Justification of the continuum hypothesis (Q1356848)

The author gives some plausible arguments to demonstrate that \(\aleph_1=2^{\aleph_0}\). He defines binary operations \(\otimes^k\) on ordinals by induction: \(m\otimes^0n=m+n\), \(m\otimes^{k+1}n= m\otimes^k(m\otimes^k(\dots(m\otimes^km)\dots))\) (\(n\)-times) and the ``limit'' operation \(m\otimes^\omega n\) should be understand in some intuitive way (it is not said how exactly) so that \(\aleph_1=\{0,1,\dots,\omega,\dots, \omega\otimes^0\omega,\dots,\omega\otimes^1\omega, \dots,\dots,\dots\}=\aleph_0\otimes^\omega\aleph_0\). Then the ordinals \(\leq\omega\otimes^k\omega\) are distributed in the unit interval so that the ``limit'' of this distribution for \(k\) to \(\omega\) should give a distribution of \(\aleph_1\) over the interval. The author generalizes this method for verification of the generalized continuum hypothesis.