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\chapter{Ideal absoluteness}
\label{chapter.ideal-absoluteness}
\thispagestyle{fancy}

Historically, the idea of considering elementary embeddings existing only in generic extensions has been around for a while, but it all started as an analysis of \textit{ideals}. \textit{Precipitous ideals} were introduced in \cite{IdealGame} and further analysed in \cite{Precipitous}, being ideals that give rise to well-founded generic ultrapowers.

\qquad In this chapter we will introduce the \textit{ideally measurable cardinals}, essentially just switching perspective from the ideals themselves to the cardinals they are on. We then proceed to show how these cardinals relate to ``pure'' generic cardinals, being proper class versions of the faintly measurable cardinals that we have considered throughout Chapter \ref{chapter.virtual-large-cardinals}. We start with a definition of the latter.

\defi[\gbc]{
  A cardinal $\kappa$ is \textbf{generically measurable} if there is a generic extension $V[g]$, a transitive class $\N\subset V[g]$ and a generic elementary embedding $\pi\colon V\to\N$ with $\crit\pi=\kappa$.
}

Note that, trivially, every generically measurable cardinal is faintly measurable. The corresponding ideal version of this is then the following:

\defi{
  A cardinal $\kappa$ is \textbf{ideally measurable} if there exists an ideal $\I$ on $\p(\theta)$ such that the generic ultrapower $\ult(V,\I)$ is well-founded in the forcing extension $V^{\mathbb P}$, where $\mathbb P := \p^V(\kappa)/\I$.
}

It should also be noted that \cite{claverie-ideal-extenders} generalised the concept of ideally measurables to \textit{ideally strong cardinals} by introducing the concept of \textit{ideal extenders} to capture the strongness properties.

\qquad Throughout this chapter we will be interested in how properties of the \textit{forcings} affect the large cardinal structure of a critical point of a generic embedding. We thus define the following:

\defi{
  Let $\theta$ be a regular uncountable cardinal, $\kappa<\theta$ a cardinal and $\Phi(\kappa)$ a poset property\footnote{Examples of these are having the $\kappa$-chain condition, being $\kappa$-closed, $\kappa$-distributive, $\kappa$-Knaster, $\kappa$-sized and so on. Formally speaking, $\Phi(\kappa)$ is a first-order formula $\varphi(\kappa)$ which is true iff there is a poset $\mathbb P$ such that $\kappa$ is a cardinal and some first-order formula $\psi(\kappa,\mathbb P)$ is true.}. Then $\kappa$ is \textbf{$\Phi(\kappa)$ faintly $\theta$-measurable} if it is faintly $\theta$-measurable, witnessed by a forcing poset satisfying $\Phi(\kappa)$. Similarly, $\kappa$ is \textbf{$\Phi(\kappa)$ generically measurable} if it is generically measurable with the associated forcing satisfying $\Phi(\kappa)$.
}

\defi{
	A poset property $\Phi(\kappa)$ is \textbf{ideal-absolute} if whenever $\kappa$ satisfies that there is a $\Phi(\kappa)$ forcing poset $\mathbb P$ such that, in $V^{\mathbb P}$, there is a $V$-normal $V$-measure $\mu$ on $\kappa$, then there is a normal ideal $\I$ on $\kappa$ such that $\p(\kappa)/\I$ is forcing equivalent to a forcing satisfying $\Phi(\kappa)$.
}

Note that this is \textit{almost} saying that $\Phi(\kappa)$ ideally measurables are equivalent to $\Phi(\kappa)$ generically measurables, but the only difference is that these definitions require well-foundedness of the target model.

\qquad If $\Phi(\kappa)$ implies $\omega$-distributivity then we \textit{do} get well-foundedness for free, however: any witness to the ill-foundedness of the ultrapower is an $\omega$-sequence in the generic extension, so $\omega$-distributivity implies that this sequence is in $V$ as well, which is contradicting $V$-normality of the measure.

\qquad A typical ideal that we will be utilising is the following:

\defi{
	Let $\kappa$ be a regular cardinal, $\mathbb P$ a poset and $\dot\mu$ a $\mathbb P$-name for a $V$-normal $V$-measure on $\kappa$. Then the \textbf{induced ideal} is
		\eq{
			\I(\mathbb P,\dot\mu):=\{X\subset\kappa\mid\norm{\check X\in\dot\mu}_{\B(\mathbb P)}=0\},
		}

		where $\B(\mathbb P)$ is the Boolean completion of $\mathbb P$.
}

Note that if the generic measure $\mu$ is $V$-normal then $\I(\mathbb P,\dot\mu)$ is also normal. This ideal will witness our first ideal-absoluteness result, which is a simple rephrasing of a folklore result.

\pagebreak
\prop[Folklore][prop.kappa-cc-ideal-abs]{
	``The $\kappa^+$-chain condition'' is ideal-absolute.
}
\proof{
	Assume $\mathbb P$ has the $\kappa^+$-chain condition such that there is a $\mathbb P$-name $\dot\mu$ for a $V$-normal $V$-measure on $\kappa$. Let $\I:=\I(\mathbb P,\dot\mu)$ --- we will show that $\p(\kappa)/\I$ has the $\kappa^+$-chain condition. Assume not and let $\bra{X_\alpha\mid\alpha<\kappa^+}$ be an antichain of $\p(\kappa)/\I$, which by normality of $\I$ we may assume is pairwise almost disjoint. But this then makes $\bra{\norm{\check X_\alpha\in\dot\mu}_{\B(\mathbb P)}\mid\alpha<\kappa^+}$ an antichain of $\mathbb P$ of size $\kappa^+$, $\contr$.
}

We next move to distributivity. This property is especially interesting in the context of our generic large cardinals, as an ideal $\I$ on some cardinal $\kappa$ is $\omega$-distributive precisely if it is precipitous\footnote{See \cite{Precipitous} and \cite{Foreman}.}, so that carrying an $\omega$-distributive ideal coincides with our definition of \textit{ideally measurable}.

\theo[N.][theo.distributive-ideal-abs]{
	``${<}\lambda$-distributivity'' is ideal-absolute for all regular $\lambda\in[\omega,\kappa^+]$.
}
\proof{
	Assume that $\mathbb P$ is a ${<}\lambda$-distributive forcing such that there exists a $\mathbb P$-name $\dot\mu$ for a $V$-normal $V$-measure on $\kappa$. Let $\I:=\I(\mathbb P,\dot\mu)$ --- we will show that $\p(\kappa)/\I$ is ${<}\lambda$-distributive. 

  \qquad Let $\gamma<\lambda$ and let $\vec\A$ be a $\gamma$-sequence of maximum antichains $\A_\alpha\subset\p(\kappa)/\I$ such that $\A_\beta$ refines $\A_\alpha$ for $\alpha\leq\beta$. We have to show that there is a maximal antichain $\A$ which refines all the antichains in $\vec\A$.

  \qquad Now define for every $\alpha<\gamma$ the sets
  \eq{
    \A^*_\alpha :=\{\norm{\check X\in\dot\mu}_{\B(\mathbb P)}\mid X\in\A_\alpha\}. 
  }

  Note that $\A^*_\alpha$ is an antichain in $\mathbb P$. They are also maximal, because if $p\in\mathbb P$ was incompatible with every condition in $\A^*_\alpha$ then, letting $X:=\triangle\{\kappa - Y\mid Y \in \A_\alpha\}$, we have that $p$ is compatible with $\norm{\check X\in\dot\mu}_{\B(\mathbb P)}$, so that $X\in \I^+$. But $X$ is incompatible with everything in $\A_\alpha$, contradicting that $\A_\alpha$ is maximal.

	\qquad By ${<}\lambda$-distributivity of $\mathbb P$ we get an antichain $\A^*$ which refines all the antichains in $\vec\A^*$. But note that for every $p\in\A^*$, if we define $s_p(\alpha)$ to be the unique $a\in\A_\alpha$ such that $p\leq a$, then it holds that $p\leq\norm{\Delta s_p\in\dot\mu}_{\B(\mathbb P)}$,\footnote{Here we are using that $\lambda\leq\kappa^+$ to ensure that the diagonal intersection is in the measure.} so that $\Delta s_p\in \I^+$. Now $\A:=\{\Delta s_p\mid p\in\A^*\}$ gives us a maximal antichain which refines all the antichains in $\vec\A$.
}

In an unpublished paper, Foreman proved the following:

\theo[Foreman][theo.foreman]{
  Let $\kappa$ be a regular cardinal such that $2^\kappa = \kappa^+$, and let $\lambda\leq\kappa^+$ be an infinite successor cardinal. If player II has a winning strategy in $\G^-_\lambda(\kappa)$ then $\kappa$ carries a $\kappa$-complete normal precipitous ideal $\I$ such that $\p(\kappa)/\I$ has a dense ${<}\lambda$-closed subset of size $\kappa^+$.
}

Here we improve that result by not relying on the \ch-assumption, reaching the conclusion for all regular infinite $\lambda$, and also showing $(\kappa,\kappa)$-distributivity of the ideal forcing. The argument follows the same overall structure as the original, with more technicalities to achieve the stronger result.

\theo[Foreman-N.][theo.hopelessideal]{
	Let $\kappa$ be a regular cardinal and $\lambda\leq\kappa^+$ regular infinite. If player II has a winning strategy in $\G_\lambda^-(\kappa)$ then $\kappa$ carries a $\kappa$-complete normal ideal $\I$ such that $\p(\kappa)/\I$ is $(\kappa,\kappa)$-distributive and has a dense ${<}\lambda$-closed subset.
}
\proof{
  Set $\mathbb P:=\text{Add}(\kappa^+,1)$ if $2^\kappa>\kappa^+$ and $\mathbb P:=\{\emptyset\}$ otherwise. If $\kappa$ is measurable then the dual ideal to the measure on $\kappa$ satisfies all of the wanted properties, so assume that $\kappa$ is not measurable.	Fix a wellordering $<_{\kappa^+}$ of $H_{\kappa^+}^V$ and a $\mathbb P$-name $\pi$ for a sequence $\bra{\N_\gamma\mid\gamma<\kappa^+}\in V^{\mathbb P}$ such that
\begin{itemize}
	\item $\N_\gamma\in V$ for every $\gamma<\kappa^+$;
  \item $\N_{\gamma+1}\prec H_{\kappa^+}^V$ is a $\kappa$-model for every $\gamma<\kappa^+$;
  \item $\N_\delta=\bigcup_{\gamma<\delta}\N_\gamma$ for limit ordinals $\delta<\kappa^+$;
  \item $\N_\gamma\cup\{\N_\gamma\}\subset\N_\beta$ for $\gamma<\beta<\kappa^+$;
	\item $\p(\kappa)^V\subset\bigcup_{\gamma<\kappa^+}\N_\gamma$.\\
\end{itemize}

Define now the auxilliary game $\G(\kappa)$ of length $\lambda$ as follows.
\game{\alpha_0}{p_0,\M_0,\mu_0,Y_0}{\alpha_1}{p_1,\M_1,\mu_1,Y_1}{\cdots}{\cdots}{}{}

Here $\bra{\alpha_\gamma\mid\gamma<\lambda}$ is an increasing continuous sequence of ordinals bounded in $\kappa^+$, $\vec p$ is a decreasing sequence of $\mathbb P$-conditions satisfying that
	\eq{
		&p_\gamma\forces\godel{\check\M_\gamma=\pi(\check\alpha_\gamma)\land\text{$\check\mu_\gamma$ is a $\check\M_\gamma$-normal $\check\M_\gamma$-measure on $\check\kappa$}}
	}

	such that $Y_\gamma=\Delta_{\xi<\kappa} X^{\mu_\gamma}_\xi$, where $\vec X^{\mu_\gamma}_\xi\in H_{\kappa^+}^V$ is the $<_{\kappa^+}$-least enumeration of $\mu_\gamma$.\footnote{We use that $\mathbb P$ is $\kappa$-closed to get the $p_\gamma$'s as well as to ensure that $\M_\gamma,\mu_\gamma\in V$.} We require that the $\mu_\gamma$'s are $\subset$-increasing, and player II wins iff she can continue playing throughout all $\lambda$ rounds. Let $\mu_\lambda:=\bigcup_{\xi<\lambda}\mu_\xi$ be the \textbf{final measure} of the play.
	
	\qquad To every limit ordinal $\eta<\kappa^+$ define the \textbf{restricted auxilliary game} $\G(\kappa)\restr\eta$ in which player I is only allowed to play ordinals ${<}\eta$ with the sequence bounded in $\eta$. Note that a strategy $\tau$ for player II is winning in $\G(\kappa)$ if and only if it is winning in $\G(\kappa)\restr\eta$ for all $\eta<\kappa^+$, simply because all sequences of ordinals played by player I are bounded in $\kappa^+$.
	
	\qquad Note that $\mu_\lambda$ is precisely the tail measure on $\kappa$ defined by the $Y_\gamma$'s; i.e. that $X\in\mu_\lambda$ iff there exists a $\delta<\lambda$ such that $\abs{Y_\delta-X}<\kappa$. From this it is simple to see that $\G(\kappa)$ is equivalent to $\G_\lambda^-(\kappa)$, so player II has a winning strategy $\tau_0$ in $\G(\kappa)$. 
	
	\qquad For any winning strategy $\tau$ in $\G(\kappa)\restr\eta$ and to every partial play $p$ of $\G(\kappa)\restr\eta$ consistent with $\tau$, define the associated \textbf{hopeless ideal}\footnote{This terminology is due to Matt Foreman.}
	\eq{
		I_p^\tau\restr\eta:=\{X\subset\kappa\mid\ &\text{For every play $\vec\alpha_\gamma*\tau$ extending $p$ in $\G(\kappa)\restr\eta$,}\\ &\text{$X$ is \textit{not} in the final measure}\}
	}

	\clai{
		\label{clai.hopelessideal}
    $\I:=\bigcap_{\eta<\lambda}I_{\bra{}}^\tau\restr\eta$ is normal and $(\kappa,\kappa)$-distributive, for every winning strategy $\tau$ in $\G(\kappa)$ for player II.
	}

	\cproof{
		For normality, if $\bra{Z_\gamma\mid\gamma<\kappa}$ is a sequence of elements of $\I$ such that $Z:=\nabla_\gamma Z_\gamma$ is $\I$-positive, then there exists a play of $\G(\kappa)\restr\eta$ in which player II follows $\tau$ such that $Z$ lies in the final measure. If we let player I play sufficiently large ordinals in $\G(\kappa)\restr\eta$ we may assume that $\bra{Z_\gamma\mid\gamma<\kappa}$ is a subset and an element of the final model as well, meaning that one of the $Z_\gamma$'s also lies in the final measure, $\contr$.

		\qquad We now show $(\kappa,\kappa)$-distributivity. Let $\U\subset\p(\kappa)/\I$ be an unrooted tree of height $\kappa$ such that every level $\U_\alpha$ is a maximal antichain of size $\leq\kappa$. We have to show that there is a maximal antichain $\mathcal A$ consisting of limit points of branches of $\U$. Pick $X\in\U$ and let $p$ be a play of $\G(\kappa)\restr\eta$ consistent with $\tau$ with limit model $\M$ and limit measure $\mu$, such that $X\in\mu$.
		
		\qquad By letting player I in $p$ play sufficiently large ordinals, we may assume that $\U\subset\M$, using that $\abs{\U}\leq\kappa$, and also that $b_X:=\U\cap\mu\in\M$. This means that $d_X:=\Delta b_X\in\p(\kappa)/\I$ is a limit point of the branch $b_X$ through $\U$, so that $\mathcal A:=\{d_X\mid X\in\U\}$ is a maximal antichain of limit points of branches of $\U$, making $\p(\kappa)/\I$ $(\kappa,\kappa)$-distributive.
	}

	Fix some limit ordinal $\eta<\kappa^+$. We will recursively construct a tree $\T^\eta$ of height $\lambda$ which consists of subsets $X\subset\kappa$, ordered by reverse inclusion. In the following we will talk of player II's moves as subsets of $\kappa$ -- here we are referring to the code of the move. During the construction of the tree we will inductively maintain the following properties of $\T^\eta\restr\alpha$ for $\alpha\leq\lambda$:\\

	\begin{itemize}
		\item \textsc{Tree strategy}: For every $\gamma<\alpha$ there is a winning strategy $\tau^\eta_\gamma$ for player II in $\G(\kappa)\restr\eta$ such that for every $\beta<\gamma$, the $\beta$'th move by $\tau^\eta_\gamma$ is an element of $\T^\eta_\beta$ and $\tau^\eta_\gamma$ is consistent with $\tau^\eta_\beta$ for the first $\beta$-many rounds.
		\item \textsc{Unique pre-history}: Given any $\beta<\alpha$ and $Y\in\T^\eta_\beta$ there is a unique partial play $p$ of $\G(\kappa)\restr\eta$ consistent with $\tau^\eta_\beta$ ending with $Y$ --- we define $I_Y^\tau:=I_p^\tau$ for $\tau$ being any winning strategy for player II in $\G(\kappa)\restr\eta$ satisfying that $p$ is consistent with $\tau^\eta_\beta$.
		\item \textsc{Cofinally many responses}: Let $\beta+1<\alpha$ and $Y\in\T^\eta_\beta$, and set $p$ to be the unique partial play of $\G(\kappa)\restr\eta$ given by the unique pre-history of $Y$. Then the $\T^\eta$-successors of $Y$ consists of player II's $\tau^\eta_\beta$-responses to $\tau^\eta_\beta$-partial plays extending $p$ such that player I's last move in these partial plays are cofinal in $\eta$.\footnote{The reason why we are dealing with the \textit{restricted} auxilliary games is to achieve this property.}
		\item \textsc{Positivity}: If $\beta<\alpha$ and $Y\in\T^\eta_\beta$ then $Y$ is $I_X^{\tau^\eta_\gamma}$-positive for every $\gamma<\beta$ and every $X\in\T^\eta\restr\gamma+1$ with $X\leq_{\T^\eta}Y$.\footnote{This actually follows from the cofinally many responses, but we include it here for transparency.}
	\item \textsc{Almost disjointness property}: Every level $\T^\eta_\beta$ consists of pairwise almost disjoint sets.\footnote{Two subsets $X,Y\subset\kappa$ are \textit{almost disjoint} if $\abs{X\cap Y}<\kappa$.}
		\item \textsc{Hopeless ideal coherence}: $I_{\bra{}}^{\tau^\eta_\beta}\cap\p(Y)=I_Y^{\tau^\eta_\beta}\cap\p(Y)$ for every $\beta<\alpha$ and $Y\in\T^\eta_\beta$.\\
	\end{itemize}

  Note that what we are really aiming for is achieving the hopeless ideal coherence, since that enables us to ensure that if $X,Y\in\T^\eta$ and $X\subset Y$ then really $X\geq_{\T^\eta}Y$ --- i.e. that we ``catch'' both $X$ and $Y$ in the same play of $\G(\kappa)\restr\eta$. This is crucial to show the ${<}\lambda$-closure of $\T$. The rest of the properties are inductive properties we need to ensure this.

	\qquad Set $\T^\eta_0:=\{\kappa\}$. Assume that we have built $\T^\eta\restr\alpha+1$ satisfying the inductive assumptions\footnote{In particular, we assume that $\tau^\eta_\alpha$ is defined.} and let $Y\in\T^\eta_\alpha$ --- we need to specify what the $\T^\eta$-successors of $Y$ are. Since $\kappa$ is weakly compact and not measurable it holds by Proposition 6.4 in \cite{Kanamori} that $\text{sat}(I_Y^{\tau^\eta_\alpha})\geq\kappa^+$, so we can fix a maximal antichain $\bra{X_\gamma^Y\mid\gamma<\eta}$ of $I_Y^{\tau^\eta_\alpha}$-positive sets. By $\kappa$-completeness of $I_Y^{\tau^\eta_\alpha}$ we can by Exercise 22.1 in \cite{Jech} even ensure that all of the $X_\gamma^Y$'s are pairwise disjoint.
	
	\qquad To every $\gamma<\eta$ we fix a partial play $p$ of even length of $\G(\kappa)\restr\eta$ consistent with $\tau^\eta_\alpha$ such that the last ordinal $\beta_\gamma^Y$ in $p$ played by player I is greater than or equal to $\gamma$ and $X_\gamma^Y$ has measure one with respect to the last measure in $p$. We then define the $\T^\eta$-successors of $Y$ to be player II's $\tau^\eta_\alpha$-responses to the $\beta_\gamma$'s (which are subsets of the $X_\gamma^Y$'s modulo a bounded set and are therefore pairwise almost disjoint).

  \qquad For limit stages $\delta<\lambda$ we apply $\tau_0$ to the branches of $\T^\eta\restr\delta$ to get $\T^\eta_\delta$.

\qquad We now have to check that the inductive assumptions still hold; let us start with the tree strategy. Assume that we have a partial play $p$ of length $2\cdot\alpha+1$ of $\G(\kappa)\restr\eta$, i.e. the last move in $p$ is by player II, consistent with $\tau^\eta_\alpha$; write $\xi_p$ for player I's last move in $p$ and $Y_p$ for player II's response to $\xi_p$, which is also the last move in $p$. We can then pick a $\zeta<\eta$ such that $\beta_\zeta^{Y_p}>\xi_p$ by the cofinally many responses property and let $\tau^\eta_{\alpha+1}(p)$ be player II's $\tau^\eta_\alpha$-response to the partial play leading up to $\beta_\zeta^{Y_p}$. After this $(\alpha+1)$'th round we just set $\tau^\eta_{\alpha+1}$ to follow $\tau_0$. It is clear that $\tau^\eta_{\alpha+1}$ satisfies the required properties.

  \qquad Before we move on to checking the remaining inductive assumptions, let us pause to get some intuition about the tree strategies. In the definition of $\tau^\eta_{\alpha+1}$ above, we took a partial play consistent with $\tau^\eta_\alpha$, applied $\tau_0$ for a while, took note of player II's last $\tau_0$-response and then included \textit{only that} response in our new $\tau^\eta_{\alpha+1}$ partial play. This means that to every $\tau^\eta_\alpha$-partial play there is an ostensibly much longer $\tau_0$-partial play into which $\tau^\eta_\alpha$ embeds; so we can look at the $\tau^\eta_\alpha$-partial plays as being ``collapsed'' $\tau_0$-partial plays.

	\qquad  Given the above tree strategy, $\T^\eta_{\alpha+1}$ clearly satisfies the cofinally many responses property and the positivity property, simply by construction. For the unique pre-history, let $Y\in\T^\eta_{\alpha+1}$ and assume it has two distinct immediate $\T^\eta$-predecessors $Z_0,Z_1\in\T^\eta_\alpha$. But then $Y\subset Z_0\cap Z_1$ and $Y$ is $I_{Z_0}^{\tau^\eta_\alpha}$-positive by the positivity assumption, contradicting that $Z_0$ and $Z_1$ are almost disjoint by the almost disjointness property. Given the unique pre-history we then also get the almost disjointness property.

\clai{
	$\T^\eta\restr\alpha+2$ satisfies the hopeless ideal coherence property.
}

\cproof{
	Let $Y\in\T^\eta_{\alpha+1}$ --- we have to show that
	\eq{
		I_{\bra{}}^{\tau^\eta_{\alpha+1}}\cap\p(Y)=I_Y^{\tau^\eta_{\alpha+1}}\cap\p(Y).\tag*{$(1)$}
	}

	It is clear that $I_{\bra{}}^{\tau^\eta_{\alpha+1}}\subset I_Y^{\tau^\eta_{\alpha+1}}$, so let $Z\in I_Y^{\tau^\eta_{\alpha+1}}\cap\p(Y)$ and assume for a contradiction that $Z$ is $I_{\bra{}}^{\tau^\eta_{\alpha+1}}$-positive. Letting $\vec\alpha_\xi*\vec Y_\xi$ be a play of $\G(\kappa)\restr\eta$ consistent with $\tau^\eta_{\alpha+1}$ such that $Z$ is in the final measure, the definition of $\tau^\eta_{\alpha+1}$ yields that $Y_\alpha\in\T^\eta_{\alpha+1}$. As $Z\in I_Y^{\tau^\eta_{\alpha+1}}$ we have to assume that $Y\neq Y_\alpha$, so that the almost disjointness property implies that
\eq{
	\abs{Y\cap Y_\alpha}<\kappa.\tag*{$(2)$}
}

By the choice of $\vec\alpha_\xi*\vec Y_\xi$ there is some $\delta\in(\alpha,\lambda)$ such that $\abs{Y_\delta-Z}<\kappa$, i.e. that $Y_\delta$ is a subset of $Z$ modulo a bounded set, since the $Y_\alpha$'s generate the final measure of the play. But then $Y_\delta\subset Y_\alpha$ by the rules of $\G(\kappa)\restr\eta$, and also that $\abs{Y_\delta -Y}<\kappa$ since $Z\subset Y$. But this means that $Y\cap Y_\alpha$ is $I_Y^{\tau^\eta_{\alpha+1}}$-positive since $Y_\delta$ is, contradicting $(2)$. This shows $(1)$.
}
 
\qquad This finishes the construction of $\T^\eta_{\alpha+1}$. For limit levels $\delta<\lambda$ we define $\tau^\eta_\delta$ as simply applying $\tau_0$ to the branches of $\T^\eta\restr\delta$ --- showing that the inductive assumptions hold at $\T^\eta_\delta$ is analogous to the above arguments, so we are now done with the construction of $\T^\eta$. Let $\tau^\eta:=\bigcup_{\alpha<\lambda}\tau^\eta_\alpha\restr{^{<\alpha}H_{\kappa^+}}$ and define\footnote{Note that the tree strategy property above ensures that the strategies \textit{do} line up, so that $\tau^\eta$ is a well-defined strategy as well.} $\I^\eta:=I_{\bra{}}^{\tau^\eta}$.

\qquad Now note that $\I^{\eta+1}\subset\I^\eta$ and $\T^\eta\subset\T^{\eta+1}$ for every $\eta<\kappa^+$ --- define the ideal $\I:=\bigcap_{\eta<\kappa^+}\I^\eta$ and the tree $\T:=\bigcup_{\eta<\kappa^+}\T^\eta$. We showed in Claim \ref{clai.hopelessideal} that $\I$ is $\kappa$-complete, normal and $(\kappa,\kappa)$-distributive.

\qquad We claim that $\T$ is dense in $\p(\kappa)/\I$.\footnote{This means that given any $\I$-positive set $X$ there is a $Y\in\T$ such that $Y-X\in\I$.} Let $X$ be an $\I$-positive set, making it $\I^\eta$-positive for some $\eta<\kappa^+$, meaning that there is a play $\vec\alpha_\gamma*\tau^\eta$ of $\G(\kappa)\restr\eta$ such that $X$ is in the final measure, which means that $\abs{Y_\delta-X}<\kappa$ for some large $\delta<\lambda$ and in particular that $Y_\delta-X\in\I$. But $Y_\delta\in\T^\eta\subset\T$ by definition of $\tau^\eta$, which shows that $\T$ is dense.

\qquad It remains to show that $\T$ is ${<}\lambda$-closed. If $\lambda=\omega$ then this is trivial, so assume that $\lambda\geq\omega_1$. Let $\beta<\lambda$ and let $\bra{Z_\alpha\mid\alpha<\beta}$ be a $\subset$-decreasing sequence of elements $Z_\alpha\in\T$. We can fix some $\eta<\kappa^+$ such that $Z_\alpha\in\T^\eta$ for every $\alpha<\beta$ by regularity of $\kappa^+$, and since the $Z_\alpha$'s are $\subset$-decreasing they must also be $\leq_{\T^\eta}$-increasing by the hopeless ideal coherence for $\T^\eta$\footnote{This is the only place in which we are using hopeless ideal coherence.}.
	
\qquad Let $\tilde Z\in\T^\eta$ be player II's $\tau^\eta$-response to the unique partial play of $\G(\kappa)\restr\eta$ corresponding to the branch containing the $Z_\alpha$'s, and pick $Z\in\T^\eta$ such that $\abs{Z-\tilde Z}<\kappa$ and $Z\geq_{\T^\eta}Z_\alpha$ for all $\alpha<\beta$, again by the density claim and the hopeless ideal coherence. Then $Z$ witnesses ${<}\lambda$-closure of $\T$.\footnote{We are using that $\lambda$ is regular to get $Z$.}
}

With a bit more work we can from this result then derive the following equivalences.

\coro[N.][coro.powerclosed]{
	Let $\kappa$ be a regular cardinal and $\lambda\in[\omega_1,\kappa^+]$ be regular. Then the following are equivalent:
	\begin{enumerate}
		\item $\kappa$ is ${<}\lambda$-closed faintly power-measurable;
		\item $\kappa$ is ${<}\lambda$-closed ideally power-measurable;
		\item $\kappa$ is $(\kappa,\kappa)$-distributive ${<}\lambda$-closed faintly measurable;
		\item $\kappa$ is $(\kappa,\kappa)$-distributive ${<}\lambda$-closed ideally measurable;
		\item Player II has a winning strategy in $\G_\lambda(\kappa)$.
	\end{enumerate}
}
\proof{
	$(v)\Rightarrow (iv)$ is Theorem \ref{theo.hopelessideal} above\footnote{Here well-foundedness of the generic ultrapower is automatic since $\lambda$ has uncountable cofinality.} and $(iv)\Rightarrow(iii)+(ii)$, $(iii)\Rightarrow(i)$ and $(ii)\Rightarrow(i)$ are trivial, so we show $(i)\Rightarrow(v)$.
	
	\qquad Assume $\kappa$ is ${<}\lambda$-closed faintly power-measurable, so there is a ${<}\lambda$-closed forcing $\mathbb P$ and a $V$-generic $g\subset\mathbb P$ such that, in $V[g]$, there exists a transitive class $N$ and a $\kappa$-powerset preserving elementary embedding $\pi\colon V\to N$. Write $\mu$ for the induced weakly amenable $V$-normal $V$-measure on $\kappa$. Now, back in $V$, define a strategy $\sigma$ for player II in $G_\lambda(\kappa)$ as follows.
	
	\qquad Whenever player I plays some model $M_\alpha$ then we let player II respond with a filter $\mu_\alpha$ such that, for some $p_\alpha\in\mathbb P$, $p_\alpha\forces\godel{\check\mu_\alpha=\dot\mu\cap\check M_\alpha}$ --- such a filter exists because $\mu$ is weakly amenable. We require the $p_\alpha$'s to be decreasing, which is possible by ${<}\lambda$-closure. Now, all the $\mu_\alpha$'s are clearly $M_\alpha$-normal $M_\alpha$-measures on $\kappa$, which makes $\sigma$ a winning strategy.
}

Note that the above results all relied on $\lambda$ being uncountable to achieve well-foundedness of the generic ultrapower. If we simply ignore this well-foundedness aspect then we get the following similar equivalence in the $\lambda=\omega$ case, which then also includes completely ineffable cardinals.

\coro[N.][coro.ineff]{
	Let $\kappa$ be a regular cardinal. Then the following are equivalent:\footnote{Points $(i)$ and $(ii)$ look a lot like the definition of faintly power-measurable and $(\kappa,\kappa)$-distributive ideally measurable, but here we are not requiring the ultrapowers to be well-founded, so that would be stretching the definition of being measurable.}
	\begin{enumerate}
		\item There exists a forcing poset $\mathbb P$ such that, in $V^{\mathbb P}$, there is a weakly amenable $V$-normal $V$-measure on $\kappa$;
		\item There exists a $(\kappa,\kappa)$-distributive forcing poset $\mathbb P$ such that, in $V^{\mathbb P}$, there is a $V$-normal $V$-measure on $\kappa$;
		\item $\kappa$ carries a normal $(\kappa,\kappa)$-distributive ideal;
		\item Player II has a winning strategy in $\G_\omega^-(\kappa)$;
		\item $\kappa$ is completely ineffable.
	\end{enumerate}
}
\proof{
	$(iv)\Leftrightarrow (v)$ was shown in Theorem \ref{theo.ineff}, and $(iii)\Rightarrow(ii)$ and $(ii)\Rightarrow(i)$ are trivial. $(i)\Rightarrow(iv)$ is as $(i)\Rightarrow(v)$ in Corollary \ref{coro.powerclosed}, and $(iv)\Rightarrow(iii)$ is Theorem \ref{theo.hopelessideal}.
}

As an immediate consequence we then get another ideal-absoluteness result.

\qcoro{
	``$(\kappa,\kappa)$-distributive ${<}\lambda$-closed'' is ideal-absolute for all regular $\lambda\in[\omega,\kappa^+]$.
}

We get the following similar results for the Cohen games\footnote{Theorem \ref{theo.hopelessideal2} is the reason for naming the $\C$-games ``Cohen games".}.

\theo[N.][theo.hopelessideal2]{
	Let $\kappa$ and $\lambda\leq\kappa$ be regular infinite cardinals such that $2^{<\theta}<\kappa$ for every $\theta<\lambda$. If player II has a winning strategy in $\C_\lambda^-(\kappa)$ then $\kappa$ carries a $\lambda$-complete  ideal $\I$ such that $\p(\kappa)/\I$ is forcing equivalent to $\text{Add}(\lambda,1)$.
}
\proof{
	We follow the proof of Theorem \ref{theo.hopelessideal} closely. Set $\mathbb P:=\text{Col}(\lambda,2^\kappa)$. Fix a wellordering $<_{\kappa^+}$ of $H_{\kappa^+}$ and a $\mathbb P$-name $\pi$ for a sequence $\bra{\N_\gamma\mid\gamma<\lambda}\in V^{\mathbb P}$ such that
	\begin{itemize}
		\item $\N_\gamma\in V$ for every $\gamma<\lambda$;
		\item $\kappa{+}1\subset\N_\gamma$ and $\abs{\N_\gamma-H_\kappa}^V<\lambda$ for every $\gamma<\lambda$;
		\item If $\delta<\lambda$ is a limit ordinal then $\N_\delta=\bigcup_{\gamma<\delta}\N_\gamma$, $\N_\delta\prec H_{\kappa^+}$ and it holds that $\N_\delta\models\zfc^-$;
		\item $\N_\gamma\cup\{\N_\gamma\}\subset\N_\beta$ for all $\gamma<\beta<\lambda$;
		\item $\p(\kappa)^V\subset\bigcup_{\gamma<\lambda}\N_\gamma$.\\
	\end{itemize}

	Define the auxilliary game $\G(\kappa)$ as in the proof of Theorem \ref{theo.hopelessideal} but where player I plays ordinals $\alpha_\eta<\lambda$ and where we use the above $\N_\gamma$'s. Here we only need ${<}\lambda$-closure of $\mathbb P$ to get an equivalence between $\G(\kappa)$ and $\C_\lambda^-(\kappa)$, since $\abs{\N_\gamma-H_\kappa}^V<\lambda$ for all $\gamma<\lambda$.

	\qquad To every limit ordinal $\eta<\lambda$ we define the restricted auxilliary game $\G(\kappa)\restr\eta$ as in the proof of Theorem \ref{theo.hopelessideal}, and to every winning strategy $\tau$ in $\G(\kappa)\restr\eta$ and partial play $p$ of $\G(\kappa)\restr\eta$ consistent with $\tau$ define the associated \textbf{hopeless ideal}
	\eq{
		I_p^\tau\restr\eta:=\{X\subset\kappa\mid\ &\text{For every play $\vec\alpha_\gamma*\tau$ extending $p$ in $\G(\kappa)\restr\eta$,}\\ &\text{$X$ is \textit{not} in the final measure}\}.
	}

	Now, if $\kappa$ is measurable then we trivially get the conclusion,\footnote{Take $\I(\text{Add}(\lambda,1),\check\mu)$ for $\mu$ the measure on $\kappa$.} so assume $\kappa$ is not measurable. Then $\text{sat}(\kappa)\geq\lambda$ since $2^{<\theta}<\kappa$ for every $\theta<\lambda$,\footnote{See Proposition 16.4 in \cite{Kanamori}.} so that we can continue exactly as in the proof of Theorem \ref{theo.hopelessideal} to construct ($\lambda$-sized) trees $\T^\eta$ and winning strategies $\tau^\eta$ for all limit ordinals $\eta<\lambda$ such that, setting $\I:=\bigcap_{\eta<\lambda}I^{\tau^\eta}_{\bra{}}$ and $\T:=\bigcup_{\eta<\lambda}\T^\eta$, $\T$ is a dense ${<}\lambda$-closed subset of $\p(\kappa)/\I$ of size $\lambda$, so that $\p(\kappa)/\I$ is forcing equivalent to $\text{Add}(\lambda,1)$.
}

\coro[N.]{
	\label{coro.cohen}
	Let $\kappa$ and $\lambda\in[\omega_1,\kappa]$ be regular such that $2^{<\theta}<\kappa$ for every $\theta<\lambda$. Then the following are equivalent:
	\begin{enumerate}
		\item $\kappa$ is ${<}\lambda$-closed faintly measurable;
		\item $\kappa$ is ${<}\lambda$-closed ideally measurable;
		\item $\kappa$ is ${<}\lambda$-closed $\lambda$-sized faintly measurable;
		\item $\kappa$ is ${<}\lambda$-closed $\lambda$-sized ideally measurable;
		\item Player II has a winning strategy in $\C_\lambda(\kappa)$.
	\end{enumerate}
}
\proof{
	$(iv)\Rightarrow (iii)+(ii)$, $(ii)\Rightarrow(i)$ and $(iii)\Rightarrow(i)$ all trivial, and $(i)\Rightarrow(v)$ is like $(i)\Rightarrow(v)$ in Corollary \ref{coro.powerclosed}, and $(v)\Rightarrow(iv)$ is Theorem \ref{theo.hopelessideal2}.
}

Again, if we ignore well-foundedness then we get the same equivalence in the $\lambda=\omega$ case:

\coro[N.]{
	\label{coro.cohen2}
	Let $\kappa$ be regular infinite. Then the following facts hold:
	\begin{enumerate}
		\item Player II has a winning strategy in $\C_\omega^-(\kappa)$; and
		\item $\kappa$ carries an ideal $I$ such that $\p(\kappa)/I$ is forcing equivalent to $\text{Add}(\omega,1)$.
	\end{enumerate}
}
\proof{
	Player II has a winning strategy in $\C_\omega^-(\kappa)$ as we are simply measuring finitely many sets without any demand for well-foundedness, showing $(i)$. Since $2^{<n}<\kappa$ for all $n<\omega$ as $\kappa$ is infinite, Theorem \ref{theo.hopelessideal2} then implies $(ii)$.
}

\qcoro[N.]{
  The property ``${<}\lambda$-closed $\lambda$-sized'' is ideal-absolute for all regular $\lambda\in[\omega,\kappa]$ such that $2^{<\theta}<\kappa$ for all $\theta<\kappa$.
}

With this corollary, our journey has come to an end. An ending, which is nothing but an oxymoron in one of these perennial mathematical journeys, so instead consider the following final chapter my baton to you, dear reader.\\

{\itshape
  \begin{center}
    \begin{tabular}{l}
      Natten bøjer sig i vækst;\\
      Og dagene vasker dit ansigt.\\
      Du træder gennem søvnen;\\
      Og åbner en dør til solen.\\
      Dér, ikke i dit eget lys;\\
      Bærer du kroppen;\\
      Strømmet igennem af blod.\\
      Dér er du menneske nu;\\
      Og du går for at finde din tid.\\[3px]
      --- Michael Strunge
    \end{tabular}
  \end{center}
}


\end{document}
