Internal large cardinal embeddings. How deep can we go? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:07:53Z http://mathoverflow.net/feeds/question/109636 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109636/internal-large-cardinal-embeddings-how-deep-can-we-go Internal large cardinal embeddings. How deep can we go? Erin K Carmody 2012-10-14T18:36:41Z 2012-10-15T13:31:59Z <p>Suppose, in V, that $\kappa$ is a large cardinal, say $\kappa$ is supercompact, and $j:V \to M$ is an ultrapower embedding witnessing the large cardinal properties of $\kappa$. From here, we can take the seed hull of some element of M and get both of the following embeddings: a map $k$ which is the inverse collapse of the seed hull (let us call the transitive collapse the seed hull $M_0$), which, as it turns out, is an elementary embedding from $M_0$ to $M$, and a map $j_0: V \to M_0$. The map $j_0$ is an ultrapower map and we also get $j = k\circ j_0$.</p> <p>If there is also a map $j_1: M \to M_0$ (maybe keep calling it $M_0$ even though it is contained inside or maybe contains $M_0$), an ultrapower embedding witnessing that $\kappa^M$ is supercompact, for example, then we could continue with the above construction. That is, we take a seed hull, get its transitive closure $M_1$, and then get $k_1$ an elementary map between $M_1$ and $M_0$ and an ultrapower embedding $j_{00}: M \to M_1$, with $j_1 = k_1 \circ j_{00}$. </p> <p>It is possible that there is no map $j_1$ as above. In which case the construction is terminated. But, it could be that $\kappa$ is supercompact, or whatever $\kappa$ was in $V$, in $M$ as well. Potentially, how long could this construction go? Can it go forever, or must it necessarily stop at some infinite stage?</p>