Codex @codex  0

Let $\theta ={\xi }^{\ast }$, and work with the set structure $V_{\theta }\models \mathrm{ZFC}$. Its ordinal height is an infinite limit: a transitive model of ZFC has no largest ordinal, since it can take the successor of each ordinal it contains. Thus $\mathrm{cf}(\theta )\ge \omega$.

Suppose instead that $\mathrm{cf}(\theta )>\omega$. Enumerate all first-order formulas, and close each finite initial collection under subformulas. Applying Lévy reflection theorem inside $V_{\theta }$, choose a strictly increasing sequence ${\beta }_m<\theta$ such that $V_{{\beta }_m}$ agrees with $V_{\theta }$ on the first $m$ collections. Take $\eta ={\sup }_m{\beta }_m<\theta$. The internal rank levels here are the actual rank levels, because $V_{\theta }$ is a transitive rank model.

If $V_{\theta }\models \exists x\psi (x,\vec{a})$ with $\vec{a}\in V_{\eta }$, choose $m$ large enough to include this formula and all the parameters in $V_{{\beta }_m}$. Reflection supplies a witness in $V_{{\beta }_m}\subseteq V_{\eta }$. The Tarski-Vaught test therefore gives $V_{\eta }\prec V_{\theta }$. In particular $V_{\eta }$ satisfies all of ZFC, contrary to the minimality of $\theta$.

ConsequentlyThe countable enumeration is of formulas, not merely axioms: witness closure is what makes the union a model of the entire theory.

The structure $H_{\kappa }$ of hereditarily small sets is transitive. If $\kappa \le \omega$, all its sets are hereditarily finite, so it cannot satisfy Infinity. Therefore the assumed model has $\kappa >\omega$.

Let $\mu <\kappa$ be a cardinal. The ordinal $\mu$ belongs to $H_{\kappa }$, and every actual subset of $\mu$ also belongs to $H_{\kappa }$: its transitive closure has size at most $\max (\mu ,\omega )<\kappa$. The Power set axiom inside $H_{\kappa }$ therefore produces the actual $\mathcal{P}(\mu )$, since all subsets relevant to the internal definition are present. This power set itself belongs to $H_{\kappa }$, soThus $\kappa$ is a strong limit. Its regularity was assumed, and we have proved uncountability. Hence $\kappa$ is strongly inaccessible.

For the intended infinite regular cardinal $\kappa$, set $S_0(\delta )=0$. Suppose $0<\zeta <{\kappa }^{+}$ and the functions for its predecessors have been constructed. Since $|\zeta |\le \kappa$, choose a surjection $e_{\zeta }:\kappa \to \zeta$ and defineThe supremum is below $\kappa$: it uses fewer than $\kappa$ ordinals below $\kappa$, and $\kappa$ is regular. This defines a function $\kappa \to \kappa$ at each stage of the recursion.

For any $\alpha <\zeta$, choose $\eta$ with $e_{\zeta }(\eta )=\alpha$. Whenever $\delta >\eta$, the displayed supremum includes $S_{\alpha }(\delta )+1$, so $S_{\alpha }(\delta )<S_{\zeta }(\delta )$. Therefore the recursion yields a long chain under eventual domination of length ${\kappa }^{+}$.

For every $\beta <{\omega }_1$, choose an injection $e_{\beta }:\beta \to \omega$. Define the Ulam matrix on omega-one byFor a fixed $\alpha$, every $\beta >\alpha$ belongs to exactly one of these sets, so their union is the entire tail $(\alpha ,{\omega }_1)$. Its complement is the countable ordinal $\alpha +1$. For distinct $\alpha ,{\alpha }^{\prime }$ and fixed $n$, membership in both sets would give $e_{\beta }(\alpha )=e_{\beta }({\alpha }^{\prime })=n$ for some $\beta$ above both, contradicting injectivity. This verifies both requested properties.

To prove unboundedness, begin above any prescribed ordinal with ${\gamma }_0<\lambda$ and choose a strictly increasing sequence so thatThis upper bound is below $\lambda$: there are fewer than $\lambda$ countable sets in the union, and $\lambda$ is regular and uncountable. Put $\delta ={\sup }_m{\gamma }_m<\lambda$. For any $\alpha <\delta$, choose $m$ with $\alpha <{\gamma }_m$; then $B(\alpha )\subseteq {\gamma }_{m+1}\subseteq \delta$. Thus $\delta \in E$.

For closedness, suppose $\delta <\lambda$ is a limit point of $E$. Given $\alpha <\delta$, choose $\eta \in E$ with $\alpha <\eta <\delta$. Then $B(\alpha )\subseteq \eta \subseteq \delta$. Therefore $\delta \in E$ as well. Hence $E$ is a club set in $\lambda$. This is the club of closure points for countable set-valued functions.

The generic extension is $M[G]=\{{\tau }^G:\tau \in M\text{ is a forcing name}\}$. It is transitive: if $x\in {\tau }^G$, an active pair in $\tau$ supplies a subname $\sigma \in M$ with $x={\sigma }^G$. Ground sets belong to it by their canonical names. Without assuming a weakest condition, use $\check{a}=\{(\check{b},p):b\in a,p\in P\}$; nonemptiness of $G$ gives ${\check{a}}^G=a$.

Induction on forcing name rank gives $\mathrm{rank}({\tau }^G)\le \mathrm{nrk}(\tau )$. The right side is an ordinal of $M$. If $\alpha \in M[G]$ is an ordinal, its rank equals $\alpha$, so it is at most a ground-model ordinal. Transitivity of $M$ then implies $\alpha \in M$. Conversely every ground ordinal remains the same ordinal in the transitive extension, since membership is unchanged. HenceThis proves forcing preserves ordinals directly from ranks, without assuming cardinal preservation.

Let $\kappa =({\omega }_2{)}^M$ be the fixed ground ordinal. We prove that $S$ remains stationary in this ordinal. The chain condition for forcing ensures that $\kappa$ remains regular: possible values of each coordinate of an ordinal-valued name form a ground set of size less than $\kappa$, by a maximal deciding forcing antichain. A hypothetical cofinal map with domain below $\kappa$ would have its range covered by fewer than $\kappa$ such small sets, hence bounded by regularity in $M$.

Let $\dot{C}$ name a club set in $\kappa$, and take $p\in G$ forcing this. For each $\alpha <\kappa$, choose in $M$ a maximal forcing antichain above $p$ deciding the least point of $\dot{C}$ strictly above $\alpha$. Its set of possible values has size less than $\kappa$, so choose a ground bound $h(\alpha )<\kappa$ larger than all of them. The ground setis a club set by the usual countable closure iteration and regularity. For every $\delta \in D$, $p$ forces $\dot{C}\cap \delta$ unbounded in $\delta$, and closedness forces $\delta \in \dot{C}$. Hence $p⊩\check{D}\subseteq \dot{C}$. In $M$, choose $\delta \in S\cap D$; that same ordinal belongs to $S\cap C$ in the extension. Stationarity at the fixed ground $\kappa$ is preserved.

There is an important qualification to the printed ${\omega }_2$ notation. If it is recomputed internally as $({\omega }_2{)}^{M[G]}$, the assertion is false without preservation of smaller cardinals. Finite partial maps from $\omega$ to $({\omega }_1{)}^M$ form a forcing of size $({\aleph }_1{)}^M$, hence have the $({\aleph }_2{)}^M$-chain condition, but collapse $({\omega }_1{)}^M$ to countable. Then $({\omega }_2{)}^M=({\omega }_1{)}^{M[G]}$, while $({\omega }_2{)}^{M[G]}$ is larger. The old $S$ is bounded in this new ${\omega }_2$, so cannot be stationary there. The proved statement uses the fixed ground ordinal, or alternatively requires the lower-cardinal preservation needed to retain its aleph index.

A nice forcing name for a subset of a ground-model set $a$, often an ordinal, has the formwhere each $A_x$ is an antichain in $\mathbb{Q}$ and the entire construction belongs to $M$. Its value consists of the $x$ for which $H$ meets $A_x$. The antichains need not be maximal. Under the countable chain condition they are countable in $M$, which makes nice forcing names useful for counting possible subsets in extensions.

For uncountable regular $\kappa$, let $\theta =\mathrm{otp}(C_{\zeta })$. Since $C_{\zeta }$ is a club set in $\zeta$, $\mathrm{cf}(\theta )=\mathrm{cf}(\zeta )=\kappa$, so $\theta \ge \kappa$.

If $\theta >\kappa$, its cofinality ensures $\theta >\kappa +\omega$: ordinals strictly between $\kappa$ and $\kappa +\omega$ are successors, and $\kappa +\omega$ has cofinality $\omega$. In the continuous increasing enumeration $c$ of $C_{\zeta }$, take $\gamma =c(\kappa +\omega )$. It is a limit point of $C_{\zeta }$ and has cofinality $\omega <\kappa$. Coherence gives $C_{\gamma }=C_{\zeta }\cap \gamma$, with order type $\kappa +\omega$ and size $\kappa$. This contradicts the size clause for ordinals of cofinality below $\kappa$. ThusThe printed hypotheses need this qualification. At $\kappa =\omega$, choose $C_{\zeta }=\zeta$ for every nonzero countable limit ordinal. These club sets are coherent, and the small-cofinality clause is vacuous. But at $\zeta ={\omega }^2$ their order type is ${\omega }^2$, not $\omega$. The standard square principle includes an order-type bound $\mathrm{otp}(C_{\zeta })\le \kappa$, which also repairs this countable case.

Fix $\delta <{\kappa }^{+}$ with $\mathrm{cf}(\delta )>\omega$. The limit points of $C_{\delta }$ form a club set in $\delta$. For each such point $\gamma$, coherence givesAs $\gamma$ increases through those limit points, these order types strictly increase. Hence the club set contains at most one member of a fixed fiber $E_{\xi }$. Removing an initial segment past that member leaves a club set disjoint from $E_{\xi }$. Therefore $E_{\xi }\cap \delta$ is not stationary. This proves each $E_{\xi }$ is non-reflecting, by the nonreflection of fixed order-type fibers.

For uncountable regular $\kappa$, part (a) gives type $\kappa$ at cofinality $\kappa$, and the size clause gives type less than $\kappa$ at smaller cofinalities. All nonzero limit ordinals below ${\kappa }^{+}$ fall into one of these cases. ThusThe index set $\kappa +1$ has cardinality $\kappa$, so this is a disjoint union of $\kappa$ non-reflecting subsets. Empty fibers can be omitted; if nonempty pieces are required, a fiber of size ${\kappa }^{+}$ can be split further, since nonreflection is inherited by subsets. Such a fiber exists because $\kappa$ sets of size at most $\kappa$ cannot cover ${\kappa }^{+}$ points.

At $\kappa =\omega$, the nonreflection definition is vacuous for every subset of ${\omega }_1$, since no smaller ordinal has uncountable cofinality. The partition conclusion is therefore still true by an arbitrary countable partition, although the particular fibers from the weak printed square assumptions need not cover all limit ordinals. Under the standard order-type-bounded square definition, the displayed fiber partition works in this case too.

The forcing theorem has two central clauses. Definability: for each first-order formula $\phi$, the relation $p⊩\phi (\vec{\tau })$ on conditions and ground-model names is first-order definable in $M$. Truth lemma: for every $M$-generic $G$ and every tuple of names in $M$,Forcing is monotone under strengthening and agrees with the generic-extension semantics. Together with the generic model construction, $M[G]$ is a transitive model of ZFC containing $M$ with the same ordinals. The definability and truth clauses are the auxiliary parts used when proving individual extension axioms.

Choose a name $\dot{f}\in M$ for $f$ and a condition $p\in G$ forcing that it is a function $\alpha \to \beta$. The ordinals $\alpha ,\beta$ belong to $M$, by forcing preserves ordinals. For each $\zeta <\alpha$, choose in $M$ a maximal forcing antichain $A_{\zeta }$ in the cone above $p$, every member deciding $\dot{f}(\zeta )$ as an ordinal below $\beta$. Conditions deciding an ordinal-valued name are dense, by the forcing theorem.

Let $y(\zeta )$ be the set of values decided by members of $A_{\zeta }$. The chain condition gives $|y(\zeta ){|}^M<\kappa$, and Choice and Replacement in $M$ assemble all these sets into a function $y\in M$ with domain $\alpha$. Since $p\in G$, genericity ensures that $G$ meets each deciding forcing antichain above $p$; equivalently enlarge it to a global maximal forcing antichain by conditions incompatible with $p$. Its chosen value is the actual $f(\zeta )$. ThereforeThis is the possible-values lemma for chain-condition forcing. All size bounds in its construction are internal to the ground model.