Why are Aleph Cardinal Numbers "strictly increasing"?
It's more accurate to say $\aleph_{\alpha+1} = \min\{\kappa \text{ cardinal } \mid (\exists f \text{ injection }: f:\kappa \to \aleph_\alpha) \land \kappa \not\simeq \aleph_\alpha \}$ so a minimum not an infimum (which is also true) because all sets/classes of cardinals (which are sets of ordinals too) have a minimum. In particular $\aleph_{\alpha+1}$ is itself a member of that set of cardinals, so is strictly greater than $\aleph_\alpha$ by definition.