Explanation of the Mathematics

There are nine universally accepted axioms in Zermelo-Frankel set theory. There is a tenth which is sometimes used called the axiom of choice which will be addressed towards the end of this section. Each of the nine main axioms will be written in the notation Bertrand Russell helped to develop and then will be addressed briefly in normal English below. See the glossary for an explanation of the symbols involved. There is a tenth which is sometimes used called the axiom of choice which will be addressed towards the end of this section.

The first axiom is called the axiom of extensionality. In formal notation it looks like this: \[\forall x \forall y[\forall z(z\in x \leftrightarrow z\in y)\rightarrow x=y] \] In more conventional language this axiom says that if two sets are composed of the exact same elements, then those two sets are equal. This axiom defines equality between sets.

The second axiom is defines what is known as the null set. It is written as follows: \[\exists x \neg \exists y(y\in x)\] This means that there exists a set which has no elements meaning that it is empty. The first axiom can be used to show that this set is unique, or that there is only one empty set. The empty set, also called the null set, is commonly denoted as {} (a set with nothing in it) or by the symbol \(\emptyset\).

The third axiom creates the notion of a pair set, which is that given two sets \(x\) and \(y\) there exists a set whose only elements are \(x\)and \(y\) denoted as \(\{x,y\}\) \[\forall x\forall y \exists z \forall w(w\in z \leftrightarrow w=x\vee w=y)\] Combined with the first axiom, it can also proven that this pair set is unique.

The fourth axiom says: \[\forall x \exists y \forall z[z\in y \leftrightarrow \forall w(w\in z \rightarrow w\in x) ]\] This defines what is known as a power set. The power set of a set \(x\) is the set of sets whose elements are in \(x\), or the power set of \(x\) is the set of all the subsets of \(x\). Similarly to before, each unique set has a unique power set.

The axiom of unions is next which says: \[\forall x \exists y \forall z[z\in y \leftrightarrow \exists w(w\in x \wedge z\in w)]\] This means that for any set\(x\) there is a set \(y\) which is composed of all the elements of all the elements of \(x\). In other words, if \(x\) is a set of sets, the union of \(x\), denoted as \(\bigcup x\), is the set composed of all the elements of the sets contained in \(x\).

The next axiom is the axiom of infinity which is a bit more complex saying: \[ \exists x[\emptyset \in x \wedge \forall y (y\in x \leftarrow \bigcup \{y,\{y\}\}\in x)]\] This axiom allows for the construction of the natural numbers as an infinite sequence of sets. By this axiom \(0 = \emptyset, 1 = \{\emptyset\}, 2 = \{\emptyset,\{\emptyset\}\}\) and so on, where each successive number is the set containing all of the sets that preceded it. This is one example of how numbers can be defined using sets and it helps to show just how foundational set theory is to math as a whole.

The next axiom is one that was touched on earlier in the paper. It is call the separation schema: \[\forall u_1\dots\forall u_k[\forall w\exists v \forall r(r\in v \leftrightarrow r\in w \wedge \psi_{x,\hat{u}}[r,\hat{u}])]\] This is the axiom that says a set can be constructed from all the elements of a given set that meet a certain criterion–denoted in this case as \(\psi\). This axiom is what resolves Russell's Paradox as it forbids creating the set \(R\) which both was and was not a member of itself.

The next axiom is somewhat similar to the separation schema and is known as the replacement schema: \[\forall u_1\dots \forall u_k[\forall x \exists!y\phi(x,y,\hat{u})\rightarrow \forall w\exists v \forall r(r\in v \leftrightarrow \exists s(s\in w \wedge \phi_{x,y,\hat{u}}[s,r,\hat{u}]))]\] This says that if we have a set \(x\) that is related to a unique set \(y\) by some function or criterion \(\phi\), we can form a new set out of all the elements of a third set \(w\) that are related by \(\phi\).

The final axiom is known as the axiom of regularity: \[\forall x[x\neq \emptyset \rightarrow\exists y(y\in x \wedge \forall z(z\in x \rightarrow \neg (z\in y)))]\] It ensures that every set is what is known as "well founded." This rules out the creation of sets that are infinitely circular and sets that are created by infinitely descending chains.

The black sheep of the ZF axioms is known as the axiom of choice. It essentially says that given and infinite set of sets, a new set can be constructed by taking one element from each of the infinite sets. This axiom is very useful in some specific circumstances, however it has some strange consequence. One is that it allows for the creation of sets which are not well defined, which is something mathematicians do not like very much. Another is that it creates a whole slew of seemingly paradoxical proofs, such as the Banach-Tarski paradox which says that a sphere can be disassembled into infinitely many pieces, which can then be reassembled into two new spheres that are identical to the original. As a result of these strange consequences, its inclusion in the Zermelo-Frankel axioms is still a matter of debate to this day.

Below is a short video that goes into a little more detail about the axiom of choice.