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?
 A: 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.
A: The objects in set theory are sets. Only sets in $\sf ZFC$ and its related theories. This means that if you want to interpret a mathematical object in set theory you need to assign it a set. 
Of course you are free to assign to it any set that you wish, as long as you have the axiom of replacement set theory is more or less interpretation agnostic (in the sense that it proves that two interpretations are generally exchangeable). But still you need to pick some interpretation, to at least show one way of doing this is possible.
So just like the standard method for interpreting ordered pairs is via the definition by Kuratowski, the standard way of assigning cardinals to well-orderable sets is by picking the least ordinal of that cardinality as a representative. So we define the cardinals for infinite [well-orderable] sets by transfinite induction, 


*

*$\aleph_0=\omega$ (the least infinite ordinal which exists from the axiom of infinity, power set and separation),

*$\aleph_{\alpha+1}$ is the least ordinal whose cardinality is not smaller than $\aleph_\alpha$ (and such ordinal exists by Hartogs theorem),

*If $\delta$ is a limit ordinal, then $\aleph_\delta=\sup\{\aleph_\alpha\mid\alpha<\delta\}$ (which exists from replacement and union). 


There is nothing more to it, and it's just one possible way of interpreting these cardinals. You can also represent cardinals in other ways (e.g. Scott's trick give you Scott cardinals, but it will fail you if choice and foundation fail, e.g. if you allow urelements and choice failed), or choose not to represent the cardinals internally and work, awkwardly I might add, with equivalence classes of sets. 
It's up to you. But the consensus of what object $\aleph_0$ is exists only in set theory, and not in analysis. So asking analysts for their opinion is quite irrelevant for this question. 
And so, if you identify in set theory $\Bbb N$ with $\omega$ you get that $|\Bbb N|=\Bbb N$. Whether you choose to identify the natural numbers with the finite ordinals is up to you, but most set theorists [working in $\sf ZF$ and the like] do. 
At the end of the day you can argue whether or not functions are sets of ordered pairs, and whether or not $0$ is a natural number or not. At the end of the day this is just missing the point of a foundational theory. To give a foundation to mathematics. We have yet to find a theory that does it as well as $\sf ZFC$ (in my opinion anyway, some might argue differently).
If you're unhappy with the current foundations of mathematics, there are only two options: (1) study it and learn to accept it's supposed flaws; or (2) find a different foundation.
