A question related to ultrapower embeddings.

Question

In Joel David Hamkins' "Forcing and Large Cardinals" a definition of $extender$ embeddings:

"An embedding $j:V \to M$ is an $extender$ embedding if every element of $M$ can be represented in the form $j(f)(\alpha)$ for some $f:\kappa \to V$ and $\alpha < j(\kappa)$, where $\kappa$ is the critical point of $j$."

Every ultra-power embedding by a measure $\mu$ on a measurable cardinal $\kappa$ is an extender embedding since $\kappa = [id]_{\mu}$ can be the seed for such an embedding and therefore generate $M$ by representing every element as a $j(f)(\kappa)$ for an $f:\kappa \to V$. Such an infinite cardinal $\kappa$ is, of course, the critical point of $j$.

However, not every extender embedding is an ultra-power embedding. For such embeddings j: M $\to$ N, how are the images of such embeddings different from ultra-power images? Or, how are the embeddings different? And, what mathematics can be done with extender embeddings which are not generated by it's critical point?

Answer 1

I'm glad you're reading my book (in preparation).

There are a variety of large cardinal notions and large cardinal embedding types that are witnessed by extender embeddings, but which cannot be witnessed by ultrapower embeddings by an ultrafilter on a measurable cardinal $\kappa$.

• Perhaps the easiest example arises from the iterations of a normal measure $\mu$ on a measurable cardinal. If $j:V\to M$ is the ultrapower by $\mu$, then in $M$ the cardinal $j(\kappa)$ is a measurable cardinal with normal measure $j(\kappa)$, and one can take the ultrapower of $M$ by $j(\mu)$. Eventually, one builds the system of embeddings $$V\to M_1\to M_2\to M_3\to\cdots,$$ where $j_{n,n+1}:M_n\to M_{n+1}$ is the ultrapower of $M_n$ by $j_{0,n}(\mu)$, and where $j_{i,j}$ is defined by composing the maps at each step. Taking the direct limit of this system, one gets a map $j_{0,\omega}:V\to M_\omega$, and one can show that this limit is well-founded. This is an elementary embedding, but it is not an ultrapower embedding by any ultrafilter on $\kappa$, because $M_\omega$ is well-founded, but is not closed under $\kappa$-sequences, indeed, not even under $\omega$-sequences, as the value $j_{0,\omega}(\kappa)=\kappa_\omega=\sup_n \kappa_n$, where $\kappa_n=j_{0,n}(\kappa)$ is the critical sequence. That is, $\kappa_\omega$ has cofinality $\omega$ in $V$, but is a measurable cardinal in $M_\omega$. Meanwhile, however, one can prove that every element of $M_\omega$ has the form $j(f)(\kappa_0,\kappa_1,\ldots,\kappa_n)$ for some function $f:[\kappa]^{\lt\omega}\to V$, and this observation can be used to show that $j_{0,\omega}$ has an extender representation. One can of course continue the iterations through the ordinals, and these will all be extender embeddings, and none of them is an ultrapower embedding by an ultrafilter on $\kappa$.

• One can prove more generally that every extender embedding is the direct limit of the induced system of ordinary ultrapower embeddings. For any $\alpha\lt j(\kappa)$, one forms the hull $X=\{j(f)(\alpha)\mid f:\kappa\to V\}$, and this is an elementary substructure of $M$ containing the range of $j$. Following $j$ with the Mostowski collapse of that structure gives rise to a factor embeding $j_0:V\to M_0$, with $k:M_0\to M$ the inverse collapse of $X$. These embeddings fit together into a directed system of embeddings, by means of Goedel pairing of the ordinals, and the extender embedding $j$ is the direct limit of the system.

• A cardinal $\kappa$ is $\theta$-strong if there is an embedding $j:V\to M$ with critical point $\kappa$ and $V_\theta\subset M$. Such kind of embeddings, for $\kappa+2\leq\theta$, cannot arise from ultrapowers by a measure on $\kappa$, since $\mu\notin M_\mu$ for any such ultrapower $j_\mu:V\to M_\mu$. But they can be represented by extender embeddings, since one simply takes the Mostowski collapse of the seed hull $\{j(f)(s)\mid f:V_\kappa\to V, s\in V_\theta\}\prec M$, and then observes that an enumeration of $V_\theta$ turns this into the form you stated.

• A cardinal $\kappa$ is $\theta$-tall if there is an embedding $j:V\to M$ with critical point $\kappa$ and $\theta\lt j(\kappa)$ and $M^\kappa\subset M$. Ultrapower embeddings by a measure on $\kappa$ cannot achieve this when $\theta$ is larger than $(2^\kappa)^+$, since one can count the number of functions to get a bound on the size of $j(\kappa)$. But meanwhile, one can realize tallness with extender embeddings.