## Exponentiation If $κ$ is a limit cardinal, and $λ ≥ \mathrm{cf} (κ)$, then $κ^λ = (lim_{α→κ} α^λ )^{\mathrm{cf} (κ)}$. **Theorem** Jech 5.15. If GCH holds and $κ$ and $λ$ are infinite cardinals then: 1. If $κ ≤ λ$, then $κ^λ = λ^+$ . 2. If $\mathrm{cf} κ ≤ λ < κ$, then $κ^λ = κ^+$ . 3. If $λ < \mathrm{cf} κ$, then $κ^λ = κ$. 4. In particular - for regular $\kappa$, $\kappa^{<\kappa}=\kappa$. - For singular $\kappa$, $\kappa^{<\kappa}=\kappa ^{+}$. ^8702eb ### Hausdorff formula For $\theta \leq \lambda$ $(\lambda ^{+})^{\theta}=\max \{\lambda ^{+},\lambda^{\theta}\}$