User blog comment:Edwin Shade/Questions About Madore's Psi Function/@comment-31663462-20180112022247


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

Intuitively, I imagine the following:

$$\psi_{\chi(0)}(x)=\text{fixed points of }\Omega_y\\\psi_{\chi(1)}(x)=\text{fixed points of }\Omega_{\chi(0)+y}\\\psi_{\chi(2)}(x)=\text{fixed points of }\Omega_{\chi(1)+y}\\\vdots\\\psi_{\chi(M)}(x)=\text{fixed points of }\chi(y)\\\psi_{\chi(M+1)}(x)=\text{fixed points of }\Omega_{\chi(M)+y}\\\vdots\\\psi_{\chi(M2)}(x)=\text{fixed points of }\chi(M+y)\\\vdots\\\psi_{\chi(M^2)}(x)=\text{fixed points of }\chi(My)\\\vdots$$

i.e. $$\chi(\dots)$$ has the last $$M$$ replaced with fixed points.

To generate fixed points of $$M^y$$, we use either $$\psi_M(x)$$ or $$\psi_{M+1}(x)$$.

To generate fixed points of $$\psi_{\chi(y)}(x)$$, replace $$x$$ with $$\Omega_{\chi(y)}=\chi(y)$$.