## What They Don’t Teach You About Sets 1

Resurrected this blog (again) for posting my postgraduate lectures/talks here. Here are the notes from LuFTER 1/2011 February 9, 2011. The Lunchtime Foundational Theory Expositions and Ruminations will consists of mainly lectures to my postgraduate students and occasionally research seminars, proposals etc.

The present ongoing lectures are meant to introduce mathematical structures in theoretical physics with perhaps Nirmala Prakash’s book as a guide of topics to be covered (but not necessarily adhering to it). The book itself is skewed towards topics for string theory but we will also cover other topics. The first topic will be on set theory. The mischievous title of the lecture is taking cue from Devlin’s book entitled “The Joy of Sets”, which is referenced.

Relationships and Sets

In most (conventional) physics, often use very rich mathematical structures (e.g. differential geometry) from the outset, which assume many things. Advances in foundational theories (e.g. quantum gravity) tend to relook at the usage basis of these structures and either generalize them or opt for primitive structures. So, what would be considered the most primitive structure? Set theory seems to fit the role with its common usage as a mathematical language. However, usually physicists are only exposed to very elementary ideas of set theory and perhaps not see its full power often only found in more advanced mathematics course. We hope to remedy this a little. In fact the deeper usage of set theory is really the following:

• understanding the infinite
• foundational subject matter of mathematics
• common mode of reasoning

So what is a set? Its fundamental idea is simply the ability to regard a collection of objects as a single entity (the set). This sounds circular. In fact, in set theory, the undefinables are really the notion of a set and the relation “is an element of”. We introduce notation:

• $x \in X$” which means “$x$ is an element of $X$“;
• $x \notin X$” which means “$x$ is not an element of $X$“.

In forming sets, one can either

• enumerate the elements or members of the set e.g. $X = \{ x_1 , x_2 , \cdots , x_n , \cdots \}$; or
• describe by using some property $P$ e.g. $X = \{ x : P(x) \}$, which is the set of all $x$ for which $P(x)$ holds. Example: $\mathbb{C} = \{ z : z \textrm{ is a complex number} \}$.

Using normal sentences to describe a set may not be best. Better, use logical statements. For this, we introduce the logical notations:

• $\Rightarrow$ means “implies”;
• $\Longleftrightarrow$ means “if and only if”;
• $\neg$ means “not”;
• $\wedge$ means “and”;
• $\vee$ means “or”;
• $\forall$ means “for all”;
• $\exists$ means “there exists”.

With these, one can start building logical statements for the set building. Some examples logical statement are given below (which also shows that some logical operations can be “derived” from others.

Example 1: $P \Longleftrightarrow Q$ is the same as $(P\Rightarrow Q) \land (Q\Rightarrow P)$
One can build the truth table to show this is true.

$\begin{array}{c|c|c} P\Rightarrow Q & Q\Rightarrow P & (P\Rightarrow Q)\land(q\Rightarrow Q)\\ \hline \textrm{False} & \textrm{False} & \textrm{False}\\ \textrm{False} & \textrm{True} & \textrm{False}\\ \textrm{True} & \textrm{False} & \textrm{False}\\ \textrm{True} & \textrm{True} & \textrm{True}\end{array}$

Note that we could have build the truth table out of atomic statements instead of compound ones but we leave this for the reader to elaborate.

Exercise 1: Show that $P\Rightarrow Q$ is the same as $(\neg P)\vee Q$.

Example 2: $P\vee Q$ is the same as $\neg((\neg P)\land(\neg Q))$.
Now we build the truth table from the atomic statements for illustration.

$\begin{array}{c|c|c|c|c|c} P & Q & \neg P & \neg Q & (\neg P)\land(\neg Q) & \neg((\neg P)\land(\neg Q))\\ \hline \textrm{False} & \textrm{False} & \textrm{True} & \textrm{True} & \textrm{True} & \textrm{False}\\ \textrm{False} & \textrm{True} & \textrm{True} & \textrm{False} & \textrm{False} & \textrm{True}\\ \textrm{True} & \textrm{False} & \textrm{False} & \textrm{True} & \textrm{False} & \textrm{True}\\ \textrm{True} & \textrm{True} & \textrm{False} & \textrm{False} & \textrm{False} & \textrm{True}\end{array}$

This example is illustrative of the fact that one does not need all the logical operations; here, the “or” operation has been replaced by a combination of a “not” and an “and”. It is in fact well known that the “nand” gate (combining “not” and “and”) is a universal gate for classical computations.

In handling or consructing sets abstractly, it is important to ponder on Quine’s dictum “No entity without identity”. How would one know that two so-called abstract entities (sets) are not one and the same? Here, we state the following axiom of extensionality telling us when two sets $X$ and $Y$ are the same:

$X=Y \iff \forall x (x\in X) \Longleftrightarrow (x\in Y)$ .

Thus, for example $\{a,b,c\} = \{c,a,b\}$. In this regard, there is one important set that we need to construct, namely the empty set $\emptyset$ or $\{\}$. It can be defined as

$\emptyset = \{ x: x \neq x \}$ .

Note from this definition, in principle we could have started with $x$ coming from different sets, say $X$ giving $\emptyset_X$ and $\chi$ giving $\emptyset_\chi$. But by virtue of axiom of extensionality, the empty set is unique, being the set with no elements (Exercise 2: Prove this).

Exercise 3: Using results of exercise 1, prove the statement $x\in\emptyset \Rightarrow P(x)$ is true for all $x$.

An easy way to define more sets is to consider another set relation, namely subsets. We define $Y$ is a subset of $X$, written as $Y\subseteq X$, if and only if every element of $Y$ is an element of $X$ i.e.

$Y\subseteq X\Longleftrightarrow\forall x((x\in Y)\Rightarrow(x\in X))$ .

Note that in this case $X$ is also called superset of $Y$ i.e. $X\supseteq Y$. In both these relations, it is possible that $X=Y$. But if you would like to consider otherwise i.e. considering $Y$ is a proper subset of $X$, then we write $Y\subset X$ where

$Y\subset X\Longleftrightarrow(Y\subseteq X)\land(Y\neq X)$ .

Another important set concept is the idea of a power set $\mathcal{P}(X)$ of the set $X$, which is the set of all subsets of $X$ i.e.

$\mathcal{P}(X) = \{ Y :Y\subseteq X\}$ .

We will illustrate the idea of power sets in a minute but before that one introduces another set relation namely the cardinality. Cardinality is a measure of the “size’ of the set particularly by looking into the “number” of its elements. We denote the cardinality of a set $X$ by $\sharp(X)$ or $\textrm{card}(X)$.

Example 3: Consider the set $X=\{\Diamond\}$. Its cardinality is $\sharp(X)=1$. The power set of $X$ is $\mathcal{P}(X)=\{\emptyset,\{\Diamond\}\}$ and hence $\sharp\mathcal{P}(X)=2$. One could proceed further to find the power set of $\mathcal{P}(X)$ itself:

$\mathcal{P}(\mathcal{P}(X))=\{\emptyset,\{\emptyset\},\{\Diamond\},\{\emptyset,\{\Diamond\}\}\}$ ;

with $\sharp\mathcal{P}(\mathcal{P}(X))=4$. In fact, one can iterate this $n$ times and find $\sharp\mathcal{P}^{(n)}(X) = 2^n$. It is interesting to note that the empty set played a role in increasing the cardinality of the nested power sets. We will later show that one can do better than this to build up what we know as numbers.

To end this part, I add a cautionary note that not all collections of objects can form a set. Consider the Russell set:

$R=\{x:x\notin x\}$ .

Let’s ask: does $R$ satisfies the property given? If yes, then $R\notin R$ but then by the set definition $R\in R$ – giving a contradiction. Suppose the answer is no then, which means $R\in R$. However the set definition implies $R\notin R$ – again, a contradiction. This is essentially known as the Russell’s paradox. It’s resolution? Perhaps, it is too much to impose $R$ to be a set, but $R$ is said to be a class (will not elaborate here). Essentially the idea is to differentiate the use of a symbol and the meaning of the symbol. Alternatively, one could axiomatize set theory using Zermelo-Fraenkel axioms, which is beyond the scope of the lecture.

References

I list here some of my reading materials without attempting to show where I have used them. Go read them, if you have interest.