User blog comment:PolyhedralMatrix102/Question about Cardinals/@comment-11227630-20180317041140/@comment-11227630-20180318012242

For a limit ordinal $$\alpha$$, the cofinality of $$\alpha$$ is the least ordinal $$\beta$$ such that there exists an increasing ordinal sequence $$\{\alpha_\xi\}_{\xi<\beta}$$ (where $$\beta$$ is the length of the sequence) such that $$\sup\{\alpha_\xi\}=\alpha$$. In addition, the cofinality of a successor ordinal is 1, and the cofinality of 0 is 0. In googology, the fundamental sequence, widely used in FGH, is one of such sequence $$\{\alpha_\xi\}_{\xi<\beta}$$.

e.g. Cofinality of $$\Omega_3$$ (denote $$\text{cf}(\Omega_3)$$) is $$\Omega_3$$ (so $$\Omega_3$$ is regular), and $$\text{cf}(\Omega_\omega)=\text{cf}(\Omega_{\Omega_{\Omega_\cdots}})=\omega$$ (so $$\Omega_\omega$$ and $$\Omega_{\Omega_{\Omega_\cdots}}$$ are not regular).

$$\aleph_1$$ is defined to be the next cardinal after $$\aleph_0$$, and $$2^{\aleph_0}>\aleph_0$$, so $$2^{\aleph_0}\ge\aleph_1$$, regardless of the continuum hypothesis. By definition, a strong limit cardinal fits that "power set of all cardinals less than it still have cardinality less than it", but now power set of $$\aleph_0$$ (i.e. $$2^{\aleph_0}$$) doesn't have cardinality less than $$\aleph_1$$, so $$\aleph_1$$ is not strong limit.