Gitik's filter system and proper-class forcing

Work in a transitive model $M$ of ZFC equipped with a class predicate $W_O$ which globally well-orders $M$, with Replacement allowed for formulas using $W_O$. Assume that the strongly compact cardinals are unbounded in the ordinals and have been thinned so that their class has no regular limit point. List them increasingly as $⟨{\kappa }_{\xi }:0<\xi \in \mathrm{Ord}⟩$ and put ${\kappa }_0=\omega$ only as a bookkeeping coordinate.

Coordinate filters

For every infinite regular cardinal $\alpha$, define ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )$ and filters as follows.

• If $\alpha <{\kappa }_1$, put ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )=\alpha$ and let ${\mathrm{\Phi }}_{\alpha }$ be the co-bounded filter on $\alpha$.
• If there is a largest strongly compact $\kappa \le \alpha$, call $\alpha$ type 1, put ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )=\alpha$, and let ${\mathrm{\Phi }}_{\alpha }$ be the $W_O$-least uniform $\kappa$-complete ultrafilter on $\alpha$.
• Otherwise put ${\beta }_{\alpha }=\sup \{\kappa :\kappa <\alpha$ is strongly compact$\}$. The no-regular-limit-point hypothesis makes ${\beta }_{\alpha }$ singular. Call $\alpha$ type 2, put ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )=\mathrm{cf}({\beta }_{\alpha })={\gamma }_{\alpha }$, and take the $W_O$-least increasing sequence $⟨{\kappa }_{\nu }^{\alpha }:\nu <{\gamma }_{\alpha }⟩$ of strongly compact cardinals cofinal in ${\beta }_{\alpha }$, with every term at least ${\gamma }_{\alpha }$. For each $\nu <{\gamma }_{\alpha }$, let ${\mathrm{\Phi }}_{\alpha ,\nu }$ be the $W_O$-least uniform ${\kappa }_{\nu }^{\alpha }$-complete ultrafilter on $\alpha$.

The type-2 sequence is “coherent” here only in the stated sense: its completeness levels follow one fixed cofinal sequence and the index used at $\alpha$ is read from the ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )$ coordinate. No projection coherence or normality of the ${\mathrm{\Phi }}_{\alpha ,\nu }$ is being asserted.

Stems

For a partial function coded as a set of triples $p\subseteq \mathrm{Reg}\times \omega \times \mathrm{Ord}$, put

${\mathrm{dom}}_1(p)=\{\alpha :(\mathrm{\exists }n,\xi )(\alpha ,n,\xi )\in p\},{\mathrm{dom}}_{1,2}(p)=\{(\alpha ,n):(\mathrm{\exists }\xi )(\alpha ,n,\xi )\in p\}\mathrm{.}$

Let $P_1$ be the definable class of such $p$ for which ${\mathrm{dom}}_{1,2}(p)$ is finite and, for every $\alpha \in {\mathrm{dom}}_1(p)$, the section $p(\alpha )$ is a finite one-to-one partial function from $\omega$ into $\alpha$. Write $r\approx p$ when $r$ and $p$ agree at every coordinate at least ${\kappa }_1$.

Let $P_2$ consist of the $p\in P_1$ satisfying:

1. ${\mathrm{dom}}_1(p)$ is closed under ${\mathrm{cf}}^{\mathrm{\prime }}$;
2. $\mathrm{dom}(p(\alpha ))\subseteq \mathrm{dom}(p({\mathrm{cf}}^{\mathrm{\prime }}(\alpha )))$ for every coordinate $\alpha$; and
3. there are unique $\alpha (p)\in {\mathrm{dom}}_1(p)$ with $\alpha (p)\ge {\kappa }_1$ and $n(p)<\omega$ such that the high coordinates below $\alpha (p)$ have section-domain $n(p)+1$, while those at or above $\alpha (p)$ have section-domain $n(p)$.

Thus $(\alpha (p),n(p))$ is the next high-coordinate slot. A $p\in P_2$ is extendable when a proper extension with the same coordinate domain and the same part below ${\kappa }_1$ remains in $P_2$. Equivalently, it is extendable iff ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha (p))=\alpha (p)$ or $({\mathrm{cf}}^{\mathrm{\prime }}(\alpha (p)),n(p))\in {\mathrm{dom}}_{1,2}(p)$. For an extendable $p$, let

${\mathrm{\Phi }}_p=\{\begin{array}{ll}{\mathrm{\Phi }}_{\alpha (p)},& {\mathrm{cf}}^{\mathrm{\prime }}(\alpha (p))=\alpha (p),\\ {\mathrm{\Phi }}_{\alpha (p),p({\mathrm{cf}}^{\mathrm{\prime }}(\alpha (p)))(n(p))},& {\mathrm{cf}}^{\mathrm{\prime }}(\alpha (p))<\alpha (p)\mathrm{.}\end{array}$

Measure-one trees and the forcing

For $r\approx p$ and $U\subseteq P_2$, write $T_r^U=\{q\in U:q\upharpoonright {\kappa }_1=r\upharpoonright {\kappa }_1\}$. A condition of Gitik's forcing $P_3$ is a pair $(p,U)$ satisfying all ten clauses below.

1. $p\in P_2$.
2. $U\subseteq P_2$.
3. $p\in U$.
4. Every $q\in U$ extends $p$ as a function and has ${\mathrm{dom}}_1(q)={\mathrm{dom}}_1(p)$.
5. If $r\in U$, $r\approx p$, $\alpha \in {\mathrm{dom}}_1(p)$, and $(\alpha ,n)$ is an unfilled slot with $\alpha <{\kappa }_1$, then $\{\xi :r\cup \{(\alpha ,n,\xi )\}\in U\}\in {\mathrm{\Phi }}_{\alpha }$.
6. If $r_1,r_2\in U$ both satisfy $r_i\approx p$, then their union belongs to $U$ whenever it belongs to $P_1$; moreover, if $r_1\subseteq r_2$, the map $q\mapsto q\cup r_2$ embeds $T_{r_1}^U$ into $T_{r_2}^U$.
7. If $q\in U$ and $a\subseteq {\kappa }_1\times \omega \times {\kappa }_1$, then $p\cup (q\cap a)\in U$.
8. If $q\in U$ is extendable and ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha (q))=\alpha (q)$, its possible next values form a member of ${\mathrm{\Phi }}_{\alpha (q)}$.
9. If $q\in U$ is extendable and ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha (q))<\alpha (q)$, its possible next values form a member of ${\mathrm{\Phi }}_{\alpha (q),q({\mathrm{cf}}^{\mathrm{\prime }}(\alpha (q)))(n(q))}$.
10. Every $q\in U$ with $q\approx ̸p$ has a predecessor $q^{-}\in U$ obtained by deleting exactly the value last inserted at $(\alpha (q^{-}),n(q^{-}))$.

These clauses are the literal tree upper-part obligations: clauses 5, 8 and 9 give measure-one successor sets; clauses 6, 7 and 10 provide amalgamation, small-coordinate closure and predecessors. For conditions $(q,V)$ and $(p,U)$, write $(q,V)\le (p,U)$, meaning stronger, when ${\mathrm{dom}}_1(q)\supseteq {\mathrm{dom}}_1(p)$ and $V\upharpoonright {\mathrm{dom}}_1(p)\subseteq U$. On a fixed coordinate domain, a direct refinement keeps the trunk fixed and shrinks the upper tree. More generally a finite support enlargement is a direct support extension when its trunk restricts to the old trunk and its projected upper tree refines the old one.

For regular $\theta$, let $P_{\theta }$ be the restriction whose coordinate domains lie below $\theta$, adjoining the trivial empty condition when the restriction has no high coordinate. Because ${\mathrm{cf}}^{\mathrm{\prime }}(\alpha )\le \alpha$, regular initial segments are closed under the dependency map. Each $P_{\theta }$ is a set forcing; $P_3={\bigcup }_{\theta \in \mathrm{Reg}}{P}_{\theta }$ is a definable proper class. The ordinary set-forcing theorem is not thereby a forcing theorem for $P_3$.