1. 역사적 서론Historical Introduction

• If $A$ is a set, its elements may themselves be sets. For instance, consider a set of lines. Then each line itself is a set of points.
• There is a possibility that $A$ may be an element of itself, i.e. $A \in A$. For instance, consider the set of all sets. Then obviously, the set of all sets is an element of the set of all sets.
• Let $S$ denote the set of all sets that are not elements of themselves. There are two possibilities that $S \in S$ or $S \notin S$.
  • (Case 1) Assume that $S \in S$. Then by the definition of $S$, $S$ is not an element of $S$, which is a contradiction to the assumption $S \in S$.
  • (Case 2) Assume that $S \notin S$. Then $S$ is a set which is not an element of $S$ itself. Therefore $S \in S$ and get a contradiction.

• Let $T$ be the set of all natural numbers that can be described in fewer than twenty words of the English language.
• Since there are only a finite number of English words, there are only finitely many combinations of fewer than twenty words. Thus $T$ is a finite set.
• Since $T$ is a finite set, there exists a natural number $N$ which is greater than all the elements of $T$. Thus by the definition of $T$, $N$ is not an natural number which can be described in fewer than twenty words of the English language.
• However, $N$ can be described with seventeen words in the way that “$N$ is an natural number which cannot be described in fewer than twenty words of the English language”. Therefore by the definition of $N$, $N \in T$, which leads to the contradiction that $N > N$.

There appeared non-Euclidean geometries such as hyperbolic geometry and elliptic geometry by replacing the parallel postulate.

• There is one primitive relation denoted by $\in$, that is, $\in$ is a undefined notion and it is understood as a membership relation intuitively.
• The expression $x \in Y$ is to be read “$x$ is an element of $Y$”.
• The variables $x$, $y$, $z$, $X$, $Y$, $Z$, $\ldots$ which are placed to the right or to the left of the symbol $\in$ are called “sets”.
• Regard elements of every set to be sets themselves. In other words, every set is considered to be a set of sets. Usually, small variables $x$, $y$, and $z$ denote elements of sets and capital variables $X$, $Y$, and $Z$ indicate sets. However, due to this identification, all expressions such as $x \in Y$, $X \in Y$, $Z \in x$ are meaningful.

How does the axiom of selection overcome Russell’s paradox? Note that the set of all sets which are not elements of themselves can be symbolized as $\{x : x \notin x\}$. However, the existence of the set $\{x : x \notin x\}$ is not guaranteed by the axiom of selection. The best candidate with the existence for the set of all sets which are not elements of themselves is given by

where $A$ is a set which can be shown to exist. Denote $S = \{x \in A : x \notin x\}$. Then Russell’s argument is not a paradox anymore. Indeed, assume $S \in S$. Then $S \in S$ implies that $S \notin S$. It is a contradiction. Therefore

Next we claim that $S \notin A$ and use the reduction to absurdity. Assume that $S \in A$. Then by (1.1), we have $S \notin S$ and $S \in A$. By the definition of $S$, we conclude that $S \in S$, which is contradiction to (1.1). Therefore, we reach the conclusion that $S \notin A$ and there is no paradox. In other words, Russell’s argument proves that for any set $A$, the set $\{x \in A : x \notin x\}$ cannot be an element of $A$.

Zermelo considered that Russell’s set such as the set of all sets that are not elements of themselves is “too large” and excluded too large sets to avoid the paradox.

Here is an another natural question. Does Zermelo’s idea overcome Berry’s paradox (Paradox 1.4)? The answer is “no”. Recall that $T$ is the set of all natural numbers that can be described in fewer than twenty words of the English language. Let $\mathbb{N}$ be the set of all natural numbers and

Then $T = \{x \in \mathbb{N} : S(x)\}$. Recall that since there are only a finite number of English words, there are only finitely many combinations of fewer than twenty words. Thus there exists a natural number $n \in \mathbb{N}$ such that $n > x$ for all $x \in T$. However, $n$ can be described with seventeen words in the following way:“$n$ is an natural number which cannot be described in fewer than twenty words of the English language”. Definitely, it is a paradox. Zermelo did not give a satisfactory answer to solve Berry’s paradox but suggested the idea that “we must place restrictions on the type of “conditions” $S(x)$. That is, the set $\{x \in A : S(x)\}$ can be formed only if $S(x)$ is meaningful for every $x \in A$. He also raised the questions: How do we understand “meaningful”? How do we determine whether $S(x)$ is meaningful?

• Zermelo used only one undefined notation called the membership relation $x \in Y$. Von Neumann added one more undefined notion called a class which is more general than a set. Von Neumann distinguished between two kinds of classes, namely “elements” and “proper classes”. Here “elements” are defined to be those classes which are elements of classes and “proper classes” are defined to be those classes which are not elements of any class.
• Zermelo excluded all too large sets. On the other hand, Von Neumann allowed too large sets to exist but they are not permitted to be elements of classes.
• Von Neumann suggested an axiom called “the class axiom” which replaces Zermelo’s axiom of selection

We will study an axiomatic theory based on Von Neumann’s idea in this lecture. You can determine by yourself if you accept axioms or not. Sometimes, it could make a meaningful progress by denying axioms. For instance, denying the parallel postulate in Euclidean geometry initiated developments of non-Euclidean geometries. However, keep in mind that most modern mathematical theories (especially analysis theories) are based on axiomatic set theories. Hilbert once wrote “We will not be expelled from the paradise into which Cantor has led us”.