First let me make a couple of general remark. First, what does it mean to show that an object $x$ satisfying property $\psi$ does not exist? It means that assuming $\exists x\;\psi(x)$ leads to a contradiction. Contradiction is often denoted by the symbol $\bot$, and it can be thought of, for example, as the conjunction of some formula and its negation: $\chi\land\neg\chi$. The important point is that a contradiction is always false and implies everything, as was discussed in post #7. So if some theory implies a contradiction, then it is useless because it implies every single statement. In ordinary mathematics, we would like to derive only true statements.
Second, once it was discovered that not all collections are sets (since if they were, that would lead to a contradiction), we must be careful which collections of objects we call sets. We could take an arbitrary formula $\varphi(x)$ with one free variable (maybe your textbook calls it a
type judging by post #1) and consider all objects $x$ that satisfy it. These objects do not automatically form a set. How do we say formally that they do? We say that there is some object $z$ such that those $x$ that satisfy $\varphi(x)$ and only they are in the relationship $\in$ with $z$. That is, the collection of object satisfying $\varphi(x)$ is a set if
\[
\exists z\;\forall x\;x\in z\leftrightarrow\varphi(x).
\]
So from the standpoint of set theory, to form a set is to exist. The following two statements mean the same:
- There exists an object $z$ that contains all $x$ satisfying $\varphi(x)$
- All $x$ satisfying $\varphi(x)$ form a set.
(There are some variations of set theory where not everything is a set: there are also so-called
classes.) Please think about this and make sure you understand this definition.
Combining these two remarks, what does it mean that the set of all sets does not exist, or, equivalently, that all sets do not form a set? It means
\[
(\exists z\,\forall x\;x\in z)\to\bot.\qquad(1)
\]
And indeed, assuming $\exists z\,\forall x\;x\in z$, we take this $z$ and using the subset axiom form a set $\{x\mid x\notin x\}$ and use it to derive a contradiction.
evinda said:
So, can't we use the definition of the generalized intersection, in order to show that $\cap \varnothing$ is not a set?
Yes. Let $A$ be fixed. We take the property $\varphi(x)$ from the definition of the generalized intersection: namely, $\varphi(x)$ is $\forall y\;y\in A\to x\in y$. We want to show that when $A=\varnothing$, all $x$ satisfying $\varphi(x)$ do not form a set, i.e.,
\[
(\exists z\;\forall x\;x\in z\leftrightarrow\varphi(x))\to\bot.\qquad(2)
\]
Well, suppose
\[
\exists z\;\forall x\;x\in z\leftrightarrow\varphi(x)\qquad(3)
\]
But $\varphi(x)$ is $\forall y\;y\in\varnothing\to x\in y$. Since $y\in\varnothing\leftrightarrow\bot$, we have $(y\in\varnothing\to x\in y)\leftrightarrow\top$, where $\top$ denotes a sentence that is always true ($\top$ can be thought of as $\neg\bot$). In other words, $y\in\varnothing\to x\in y$ and its universal closure are always true. So (3) says
\[
\exists z\;\forall x\;x\in z\leftrightarrow\top
\]
or simply
\[
\exists z\;\forall x\;x\in z.
\]
Then we use (1) to derive $\bot$. Thus, (2) is proved.