Is $\aleph_0 = \mathbb{N}$? Some very wise people here have just told me that  $\aleph_0 = \mathbb{N}$, i.e. that the cardinality of the set of natural numbers is just the set of natural numbers itself. Is this now the general consensus in mathematics, real analysis in particular? Or have I got completely the wrong end of the stick as usual?
There's nothing deep going on here. Its just that:


Its often convenient to identify $\mathbb{N}$ with the least infinite ordinal $\omega$.
Its often convenient to identify each well-orderable cardinal number $\kappa$ with the least ordinal $\alpha$ such that $|\alpha| = \kappa$. This is called the von Neumann cardinal assigment.


Under these identifications, we find that $\mathbb{N} = \omega = \aleph_0 = |\mathbb{N}|.$ I wouldn't read too much into it though.
You have N≠ℵ0, Infact one is the set (N), while ℵ0 is a transfinite number.
Asking  ℵ0=N, is somewhat.. you are asking.. Is 2={1,2} ? which is obviously not.