Jech, Introduction to Set Theory: Exercise 3.3 (a)

Question

I am a total beginner in set theory and currently work through Jech's "Introduction to Set Theory." In the first Chapter, Jech introduces the axioms of Existence, Extensionality, Comprehension, Pair, Union, and Power Set. However, he does not really explain what elements of sets are. Given this background, the following exercise is difficult for me to solve:

Exercise: Proof that the set of all sets does not exist. (Hint: Assume that a set $V$ exists such that $V = \{x \in V: x \notin x\}$).

What I am confused about:

What does $x \notin x$ even mean? To go deeper, two more general questions:

1. What is an element of a set, and how is it defined? Is an element of a set always a set?

2. And if an element of a set is not necessarily a set , then how can I even say something like "$x \notin x$"?

One way I tried to explain 1. and 2. to myself is as follows:

Given the axioms I mentioned above, I tried to construct a set that is not the empty set. To do so, I used the axiom of pair: Let $A, B$ be empty sets. Then, using the Pair Axiom, there must exist a set $B$ s.t. for all elements of $B,$ either $x = A$ or $x = B.$

Hence, $B = \{\emptyset\} \not= \emptyset.$ I also realise that the elements of sets created by the Pair Axiom must be sets.

Is that a solid start? How can I deduce from this that for all sets, it is true that all elements are sets? Maybe, by showing that any set that is not the empty set must have been generated via the Pair Axiom?

And once I have that, how can I interpret a statement like $x \notin x$?

Answer 1

1. What does x∉x mean?

That x is not an element of x

Ex. ∅∉∅ - As ∅ ( empty set) contains no elements

2. We think of ∈ as a psuedo-relation between sets. So a∈b, means that a is a set, b is a set, and a is an element of b.

∈ - We leave it as a primitive, undefined notion besides the fact that it is binary ( acts on only 2 sets)

Another example of a psuedo-relation is =, It is also binary, and acts between sets in set theory.

We call them psuedo-relations because, we later use sets to define relations as sets of ordered pairs. However, =, ∈ are themsleves, not sets.

One can think of them as sets in the metatheory, meta-theoretical sets. However, that is not really an area that I will go down in this answer. If you want to learn more, you should consider a Introduction to Mathematical Logic, book.

3. In Set Theory, our domain of discourse is just that of sets.

So we cannot consider sets of dogs or cats.

This restricts us, in a sense. However, it frees us, in that we can freely use =,∈ without worry.

Elements of sets aren't proved to be sets. that's a backwards way of looking at it. It's not an application of the Axioms, it's a feature of how the Language of Set Theory is Defined.

All Variables range over sets.

One might wonder how we can do mathematics without mention of numbers, functions, etc... well, it turns out our restriction to just sets isn't that big of an issue. We can, define all standard mathematical objects, as particular sets. So, our restriction to just sets, is not too prohibitive.

∈, = are ways of relating a set to another set, in Set Theory.

It's similar to if I have that a < b, and b is a number. How do I know that a is a number? Because < is a relation between numbers.

Also, If I am working in Peano Arithmetic ( The Theory of Natural Numbers), and I say x + y = z, there is no question that x,y,x are natural numbers.. because all the variables in Peano Arithmetic represent natural numbers.

You are welcome to think of {cat} as a set, however when we do mathematics we just restrict our attention to just the so called "hereditary sets". That is, sets whose elements are sets, and their elements are sets, etc... all the way down.

So, if you wish, whenever you see "x is set" in a set theory textbook, see it as a shorthand for: hereditarily, x is a set.

Note: X∈X is weird. We can consider cases like x = {x}, but they are certainly strange, so your hesitence around them is well-founded. ( Paul Cohen called x∈x a monstrosity) However, it is a perfectly valid statement in the Language of Set Theory, but you don't need to consider it really. We can show that all of the axioms of set theory hold, for sets that aren't like that. So, we can basically just work in a well-founded set theory ( one without any cases where x∈x). And so, even if cases like that exist, we can ignore them, in Standard Set Theory.