This is a total noob question.
I am reading Naive Set Theory by Paul R. Halmos, and I'm having difficulty to understand something which seems to be trivial.
In the first chapter he writes:
If $x$ belongs to $A$ ($x$ is an element of $A$, $x$ is contained in $A$), we shall write
$x\in A$
I understand this.
Then, he write:
If $A$ and $B$ are sets and if every element of $A$ is an element of $B$, we say that $A$ is a subset of $B$, or $B$ includes $A$, and we write:
$A \subset B$
I understand this too.
Then he says:
The working of the definition implies that each set must be considered to be included in itself ($A \subset A$); this fact is described by saying that set inclusion is reflexive.
I understand this too.
But then:
Observe that belonging ($\in$) and inclusion ($\subset$) are conceptually very different things indeed. One important difference has already manifested itself above: inclusion is always reflexive, whereas it is not at all clear that belonging is ever reflexive. That is: $A \subset A$ is always true; is $A\in A$ ever true? It is certainly not true of any reasonable set that anyone has ever seen.
And this is where I don't think I understand anything. There is not more elaboration on this point in the text.
I tried to skip this but it seems it is quite fundamental for understanding what follows in the book.
Could someone explain what is meant here?
Best Answer
Whenever you come across something like this and it trips you up, you might want to look at particular examples. For instance, consider $\{4\}$. $\{4\} \subset \{4\}$, but $\{4\} \in \{4\}$ is false, since the only member of $\{4\}$ is $4$, not $\{4\}$. It may have tripped you up that "includes" and "contains" in everyday language usually qualify as synonyms. They don't here, and the terms get defined by the definitions for $\in$ and $\subset$. You might want to prove that $A \subset A$ for any set $A$ as it can get proven in a line or two.