Bernstein's proof of the continuum hypothesis In the paper The Continuumproblem, Felix Bernstein introduces a new axiom and uses it to conclude the continuum hypothesis.
(1) As the paper is relatively old and the writing style is somehow informal, I am wondering if there is a more exact and concrete proof of the result? (or may please give a better presentation of the argument!)
(2) Is the system introduced by Bernstein consistent? (he does not discuss this in the paper)
 A: This is really a long comment. This paper has been reviewed twice by zbMATH: one by H. B. Curry, which is not informative; another by W. Ackermann, which is in German. The following is the (manipulated) translation of part of his review by DeepL.

... However, a certain sense cannot be connected with this axiom, since it is not said what is to be understood by the existence of an element. If one tries to find this sense in the application of the axiom, one is also disappointed. The point of the proof is that a subset of power $\aleph_1$ is extracted from a certain set of the power of the continuum by multiple application of the axiom of choice, and it is claimed that the selection can be made so arbitrarily that it is impossible to determine whether the subset is a real subset or not without giving any reason for it. (Indeed, in the presence of a proof for $2^{\aleph_0}\neq\aleph_1$ this statement would be possible). In any case, the validity of Cantor's hypothesis remains just as indeterminate after the present remarks as before.


Edit: It seems, in this paper (1905), Bernstein has sketched a proof of the Continuum Hypothesis for a submission to Math. Ann.
