Reflection principle: Difference between revisions

: The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)).