User blog:Ecl1psed276/Question about standard notation

In standard notation, \(\Omega_\omega\) is the limit of \(\Omega_n\) for finite n. However, in standard notation, \(I_\omega\) is NOT the limit of \(\I_n\) for finite n, it is instead the \(\omega\)th inaccessible cardinal. So why isn't \(\Omega_\omega\) defined to be the \(\omega\)th regular cardinal? If we defined it that way, then the limit of \(\Omega_n\) for finite n would actually be \(\psi(\psi_{\omega}(0))\) (I think).

My question is this:

Why did we define inacessibles this way? Why didn't we just define \(\I_\omega\) to be the limit of \(\I_n\), just as \(\\Omega_\omega\) is the limit of \(\\Omega_n\) ?

This question has been bugging me for a long time and I would appreciate an answer to it ;)

(also, this is the reason why, UNOCF is defined so that \(\I_\omega\) actually is the limit of \(\I_n\), to keep consistency)

thank