User blog comment:Alemagno12/Some more set theory questions/@comment-32876686-20180102221118

For question number three I believe the answer to be \(\beth_1\), or the cardnality of the real numbers; for question number four there is such an ordinal corresponding to the first fixed point of \(F(\Delta)\mapsto\Delta\), or \(\aleph_{\aleph_{\aleph_{._{._.}}}}\). It can be expressed as \(\psi_I(0)\), or incompletely, (yet more intuitively), as \(\Omega_{\Omega_{\Omega_{._{._.}}}}\), and is called the Omega fixed point. it is a fairly large cardinal.