea is a mental facility (of a particular kind). For example, I just thought “I am hungry”, which is therefore a thought. That thought is not an idea, however, because there is absolutely no goal that motivated its existence. Now I have thought “I should eat”, which is an idea, because it was motivated by my wanting to accomplish the goal of no longer being hungry.
Clearly there is a lot of wiggle room here, and this is open to much (welcomed) debate and scrutiny. The point that I’m trying to show, however, is that we sometimes need to work really hard to make definitions that are precise enough for us to actually be able to work with. Most mathematical definitions are easier than these two!
Exercise 2) Can you think of a set that contains itself as an element? (If you can’t, don’t worry, this one is tricky and it’s something we’ll be addressing more in a little while).
Solution: If you haven’t already guessed it, an answer is actually in the solution to exercise 1 above. After a little thought, we notice that the set of all sets contains itself! It is, after all, a set, and since it’s the set of all sets, it must have itself as an element. On the face of it there’s nothing logically wrong with a set containing itself, but as we explore sets of sets more deeply in lesson 4 and lesson 5 we’ll find that there is indeed something very wrong with this fact. Nonetheless, it works to solve this problem and so we’ll go with it. If this doesn’t make all the sense in the world at the moment, don’t worry, because we’ll be addressing this issue more slowly in the coming lessons.