I'm studying Moretti's introduction to Spectral Theory and Quantum Mechanics in which he makes the following claim: (as far as I understand, I have reworded this to avoid discussing the irrelevant parts)
Suppose $M$ is an infinite set. Then $\text{card } M = \text{card } M \times \mathbb{N}$.
Is this true? I realized that this is equivalent to stating that all infinite sets can be partitioned into a countable number of sets, whose partitions all have the same cardinality as the set itself. How can this be proven?
The claim above is easily verifiable on sets such as $\mathbb{N}$ (partition it into multiples of primes) or in $\mathbb{R}$ (partition it into reals in the interval $(n, n+1]$ for $n \in \mathbb{Z}$).
This is because then, we can easily generate a surjection from $M$ to $M \times \mathbb{N}$ by mapping the $i$th partition of $M$ (which has a cardinality of $M$) trivially to the elements in $M \times \mathbb{N}$ of the form $(., i)$.
A surjection from $M \times \mathbb{N}$ to $M$ is trivial. So by Schroder-Bernstein they must have the same cardinality.
Possible idea: I think for any infinite set $M$ we know $\text{card } M = \text{card } M \times M$ by the axiom of choice. source
Since $M \times M$ can be easily partitioned into two parts each with cardinality $M$, then we now $M$ can be partitioned into two infinite sets with the same cardinality as M$.
We pick one of these two partitions and partition it again to two parts. We keep doing this and construct a countably infinite partition of sets with the cardinality of $M$.
Best Answer
You've missed a point. Since $M$ is infinite, it contains a countably infinite subset. Moreover, $M\times M$ can be partitioned into $|M|$ parts, each with cardinality $|M|$, which is way more than $2$, and unless $M$ was countable, way more than $\aleph_0$ as well.
What is true, however, is that even without the axiom of choice we have $|M\times\Bbb N|=|M|$ if and only if $|M\times\{0,1\}|=|M|$. This is accomplished by fixing a bijection between $M$ and $M\times\{0,1\}$, and then "iterating it" to define the necessary partition into $\aleph_0$ different parts.