# Why are these two definitions of extendible cardinal equivalent?

I have encountered two different definitions of extendible cardinals.

Definition 20.22 of Jech's 2003 Set Theory says:

Definition. A cardinal $$\kappa$$ is extendible if for every $$\alpha > \kappa$$ there exist an ordinal $$\beta$$ and an elementary embedding $$j : V_\alpha \to V_\beta$$ with critical point $$\kappa$$.

Meanwhile, Page 311 of Kanamori's 2003 The Higher Infinite says:

Definition. A cardinal $$\kappa$$ is extendible for all $$\eta > \kappa$$, there is a $$\zeta$$ and a $$j : V_{\kappa + \eta} \to V_\zeta$$ with $$\operatorname{crit}(j) = \kappa$$ and $$j(\kappa) > \eta$$ (i.e. $$\kappa$$ is $$\eta$$-extendible for all $$\eta > \kappa$$).

Why are these two definitions equivalent? In particular, it's unclear to me how in Jech's definition, it implies that there exist elementary embeddings with arbitrarily large $$j(\kappa)$$.

Assume ZFC. Suppose $$\kappa$$ is extendible in Jech's sense, but not Kanamori's. In what follows, all embeddings mentioned will have critical point $$\kappa$$, without explicitly mentioning it.

Claim 1: There is some $$\gamma$$ such that for all sufficiently large $$\alpha$$, if $$j:V_\alpha\to V_\beta$$ then $$j(\kappa)<\gamma$$.

Claim 1 is easy to see. Let $$\gamma$$ be the least witness.

Claim 2: $$\gamma$$ is a limit ordinal and for all $$\xi<\gamma$$ and all $$\alpha>\kappa$$ there is $$j:V_\alpha\to V_\beta$$ with $$j(\kappa)>\xi$$.

Proof of Claim 2: Suppose $$\gamma=\xi+1$$. Then for cofinally many $$\alpha\in\mathrm{OR}$$, there is $$j:V_\alpha\to V_\beta$$ with $$j(\kappa)=\xi$$. It follows that this in fact holds for all $$\alpha>\kappa$$ (by restricting some embedding with larger domain). Fix an $$\alpha>\kappa$$ sufficiently large that it witnesses Claim 1, and let $$j:V_\alpha\to V_\beta$$ be an embedding such that $$j(\kappa)=\xi$$. Since $$\kappa<\alpha\leq\beta$$ we can also fix an embedding $$k:V_\beta\to V_\delta$$ with $$k(\kappa)=\xi=j(\kappa)$$. Then since $$k(\kappa)=\xi<\beta$$, we have $$k(\xi)>\xi$$. But then $$\ell=k\circ j:V_\alpha\to V_\delta$$ is such that $$\ell(\kappa)>\xi$$, a contradiction.

Claim 3: If $$\beta$$ is sufficiently large and $$j:V_\beta\to V_\delta$$, then $$j\gamma\subseteq\gamma$$.

Proof of Claim 3: Suppose not. Then we can fix $$\xi<\gamma$$ such that for cofinally many $$\beta$$, hence for all $$\beta>\xi$$, there is $$j:V_\beta\to V_\delta$$ such that $$j(\xi)\geq\gamma$$. Let $$\alpha>\xi$$ be such that there is some $$k:V_\alpha\to V_\beta$$ with $$k(\kappa)>\xi$$, and such that whenever $$\ell:V_\alpha\to V_\delta$$, then $$\ell(\kappa)<\gamma$$. Now let $$j:V_\beta\to V_\delta$$ be such that $$j(\xi)\geq\gamma$$. Let $$\ell=j\circ k:V_\alpha\to V_\delta$$. Then $$\ell(\kappa)\geq\gamma$$, contradiction.

Now let $$\alpha$$ be sufficiently large and with $$\alpha\geq\gamma+2$$ and $$j:V_\alpha\to V_\beta$$. By Claim 3, we have $$j\gamma\subseteq\gamma$$. So if $$\mathrm{cof}(\gamma)=\omega$$ then $$j(\gamma)=\gamma$$ and $$j\upharpoonright V_{\gamma+2}$$ contradicts Kunen. But if $$\mathrm{cof}(\gamma)>\omega$$ then $$\lambda=\kappa_\omega(j)$$ (the sup of the critical sequence of $$j$$) is such that $${\lambda<\gamma}$$, so $$j\upharpoonright V_{\lambda+2}$$ contradicts Kunen.

• Thank you. Can you explain why does $\operatorname{cf}(\gamma) = \omega$ imply that $j(\gamma) = \gamma$? Mar 15 at 5:57
• Let $f:\omega\to\gamma$ be cofinal. Then $j(f):j(\omega)\to j(\gamma)$ is cofinal. But $j(\omega)=\omega$. So $j(f)[\omega]$ is cofinal in $j(\gamma)$. But for $n<\omega$, we have $j(f(n))=j(f)(n)$. So $j(f)[\omega]\subseteq\gamma$. So $j(\gamma)=\gamma$. Mar 15 at 10:12