From PhilPapers forum Logic and Philosophy of Logic:

2016-03-17
Set Theory: Mathematics or Metaphysics?
Cantor-Bernstein Theorem (CBT) viewed by Hessenberg

My analysis of this theorem is apparently far from original. In fact, it had already been presented 110 years ago by Hessenberg in his "Grundzüge der Mengentheorie", 1906). What makes this author very interesting, besides his other mathematical achievements, is that, two years before Zermelo's second proof and first axiomatization of set theory he tried to build such a system. 
I liked his warning about mysticism in Mathematics concerning Set Theory (see the introduction in "Das Unendliche in Mathematik", 1904), I just regret that he was not more critical of the whole project.
I will certainly come back to Hessenberg and his presentation of set Theory, let me just sum up, without going into any details, his view of CBT:
1) If one accepts the principle of the whole greater than the parts, then CBT is invalid.
2) When one takes said principle as not applicable to all cases, especially infinite sets, then CBT becomes valid.
3) A and B are equivalent when A equivalent to B1 subset of B, and B equivalent to subset A1 of A.
4) Principle (3) can be replaced by: 
If A1 is a part of A, A2 a part of A1, and A2~A, then A1~A.

He considers (4) as a better formulation of CBT than (3), since it involves only one set and its subsets.
What we can learn from this is that CBT can never be used as an argument in favor of Infinity since it can only be considered valid if we accept beforehand the definition of an infinite set as one that is equal to its subset. It cannot therefore be used as an argument in favor of any Cantorian conception since it is itself a consequence of Cantorian assumptions concerning the infinite.

Also, it is interesting to see that all the authors that I have analyzed so far make it sound like CBT is a product of pure logic, instead of being the result of a metaphysical assumption!