Preservation, Cohen Forcing, and the Continuum

Suppose $P$ is $\kappa$-closed. Given $p$ and dense open $D_{\xi }$ for $\xi <\gamma <\kappa$, recursively choose $p_{\xi +1}\le p_{\xi }$ in $D_{\xi }$ and at each limit take a lower bound. Regularity keeps every stage below $\kappa$; a final lower bound belongs to every $D_{\xi }$. Thus $P$ is $\kappa$-distributive. AC is used for the recursive choices.

Suppose $P$ is $\kappa$-distributive and $p⊩\dot{f}:\check{\gamma }\to \check{V}$ for $\gamma <\kappa$. For each $\xi <\gamma$, the set of conditions deciding $\dot{f}(\xi )$ is dense open. A common extension $q$ decides every coordinate, say as $x_{\xi }\in V$. Replacement forms $f=⟨x_{\xi }:\xi <\gamma ⟩$ in the ground model and $q⊩\dot{f}=\check{f}$. Conversely, assume that separative $P$ adds no such sequence. Given maximal antichains $A_{\xi }$ for $\xi <\gamma$, let $\dot{f}(\xi )$ be the unique member of $A_{\xi }$ met by the generic. This is a name for a $\gamma$-sequence of ground-model conditions, hence conditions deciding its whole ground-model value are dense. If $q$ decides that value as $f_q$, then $q\le f_q(\xi )$ for every $\xi$: otherwise separativity gives $r\le q$ incompatible with $f_q(\xi )$, while a generic through $r$ must both realize the decision and meet $A_{\xi }$, a contradiction. A maximal antichain of such $q$ therefore refines every $A_{\xi }$. Replacing each dense open set by a maximal antichain contained in it proves that the intersection of the original family is dense. AC is used to choose the maximal antichains.

A subset of $\gamma <\kappa$ is a $2$-valued sequence of length $\gamma$, so closure adds none. A new cofinal map into a ground-model ordinal of cofinality at most $\kappa$, or a collapse of a cardinal at most $\kappa$, would yield after restricting to a cofinal domain a new sequence of ground-model ordinals of length below $\kappa$. Hence those cofinalities and cardinals are preserved.

If $p⊩\dot{f}:\check{\mu }\to \check{\lambda }$, choose for each $\xi <\mu$ a maximal antichain below $p$ deciding $\dot{f}(\xi )$. Each has size $<\theta$, so the ground-model set $B_{\xi }$ of possible values has size $<\theta$. Then $p⊩\mathrm{ran}(\dot{f})\subseteq {\bigcup }_{\xi <\mu }{B}_{\xi }$. AC is used for maximal antichains and their simultaneous selection.

Let $\lambda \ge \theta$ be regular and $\mu <\lambda$. If a condition forced $\dot{f}:\mu \to \lambda$ cofinal, step 1.1 and regularity would put its range inside a ground set of size $\max (\mu ,<\theta )<\lambda$, which is bounded in $\lambda$, contradiction. Thus regular cofinalities at least $\theta$ are preserved; F2 transfers this to every ground cofinality at least $\theta$.

Suppose a ground cardinal $\lambda \ge \theta$ were collapsed. By F4, some $\mu <\lambda$ and a condition $p$ would force a surjection $\dot{f}:\mu \to \lambda$. Step 1.1 puts its range inside the ground set $U={\bigcup }_{\xi <\mu }{B}_{\xi }$, with $\mathrm{\mid }{B}_{\xi }\mathrm{\mid }<\theta$. If $\lambda =\theta$, regularity of $\theta$ gives $\mathrm{\mid }U\mathrm{\mid }<\theta$; if $\lambda >\theta$, cardinal arithmetic under AC gives $\mathrm{\mid }U\mathrm{\mid }\le \mu \cdot \theta =\max (\mu ,\theta )<\lambda$ (with the finite cases immediate). Either way $p$ cannot force $\dot{f}$ onto $\lambda$. Thus every ground cardinal at least $\theta$ remains a cardinal. For ccc, $\theta ={\mathrm{\aleph }}_1$; finite and countable cardinals and cofinalities are absolute, so steps 2.1 and 3.1 cover all of them.

For every $a\in A$, choose a maximal antichain $A_a$ below $p$ consisting of conditions deciding $\check{a}\in \dot{x}$, retain its positive members $B_a$, and let $\dot{y}=\{⟨\check{a},q⟩:a\in A,q\in B_a\}$. Fix $q\le p$ and $a\in A$. Maximality gives a common extension $r\le q,s$ for some $s\in A_a$. If $s$ is positive, persistence makes $r$ force membership in $\dot{x}$, while the coefficient $s\in B_a$ makes $r$ force membership in $\dot{y}$ by F4. If $s$ is negative, persistence makes $r$ force nonmembership in $\dot{x}$, and incompatibility with every member of $B_a$ leaves no extension of $r$ forcing membership in $\dot{y}$, so the negation clause makes $r$ force nonmembership there. Thus conditions agreeing on the membership of each ground element are dense below $p$. Since $p$ forces $\dot{x}\subseteq \check{A}$ and the displayed coefficients make $\dot{y}$ a name for a subset of $\check{A}$, density closure and the two extensional clauses in F4 give $p⊩\dot{x}=\dot{y}$. The simultaneous maximal-antichain choice is the first use of AC.

Under ccc, each $A_a$ is countable. There are at most ${\mu }^{{\mathrm{\aleph }}_0}$ countable subsets of $P$, so a nice name is coded by a $\lambda$-sequence of such subsets and their number is at most $({\mu }^{{\mathrm{\aleph }}_0}{)}^{\lambda }$. For $A=\omega$, cardinal exponentiation gives $({\mu }^{{\mathrm{\aleph }}_0}{)}^{{\mathrm{\aleph }}_0}={\mu }^{{\mathrm{\aleph }}_0}$. These counts use AC to identify all sets with cardinals.

Well-order the family as $⟨x_{\xi }:\xi <\theta ⟩$. Its union has cardinality at most $\theta \cdot \kappa =\theta$, so transport its elements into $\theta$. Since $\theta$ is regular and there are only $\kappa <\theta$ possible order types below $\kappa$, thin to constant order type $\eta <\kappa$ and enumerate each remaining set increasingly as $x_{\xi }=\{x_{\xi }(i):i<\eta \}$.

Fewer than $\theta$ members of the thinned family can lie wholly below any fixed $\alpha <\theta$, because there are only $\mathrm{\mid }\alpha {\mathrm{\mid }}^{<\kappa }<\theta$ such sets. Its union is therefore unbounded in $\theta$, so some coordinate $i<\eta$ has unbounded values; let $i_0$ be least. For every $i<i_0$ the coordinate values are bounded, and regularity gives ${\alpha }_0=\sup \{x_{\xi }(i)+1:\xi <\theta ,i<i_0\}<\theta \mathrm{.}$ Recursively choose $x_{{\xi }_{\mu }}$ for $\mu <\theta$ so that $x_{{\xi }_{\mu }}(i_0)$ exceeds ${\alpha }_0$ and every element of the earlier chosen sets. The unboundedness of the $i_0$th values makes each choice possible. Consequently distinct chosen sets meet only below ${\alpha }_0$.

There are at most $\mathrm{\mid }{\alpha }_0{\mathrm{\mid }}^{<\kappa }<\theta$ possible intersections $x_{{\xi }_{\mu }}\cap {\alpha }_0$. Regularity therefore yields a set $J\subseteq \theta$ of size $\theta$ on which this intersection is one fixed $r$. Step 1.2 then gives $x_{{\xi }_{\mu }}\cap x_{{\xi }_{\nu }}=r$ for distinct $\mu ,\nu \in J$. AC was used to well-order, enumerate, choose recursively, and thin.

Suppose now that $\kappa$ is regular, $\rho =2^{<\kappa }$, and $\theta ={\rho }^{+}$. For every $\mu <\kappa$, a function $\mu \to \rho$ uses fewer than $\kappa$ members of the union $\rho ={\bigcup }_{\nu <\kappa }{}^{\nu }2$; regularity bounds all their lengths below one $\nu <\kappa$. Coding the function by a binary array of size below $\kappa$ shows ${\rho }^{\mu }\le 2^{<\kappa }=\rho$. Thus ${\rho }^{<\kappa }=\rho$, and for $\alpha <\theta$ one has $\mathrm{\mid }\alpha {\mathrm{\mid }}^{<\kappa }\le {\rho }^{<\kappa }=\rho <\theta$. The main clause applies.

The union of a descending chain of length $\gamma <\kappa$ is a function. Regularity makes its domain, a union of $\gamma$ many sets of size below $\kappa$, again have size below $\kappa$. It is therefore a common lower bound.

Let $\rho =2^{<\kappa }$ and take ${\rho }^{+}$ conditions. For $\kappa >\omega$, F2 thins their domains to a ${\rho }^{+}$-sized delta system with root $r$; for $\kappa =\omega$, use F3. There are at most $2^{\mathrm{\mid }r\mathrm{\mid }}\le \rho$ root restrictions, so two conditions agree on $r$. Their union is a function and a common extension, contradicting antichainhood. Thus the order is ${\rho }^{+}$-cc.

At $\kappa =\omega$, finite domains and the finite delta-system theorem give ccc directly. If $2^{<\kappa }=\kappa$, step 1.2 reads ${\kappa }^{+}$-cc. The thinning and pigeonhole steps use AC.

A descending sequence of fewer than $\kappa$ collapse conditions has union of domain size below $\kappa$, so $\mathrm{Col}(\kappa ,\lambda )$ is $\kappa$-closed. For each $\xi <\kappa$ and $\beta <\lambda$, the sets requiring $\xi$ in the domain and $\beta$ in the range are dense (using a fresh coordinate for the latter). Hence the generic union is a total surjection $\kappa \twoheadrightarrow \lambda$.

Given $\theta$ many Lévy conditions, F3 thins their finite domains to a delta system with a fixed finite root. At each root coordinate $(\alpha ,n)$ there are only $\mathrm{\mid }\alpha \mathrm{\mid }<\theta$ possible values; regularity and finiteness of the root therefore leave fewer than $\theta$ possible root assignments. Thin the $\theta$ conditions until their root restrictions agree. Two remaining conditions then have compatible union, so $\mathrm{Lv}(\theta )$ is $\theta$-cc.

For every $0<\alpha <\theta$, the union of the generic restrictions to $\{\alpha \}\times \omega$ is total by coordinate dense sets and hits every $\beta <\alpha$ by range dense sets. It is a surjection $\omega \twoheadrightarrow \alpha$. step 1.2 and F2 preserve the cardinal and regularity of $\theta$, while all smaller infinite ordinals become countable; hence the extension identifies $\theta$ with ${\mathrm{\aleph }}_1$.

For fixed $(\xi ,n)$, conditions defining that bit are dense, so $c_{\xi }$ is total. For $\xi \mathrm{\ne }\eta$, below any condition choose a fresh $n$ and assign opposite bits at $(\xi ,n)$ and $(\eta ,n)$; this dense set proves $c_{\xi }\mathrm{\ne }{c}_{\eta }$.

Restriction is an order isomorphism $\mathrm{Add}(\omega ,\lambda )\cong \mathrm{Add}(\omega ,I)\times \mathrm{Add}(\omega ,J)$, with inverse union, so each projection is ground-model generic. Let $D={\dot{D}}_{G_I}$ be dense open in the second factor in $M[G_I]$. Choose $p\in G_I$ forcing that $\dot{D}$ is dense. Below $(p,1)$, pairs $(r,t)$ for which $r⊩\check{t}\in \dot{D}$ are dense: below any $(r,s)$, forced density supplies an extension $t\le s$ in $\dot{D}$, and one may strengthen $r$ to decide a witnessing ground-model $t$. Adjoining all pairs whose first coordinate is incompatible with $p$ makes this a ground-model dense subset of the full product. The product generic meets it, and directedness with $p\in G_I$ excludes the incompatible branch; its second coordinate is the actual $t\in G_J\cap D$. Thus $G_J$ is generic over $M[G_I]$, and symmetry gives the reverse direction. If $I$ or $J$ is empty, its factor is the one-condition forcing, its projection gives the unique filter, and the same statement is literal.

Evaluation of a product name can be performed successively in either coordinate, and the full generic is recovered as $G_I\times G_J$. The three extension models are therefore equal. No choice beyond the stated ZFC background is hidden in the coordinate construction.

By F1, F2 all cardinals are preserved. F3 supplies an injection $\lambda \hookrightarrow \mathcal{P}(\omega )$ in the extension, so $2^{{\mathrm{\aleph }}_0}\ge \lambda$.

The forcing has size $\lambda$. Under the additional hypotheses F4 gives at most ${\lambda }^{{\mathrm{\aleph }}_0}=\lambda$ nice names for reals. Every real has one, so $2^{{\mathrm{\aleph }}_0}\le \lambda$; combine with step 1.1.

Under GCH, the ground continuum is ${\mathrm{\aleph }}_1$ and $({\mathrm{\aleph }}_2{)}^{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_2$. Apply step 2.1 with $\lambda ={\mathrm{\aleph }}_2$; preservation ensures that this remains the extension's ${\mathrm{\aleph }}_2$. AC is used in F2, F4, and the cardinal computations.

F1, F2 preserve cardinals at most $\kappa$ by closure and at least ${\kappa }^{+}$ by the chain condition, hence all cardinals. The coordinate-domain and bit-separation dense sets from the Cohen calculation give $\lambda$ distinct subsets of $\kappa$, so $2^{\kappa }\ge \lambda$.

Replace the countable antichains in the nice-name proof by antichains of size at most $\kappa$. A nice name for a subset of $\kappa$ is coded by $\kappa$ many subsets of $P$ of size at most $\kappa$. Since $\mathrm{\mid }P\mathrm{\mid }=\lambda$ and ${\lambda }^{\kappa }=\lambda$, there are at most $({\lambda }^{\kappa }{)}^{\kappa }=\lambda$ such names. Thus $2^{\kappa }\le \lambda$, proving equality.

If $\lambda \ge {\kappa }^{++}$, equality gives $2^{\kappa }>{\kappa }^{+}$, so GCH fails. Under GCH with $\kappa ={\mathrm{\aleph }}_1$ and $\lambda ={\mathrm{\aleph }}_3$, the hypotheses hold; $\kappa$-closure adds no reals, so CH remains true, while step 1.2 gives $2^{{\mathrm{\aleph }}_1}={\mathrm{\aleph }}_3$.

In ZFC let $\lambda =(2^{{\mathrm{\aleph }}_0}{)}^{+}$ and use $P=\mathrm{Add}(\omega ,\lambda )$. The empty condition witnesses nonemptiness. The finite-partial-function definition gives the preorder and generic-coordinate names as sets. The delta-system ccc argument and the maximal-antichain cardinal-preservation argument in F2 are ZFC proofs; for this fixed $\mathrm{\Delta }$, collect the finitely many axioms and schema instances they use. AC is used for the cardinal successor and the maximal-antichain argument.

For distinct $\xi ,\eta <\lambda$, extending a condition at a fresh bit forces the $\xi$ and $\eta$ coordinate reals to differ. Thus $\lambda$ injects into the reals of the extension. Since $P$ preserves ${\mathrm{\aleph }}_2$, it forces $2^{{\mathrm{\aleph }}_0}\ge {\mathrm{\aleph }}_2$, hence $\mathrm{\neg }\mathrm{C}\mathrm{H}$. GCH implies CH at $\omega$, so the same extension forces $\mathrm{\neg }\mathrm{G}\mathrm{C}\mathrm{H}$. No ground-model CH or equality for the continuum is used.

For the chosen $\mathrm{\Delta }$, include its finitely many ZFC axioms and the finite instances needed to verify the forcing relation, the generic extension, cardinal preservation, and step 2.1. The forcing theorem supplies a formal derivation that every condition forces each target member; the quantified schema instances in $\mathrm{\Delta }$ are handled one at a time as their actual formulas, with their translated forcing instances included in the finite source fragment. F1 now yields a ZFC proof that a model of this fixed $\mathrm{\Delta }$ exists. The choices of proofs and finite support may depend on $\mathrm{\Delta }$; no arithmetic uniformity or PA checker theorem follows.

If $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{\neg }\mathrm{C}\mathrm{H}$ were inconsistent, a contradiction proof would use only a finite set $\mathrm{\Delta }$ of its axioms. Fix that $\mathrm{\Delta }$ externally. F1 gives a ZFC proof of the existence of a model of $\mathrm{\Delta }$. Soundness for the finite proof shows that no such model exists, so ZFC itself would be inconsistent. Contraposition gives the first consistency implication.

The same argument with a finite refutation of $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{\neg }\mathrm{G}\mathrm{C}\mathrm{H}$ and the second F1 construction gives the other implication. Both are external consequences of fixed-fragment proofs; the positive consistency counterparts are separate results.