My question is really close to this one Meaning of Fock Space
In fact I don't understand why we define Fock space as :
$$\Gamma(H):=\mathbb C\oplus H\oplus H^2\oplus\cdots\oplus H^N\oplus\cdots$$
Indeed, I would define the Fock space as a Hilbert space which is an infinite tensorial product of Hilbert space.
Because for me to write :
$$ |01\rangle = |0\rangle|1\rangle $$ is physically the same state as $$ |1\rangle$$
Indeed (for me but I may be wrong), to say "first state is vacuum and second one is with one particle" is physically the same as saying "we have one particle"
Where is my mistake?