Shelah's CH-length homogeneous sweet construction

Well-order all countable task codes: complete isomorphisms between countably generated complete subalgebras, pairs $C_0\subseteq C_1$ of such subalgebras requiring a free copy over $C_0$, and canonical UM-extension requests. Under CH the set of countable sequences from ${\omega }_1$ has size ${\omega }_1$ by [F2], so use ${\omega }_1$ bookkeeping that repeats every task cofinally often. A homogeneity task retains its last partial extension; later occurrences extend that coherent map over the then-current stage.

Begin with the trivial sweetness model. At a successor, perform the named task only after all of its data have appeared. For an isomorphism task use [F8] to extend the current coherent map over the whole current completion. For $C_0\subseteq C_1$, first use the named amalgam [F4] to create a second canonical copy of $C_1$ freely amalgamated with the first over $C_0$, then use [F3] to extend the copy isomorphism to the required automorphism. For a UM task use the fixed-model composition in [F4]. Every successor is therefore an extension of sweetness models.

At a nonzero limit $\lambda <{\omega }_1$, put $P_{\lambda }={\bigcup }_{\alpha <\lambda }{P}_{\alpha },D_{\lambda }={\bigcup }_{\alpha <\lambda }{D}_{\alpha },E_n^{\lambda }={\bigcup }_{\alpha <\lambda }{E}_n^{\alpha }\mathrm{.}$ Since every such limit is countable and has countable cofinality, [F5] makes this a sweetness model extending every earlier stage. Only after forming this direct union forcing set $B_{\lambda }=BA(P_{\lambda })$. In general $B_{\lambda }$ is not asserted to equal ${\bigcup }_{\alpha <\lambda }{B}_{\alpha }$; completing the direct union is a separate operation.

The recursion is well defined, every $P_{\alpha }$ is sweet, and each earlier $P_{\alpha }$ is complete in every later $P_{\beta }$. Consequently the induced maps make $B_{\alpha }$ a complete subalgebra of $B_{\beta }$. The word "continuous" refers to the forcing/sweetness-model chain of step 1.3, not to an unproved direct union of complete algebras.

Inductively keep $\mathrm{\mid }{P}_{\alpha }\mathrm{\mid }\le {\omega }_1$: each successor construction is made from at most ${\omega }_1$ conditions, and each limit below ${\omega }_1$ is a countable union. Every $P_{\alpha }$ is ccc by [F6]. A completion element is the join of a maximal antichain from the dense image of $P_{\alpha }$, that antichain is countable, and [F6] gives at most $({\omega }_1{)}^{{\mathrm{\aleph }}_0}={\omega }_1$ such codes. Hence $\mathrm{\mid }{B}_{\alpha }\mathrm{\mid }\le {\omega }_1$ at every stage without treating the construction as a finite-support iteration.

Every countable task has a bounded set of birth stages, hence is active at all sufficiently late occurrences of its code. Repeated occurrences of an isomorphism task form a coherent cofinal chain of extensions. A free-amalgamation requirement, once realised, remains realised in all later complete extensions. UM requests occur unboundedly often, so arbitrarily late successor quotients are the canonical UM forcing.

Let $P={\bigcup }_{\alpha <{\omega }_1}{P}_{\alpha }$ and put $B=BA(P)$. This is an ${\omega }_1$-length union, so [F5] does not apply. Instead the special final-union conclusion recorded in [F7] proves the identity $B={\bigcup }_{\alpha <{\omega }_1}BA(P_{\alpha })$ and proves that $B$ is ccc. The identity and step 2.2 give $\mathrm{\mid }B\mathrm{\mid }\le {\omega }_1$. The unboundedly many nontrivial UM quotients make the chain strictly increase unboundedly often, so $\mathrm{\mid }B\mathrm{\mid }\ge {\omega }_1$. Hence $\mathrm{\mid }B\mathrm{\mid }={\omega }_1$.

Let $f:A\to A^{\mathrm{\prime }}$ be a complete isomorphism between countably generated complete subalgebras of $B$. By the final identity in step 3.1, the countable generating data occur at a bounded stage. Its repeated bookkeeping thread extends coherently over unboundedly many later stages, and its union is an automorphism of $B$ extending $f$. This is clause (b) of [F7], not the unsupported extension of a single-stage automorphism. Clause (c) gives the final free-amalgamation property, and clause (d) gives the arbitrarily late UM quotients.

Steps 1.1--2.3 construct the continuous chain of sweetness models; step 3.1 performs the distinct final Boolean-completion argument; and step 4.1 gives the union-level homogeneity, free-amalgamation and UM clauses. This proves the Statement.