Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}_w$. Specifically, $\mathbb{R}_w$ is the Radin forcing defined using the measure sequence $w=j_{\alpha}(u)$, where $j_\alpha$ is the $\alpha^{th}$-iterated of $j$.

My question is the following: Which are the measures that we use to carry out this process? Are the second coordinates (i.e. $u_\gamma(1)$) of the corresponding measure sequence $u_\gamma$? Are the whole measure sequences $u_\gamma$? In this latter case, how looks like the definition of a ultrapower given by a measure sequence $u_\gamma$?


See section 5 (called "Generating generic sequences by iterating j") of Radin's paper Adding closed cofinal sequences to large cardinals.

More details: The iteration is the usual one, using $j$ (see my comment below). Now the point is that is $u$ is the measure sequence in $V$, then there is $\sigma: ORD \to V$ so that $\sigma(\alpha)=j_{0, \alpha}(u)\restriction \beta_\alpha,$ for some suitable $\beta_\alpha$ and such that for any $\alpha, \sigma \restriction \alpha+1$ is $\mathbb{R}_{\sigma(\alpha)}$-generic over $M_\alpha$.

  • $\begingroup$ Thanks for the reference! Anyway I'm still having some doubts on it. I may understand that if $j:V\rightarrow M$ is an elementary embedding then what we are iterating not the measures $u_\gamma(1)$ but the extender generating $j$. More precisely, if $j:V\rightarrow M_1$ is generated by $E$ then $V$ thinks $u$ is generated by $E$, hence $M$ thinks that $j(u)$ is generated by $j(E)$. Now as $M_2$ we take the ultrapower of $M_1$ by the extender $j(E)$ so that $M_2$ thinks $j_{2}(u)$ is generated by $j_2(E)$. Does it make sense? Thank you in advance! $\endgroup$ – Cesare Apr 4 '19 at 8:05
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    $\begingroup$ The iteration by j may be considered by applying j at successor steps $(j_{0,\alpha+1}=j(j_{0,\alpha}) \circ j)$, and taking direct limits at limit ordinals $\endgroup$ – Mohammad Golshani Apr 7 '19 at 1:55

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