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Shelah amalgamation preserves sweetness

Statement

Let $Q_{1}$ and $Q_{2}$ be sweet forcings and let $P_{0}$ be completely embedded in $B A (Q_{1})$ and in $B A (Q_{2})$ . Their Boolean amalgam $Q_{1} *_{P_{0}} Q_{2}$ is sweet and contains complete canonical copies of $Q_{1}$ and $Q_{2}$ . If the sweetness model on $Q_{2}$ extends a fixed model on $Q_{1}$ in the sense of the extension clauses, the amalgam can be equipped with a sweetness model extending that fixed model. The formulation also permits two named complete embeddings of $P_{0}$ , by identifying their images before taking the amalgam.

Facts & Assumptions

Given: Work in ZFC. Let $(Q_{ℓ}, D_{ℓ}, (E_{n}^{ℓ})_{n < ω})$ , $ℓ = 1, 2$ , be sweetness models, and let $i_{ℓ} : P_{0} \to B A (Q_{ℓ})^{+}$ be named complete embeddings. Identify the two images of $P_{0}$ . A pair $(q_{1}, q_{2}) \in Q_{1} \times Q_{2}$ is admitted when some $p_{0} \in P_{0}$ satisfies $p_{0} ⊩ q_{ℓ} \in Q_{ℓ} / P_{0}$ for both $ℓ = 1, 2$ ; write $O = Q_{1} *_{P_{0}} Q_{2}$ for the admitted pairs with the coordinatewise order. This is the source's Definition 7.1, so admission is an intrinsic existential property of the pair, not extra witness data carried by a condition.

[F1]

Shelah sweetness models for forcing: both $(Q_{ℓ}, D_{ℓ}, (E_{n}^{ℓ}))$ satisfy the sequential and transfer clauses with downward directed classes, and the extension relation between sweetness models is the five-clause relation of the Definition.

[F2]

Sweet forcings are countable unions of directed sets and ccc: a sweet forcing itself is a countable union of directed sets and is ccc.

[F3]

Shelah, Claim 7.3(1): if $P < B A (Q)$ and $Q$ is sweet, then $P$ is a countable union of directed subsets. Applied to $P_{0} < B A (Q_{1})$ , this supplies $(A_{j})_{j < ω}$ with $P_{0} = ⋃_{j} A_{j}$ and every $A_{j}$ directed. Definition 7.1 also states the standard amalgam facts: $O$ is forcing-equivalent to $P_{0} * (Q_{1} / P_{0} \times Q_{2} / P_{0})$ and the maps $q_{1} \mapsto (q_{1}, 1_{Q_{2}})$ and $q_{2} \mapsto (1_{Q_{1}}, q_{2})$ are complete embeddings. The weak-coordinate symbols are the distinguished weakest conditions from [F1].

[F4]

Sweet density transfers along complete suborders: the two-part uniform conclusion of the corresponding source claim, in the form that for a condition $q \in D_{ℓ}$ and $p_{0} ⊩ q \in Q_{ℓ} / P_{0}$ there are $j$ and $k$ such that every $q^{'} E_{k}^{ℓ} q$ has some $p_{0}^{'} \in A_{j}$ with $p_{0}^{'} \leq p_{0}$ and $p_{0}^{'} ⊩ q^{'} \in Q_{ℓ} / P_{0}$ , and the union of those $A_{j}$ having this property is dense below $p_{0}$ .

[F5]

The Axiom of Choice: the ambient ZFC assumption permits the witness choices used in Claims 7.3 and 7.4.

[F6]

Shelah's Lemma 7.5 proves the sweetness construction from the displayed least modulus, and Claim 7.12 proves the fixed-model extension construction. Both are cited in the source locator above. [source]

Proof

1.1

Put $D = {(q_{1}, q_{2}) \in O : q_{1} \in D_{1} and q_{2} \in D_{2}} .$ The source amalgamation lemma recorded in [F6] first proves that $D$ is dense. Its proof keeps the admission witness until after both coordinates have been strengthened: below a witness $p_{0}$ , apply the dense conclusion of [F4] to the first coordinate; reindex the resulting subfamily of the directed cover $(A_{j})$ , whose union remains dense below $p_{0}$ ; then apply [F4] to the second coordinate. Directedness inside one $A_{j}$ produces a common strengthening of the two quotient witnesses. Thus the result is still an admitted pair, rather than merely two independently dense coordinates.

F3F4F5F6

1.2

The same two applications prove the intrinsic assertion

$(*)_{x} \exists m < ω \forall q_{1}^{'} E_{m}^{1} q_{1} \forall q_{2}^{'} E_{m}^{2} q_{2} (q_{1}^{'}, q_{2}^{'}) \in O$

for every $x = (q_{1}, q_{2}) \in D$ . Define $m (x)$ to be the least such $m$ . No chosen witness, index $j$ , or reduction is part of $m (x)$ . If $q_{ℓ}^{'} E_{m (x)}^{ℓ} q_{ℓ}$ for both coordinates and $x^{'} = (q_{1}^{'}, q_{2}^{'}) \in D$ , then the $E_{m (x)}^{ℓ}$ -classes of the old and new coordinates coincide, so $(*)_{x^{'}}$ holds at $m (x)$ . Conversely, if it held at some $r < m (x)$ for $x^{'}$ , refinement gives $q_{ℓ}^{'} E_{r}^{ℓ} q_{ℓ}$ , hence the two $E_{r}^{ℓ}$ -classes coincide and it would hold for $x$ , contradicting minimality. Therefore

$(* *)_{x} m (x^{'}) = m (x) .$

This is exactly the least-modulus convention and the displayed observation in the source amalgamation lemma [F6]. [F1, F3, F4, F5, F6]

2.1

On $D$ define $x E_{n} x^{'} ⟺ m (x) = m (x^{'}) = : m and q_{ℓ} E_{m + n}^{ℓ} q_{ℓ}^{'} (ℓ = 1, 2) .$ The relation is intrinsic because $m$ is intrinsic. The observation $(* *)$ gives reflexivity and makes the common- $m$ condition stable under the coordinate equivalences; symmetry and transitivity then follow from the old relations. Refinement is immediate. An $E_{n}$ -class is encoded by $m$ and one $E_{m + n}^{1}$ -class and one $E_{m + n}^{2}$ -class, so there are countably many classes.

F1step 1.2

3.1

The remaining sweetness checks are those of the source amalgamation lemma [F6] with precisely this definition. For directedness, take coordinatewise common lower bounds inside the two $E_{m + n}^{ℓ}$ -classes; since they still lie in the corresponding $E_{m}^{ℓ}$ -classes, $(*)_{x}$ admits the resulting pair, and $(* *)_{x}$ keeps its modulus equal to $m$ . For a diagonal sequence, apply the sequential clause in each coordinate; the tail bounds lie in the required $E_{m + n}^{ℓ}$ -classes, so the same $(*), (* *)$ argument makes them admitted $E_{n}$ -bounds. For transfer, use the two coordinate transfer moduli and then the common-admission conclusion obtained in step 1.1 from [F4]; $(* *)$ again keeps the output in the prescribed amalgam class. These checks prove the directed, sequential and transfer clauses without selecting a witness as part of a condition. Thus $(O, D, (E_{n}))$ is a sweetness model.

F1F4F6step 1.1step 1.2step 2.1

4.1

By the standard amalgam facts recorded in [F3], the weak-coordinate maps $q_{1} \mapsto (q_{1}, 1_{Q_{2}})$ and $q_{2} \mapsto (1_{Q_{1}}, q_{2})$ are complete embeddings into $O$ . This conclusion is about the canonical copies in the full amalgam; it does not require those copies to lie in the particular dense presentation $D$ . Sweetness also implies ccc by [F2].

F2F3step 3.1

4.2

Now suppose $(Q_{1}, D_{1}, (E_{n}^{1})) < (Q_{2}, D_{2}, (E_{n}^{2}))$ in the exact five-clause sense of [F1]. The fixed-model source result [F6] applies to the same named embeddings and the preceding amalgam. It first replaces the initial dense presentation by an equivalent dense-open presentation $D^{*}$ separated from the canonical old dense set, then sets $D^{'} = D^{*} \cup D_{1}$ . On $D^{*}$ it uses the new relations and on $D_{1}$ it uses exactly the old $E_{n}^{1}$ ; there are no cross-piece classes. The mixed case of the transfer clause is checked by the comparable transfer form in [F1], as in the source's preceding fixed-model argument. The source result then verifies: the canonical $Q_{1}$ is a complete suborder, $D_{1} \subseteq D^{'}$ , the old relations are the restrictions, every new class meeting $D_{1}$ remains in $Q_{1}$ , and if an old condition strengthens a member of $D^{'}$ , that member already lies in $D_{1}$ . Hence this is a sweetness model on $O$ extending the fixed model on $Q_{1}$ .

F1F6step 3.1

5.1

Steps 3.1--4.2 prove every assertion in the Statement, including the named canonical-copy and fixed-model interfaces.

step 3.1step 4.1step 4.2∎

Depends on

Used by

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Sources

Saharon Shelah, Can You Take Solovay's Inaccessible Away? (standard reference, not scraped)