User blog:Edwin Shade/Questions About Madore's Psi Function

''This blog post need not litter the main space with another beginner question, so as soon as I have received a satisfactory answer I will remove the contents of this post and request it be deleted. I hope though before then anyone reading this will not mind if I ask them additional questions that are not in this post, as I find I often require a more thorough explanation than mere equations can offer to understand a matter.''

'''Question 1: What is the difference between \(\psi(\omega)\) and \(\psi(\Omega_1)\) ? Or more generally, how are different uncountable ordinals to be treated ? For instance, if the fundamental sequence of \(\psi(Omega)\) is \(\{\psi(0),\psi(\psi(0)),\psi(\psi(\psi(0))),...\}\), then how is a fundamental sequence to be assigned when \(\Omega_5\) or \(\Omega_{\Omega}\) is invoked ?'''

'''Question 2: Is \(\psi_{\psi_{\psi_{._{._{._{\psi(0)}.}.}.}.}.}(0)=\psi_{\Omega}(0)\) ? - Why or why not ?'''

Question 3: If the ordinal which represents the fixed point of \(\alpha\mapsto\Omega_{\alpha}\) is denoted \(I\), then why is \(\psi_I(1)\) equal to the second fixed point of \(\alpha\mapsto\Omega_{\alpha}\) instead of \(\epsilon_{I+1}\), as the definition of Madore's psi function would seem to imply ?

Question 4: What is an easily graspable explanation of Mahlo cardinals, (one that can be thought of in terms of fixed points) ?