In a previous post Axioms: Sets (Russell’s Paradox) we discussed axiomatic systems and we explored an example of why we might need one. Here we will view the axiomatic system that “fixed” naive set theory. This material will be a little heavy so in between each axiom We will see a little intuition to better help understand the system. This list is known as the Zermelo-Fraenkel axioms, and is the most widely used system in modern mathematics. We cannot however just dive right in. As seen in the tree analogy, we need to know our undefined terms and definitions. Recall that we are assuming the logical system is understood.
Our undefined terms will be set and the binary relation of membership, when we write it will read as “ is a member of . We will use capital letters and to represent sets.
(1) AXIOM (of Extensionality) : For every and for every and for every if implies and is implies then . This can also be stated as
We have here what it means for two sets to be equal. That is, two sets are equal if they contain the same members. The next axiom guarantees the existence of subsets in a way.
(2) AXIOM (Schema of Comprehension): Suppose is a property of z. For every there exists so that if and only if and is true. We see this as,
DEFINITION: The set is a subset of the set if and only if implies , in this case we write .
Suppose we are given . Then for any we are guaranteed, by the Axiom Schema of Comprehension, that there is some whose elements all satisfy . But since there is nothing that satisfies we conclude contains no elements. Notice, as of right now, our depends on a given . Suppose there is some other set, with no members. Then the statement is vacuously true as the antecedent is always false. We use the same reasoning to see that the statement . We conclude then by the Axiom of Extensionality that . (this is actually a proof of the proposition: “There exists a unique set with no elements”) Hence, the set with no elements exists and is unique, so we obtain the following definition.
DEFINITION: The set with no elements will be called the empty set and will be denoted, .
Notice the empty set is a subset of every set since the statement is vacuously true. The next axiom allows us to “build” sets in a way. It tells us that we can find a set containing two given sets.
(3) AXIOM (of Pairing): For any and , there exists so that and .
Notice that this axiom does not give us that contains only and . For this we have the following proposition.
PROPOSITION: For any pair there exists a unique set containing exactly and as its only members.
Proof: Let be a set containing and by the Axiom of Pairing. Let be the statement “ or “. By logic and the Axiom Schema of Comprehension we find that exists. We conclude that is unique by the Axiom of Extensionality.
From the Axiom of pairing we also get for any set the set is valid. And, we get that . This will come in handy later but first,
(4) AXIOM (of Union): For any set there exists a set whose elements contain all of those elements in the members of .
Let and be sets. We can use the Axiom of Pairing to obtain a set . Then the axiom of union guarantees exists. But what about intersections? We have actually had the definition of intersection all along. Let be the statement ““. Then we obtain by the Axiom Schema of Comprehension, there is a containing elements of for which is true. But this makes exactly the set of all for which and . Hence . We obtain the following definitions.
DEFINITION: Let and be sets. The union of and is the set where if and only if or $latex z\in y.
The intersection of and is the set for which if and only if and .
(5) AXIOM (of Replacement): Let be a property such that for every there is a unique so that holds. Then for every there exists a such that for every there is a so that is true.
We can use this axiom along with the Axiom Schema of Comprehension to show that there is a set containing exactly the that satisfy . We can show it is unique by the Axiom of Extensionality. (Try this!)
(6) AXIOM (of Infinity): There is a set so that and whenever , .
From this we find , , and , and so on. Let be the property that is one of these sets. Then we define the set to be the set of all so that is true. This is no mistake for this set is exactly what we will call the natural numbers. In this way we replace these sets with notation, There are many more intricacies of this process (such as uniqueness) that we will not delve into in this post. For our subject is that of Zermelo-Fraenkel axioms. So we shall proceed from this topic.
(7) AXIOM (of Power Set): For every set there exists a set so that implies .
From this axiom (and the others we have used) we can define the unique power set of a set (prove it!).
DEFINITION: Let be a set. The power set of , , is the set which contains exactly all of the subsets of .
The next axiom will complete the list for Zermelo-Fraenkel (ZF) set theory.
(8) AXIOM (of Foundation): Every nonempty set (a set so that ) contains a member so that .
The Axiom of Foundation tells us that if is a set then . (Why?) Then we see that in this system Russell’s Paradox is not possible. The final axiom is the Axiom of Choice. This is a very controversial axiom, so in the interest of space we will simply state it in this post.
(9) AXIOM (of Choice): Suppose is a collection of nonempty sets. There exists a function that assigns to each set an element .
The axiomatic system formed by (1)-(9) is known as ZF+C or ZFC and is the most widely used and accepted system in mathematics today. In the next post we will explore the Axiom of Choice and its equivalents. We have seen that, by assuming the Axiom of Choice, it is possible to split a sphere into two spheres that are congruent to the original. The next axiomatic system we will look at is Bernays-Gödel-von Neumann axiomatics as presented in Dugundji’s Topology. This will be the last set theory system we will explore for a while.