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I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is interpreted as the extender sequence at $\alpha$. Let $\dot F^{\mathcal{M}}$ be a 3-ary predicate interpreted as the weakly amenable coding of $E_{\alpha}$.

Mitchell and Steel define the ultrapower in the case $\mathcal{M}$ is active. In the first page in http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.lnl/1235423433&view=body&content-type=pdf_1 there is case 1 where $\mu < \kappa$ ($\mu$ is the critical point of $\dot F^{\mathcal{M}}$).

The authors claim this directly implies that $g$ is constant almost everywhere ($g$ is defined a couple of line before the argument). I don't understand why that is so. Thanks for any help.

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    If you have an answer you can post it as an answer to your own question. Give it a day or two for some feedback, and then accept it if you are certain it is correct.2012-10-19

1 Answers 1

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Suppose $ E $ is an extender on the sequence of some premouse $\mathcal{N}$ such that $ \mathcal{P}^{\mathcal{N}|lh(E)}(\kappa) = \mathcal{P}^{\mathcal{N}}(\kappa) = \mathcal{P}^{\mathcal{M}}(\kappa),$ where $crit(E) = \kappa$.

  1. If $\mathcal{M}$ and $\mathcal{N}$ are premice and $ \mathcal{P}^{\mathcal{N}}(\kappa) = \mathcal{P}^{\mathcal{M}}(\kappa)$, then $\kappa^{+\mathcal{M}} = \kappa^{+ \mathcal{N}} =: \theta $ and $ \mathcal{M}|\theta = \mathcal{N}|\theta$.

This implies the following:

  1. $\pi^{\mathcal{M}}_{E}(\kappa) = \pi^{\mathcal{N}}_{E}(\kappa)$, $\pi^{\mathcal{M}}_{E}(\kappa^{+}) = \pi^{\mathcal{N}}_{E}(\kappa^{+})$ and $Ult_{0}(\mathcal{M},E)|\pi^{\mathcal{M}}_{E}(\kappa^{+}) = Ult_{0}(\mathcal{N},E)|\pi^{\mathcal{N}}_{E}(\kappa^{+})$

We also have from the coherence of $E$ and the fact that $\kappa$ is a cardinal in $\mathcal{N}$ that

  1. $ Ult_{0}(\mathcal{N},E)|\pi^{\mathcal{N}}_{E}(\kappa) \models \kappa \ \text{is a cardinal}$

Thus by 2. we have

  1. $ Ult_{0}(\mathcal{M},E)|\pi^{\mathcal{M}}_{E}(\kappa) \models \kappa \ \text{is a cardinal}.$

Since $Ult_{0}(\mathcal{M},E) \models \pi(\kappa) \ \text{is a cardinal} $, it follows from 4. and acceptability that

  1. $Ult_{0}(\mathcal{M},E) \models \ \kappa \ \text{is a cardinal}$

From 5. and the fact that $\mathcal{M} \models (\kappa \ \text{is a regular caridnal} )$ it follows that

  1. $\mathcal{M} \models \ \kappa \ \text{is a inaccessible cardinal}$

We have $Ult(\mathcal{M},E) \models h:\pi^{\mathcal{M}}_{E}(\mu) \rightarrow \bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\pi^{\mathcal{M}}_{E}(\mu)]^{n}) $ and $h=[b,g]_{E}^{\mathcal{M}}$, by Los, we can assume that $g:\kappa^{|b|} \longrightarrow \bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\mu]^{n})^{\mu}$. We are in the case $ \mu < \kappa$ so from 6. we have

$\mathcal{M} \models |\bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\mu]^{n})|^{\mu} < \kappa $

So we can assume that $g$ is constant a.e. $E_{b}$.