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?
