---

Source. Richard Laver, An $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$ (1982): the definitions and the statement on p. 173, the proposition $\Phi$ and Kunen's construction on p. 174, and the proof of § 1 on pp. 174--176. The edition is identified on the source card.

Definitions (p. 173). For a regular uncountable cardinal $\kappa$, a $\kappa$-ideal is a nontrivial $\kappa$-complete ideal $\mathcal J$ of subsets of $\kappa$, and $\mathcal J^+$ is the family of subsets of $\kappa$ not in $\mathcal J$. An $\omega_1$-ideal $\mathcal J$ is $(\aleph_2,\aleph_2,\aleph_0)$-saturated when every family of $\aleph_2$ sets in $\mathcal J^+$ has a subfamily of $\aleph_2$ sets in which every countable subfamily has its intersection in $\mathcal J^+$; the paper takes the notion from its author's earlier paper (its [10]).

Statement. If the existence of a huge cardinal is consistent, then it is consistent that there is a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated $\omega_1$-ideal (p. 173). In detail (pp. 174--175): let $j:V\to M$ witness that $\kappa$ is huge, with $j(\kappa)=\lambda$. Laver modifies Kunen's forcing to a two-step forcing $\bar{\mathbb B}=\bar{\mathcal P}\otimes\bar{\mathcal Q}$ after which $\kappa=\omega_1$ and $\lambda=\omega_2$. The extension $V^{\bar{\mathbb B}}$ is a model of Kunen's proposition $\Phi$ below, with a stronger form of its clause (2), and in it an ideal $\mathcal J$ on $\kappa$, defined on p. 175 from an ultrafilter on $\kappa$ in a further extension, is a normal nontrivial ideal on $\omega_1$ that is $(\aleph_2,\aleph_2,\aleph_0)$-saturated.

The proposition $\Phi$ (p. 174). $\Phi$ says that $2^{\aleph_0}=\aleph_1$, that $2^{\aleph_1}=\aleph_2$, and that there are a poset $\mathbb C$ and, in $V^{\mathbb C}$, an elementary embedding $\tilde j:V\to N$ into a transitive class with $\tilde j(\alpha)=\alpha$ for $\alpha<\omega_1^V$ and $\tilde j(\omega_1^V)>\omega_1^V$, such that (1) $\mathbb C$ is a two-step forcing whose first factor has the $\aleph_2$-chain condition and whose second is $\aleph_2$-closed, and (2) $\{\tilde j(\alpha):\alpha<\omega_2^V\}\in N$ and $\tilde j(\omega_1^V)=\omega_2^V$. The paper recalls that Kunen proved $\Phi$ consistent relative to a huge cardinal, that clause (1) gives an $\aleph_2$-saturated ideal on $\omega_1$, and that clause (2) gives Chang's conjecture.

What the paper says about GCH in this model. The paper does not say that the generalized continuum hypothesis holds in its model. Its statements about the continuum function there are three. The model satisfies $\Phi$, whose first two clauses are $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$ (p. 174: Laver shows "how to modify it to get a model" of $\Phi$). The introduction says that $2^{\aleph_0}=\aleph_1$ "holds in the model of §1" (p. 173). The proof uses that $2^\kappa=\lambda$ in $V^{\bar{\mathbb B}}$ (p. 175), that is, $2^{\aleph_1}=\aleph_2$. Nothing is said about $2^{\aleph_2}$ or larger powers, and the paper names no hypothesis on the ground model beyond the huge embedding, assuming familiarity with Kunen's construction (p. 174). The abbreviation GCH occurs only on p. 177, in the question how strong the two polarized relations of § 2 are "assuming GCH", and in the remark that, relative to ZFC alone, $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}>\aleph_2$ are consistent with the first of them (start with a model of GCH and add more than $\aleph_2$ subsets of $\omega_1$ with countable conditions). Komjáth's survey reports this theorem as making GCH consistent with an $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$ (Problem 24, p. 431, of his problem-list survey); that is the survey's wording, not the paper's.

Normality. The ideal is normal: the introduction says so (p. 173), and the proof states that in $V^{\bar{\mathbb B}}$ the ideal $\mathcal J$ "is a normal nontrivial ideal on $\omega_1$" (p. 175).

Proof pointer. Pages 174--176. Kunen's forcing is a two-step product $\mathcal P\otimes\mathcal Q$, with $\mathcal P$ of size $\kappa$ and the $\kappa$-chain condition and $\mathcal Q$ of size $\lambda$ and the $\lambda$-chain condition, together with a regular embedding of it into $j(\mathcal P)$ that fixes $\mathcal P$ and has a master code. Laver replaces $\mathcal P$ by a $\kappa$-stage iteration $\bar{\mathcal P}\subseteq V_\kappa$ of Silver-type collapses with Easton supports, and $\mathcal Q$ by the collapse $\mathcal E_{\kappa\lambda}$. A $\Delta$-system argument using the Mahloness of $\kappa$ shows that for every $\beta<\kappa$, every $\kappa$ conditions of $\bar{\mathcal P}$ include $\kappa$ of which any fewer than $\beta$ have a common extension (the "$(\kappa,\kappa,<\beta)$-cc", p. 175). The same argument, applied to $j(\bar{\mathcal P})$ in $M$, yields the $(\aleph_2,\aleph_2,\aleph_0)$-saturation of $\mathcal J$. Not reconstructed here.

Read depth. Claims checked: the definitions and statements of p. 173, the proposition $\Phi$ with Kunen's clauses (a)--(c) on p. 174, and the assertions on p. 175 that $\bar{\mathcal P}$ has the $(\kappa,\kappa,<\beta)$-cc, that $2^\kappa=\lambda$ in $V^{\bar{\mathbb B}}$ and that $\mathcal J$ is a normal nontrivial ideal on $\omega_1$, were read clause by clause on the page images. The proof on pp. 174--176 was looked through on the page images and no step was checked. Nothing here is independently reviewed.

Dependencies. External, and assumed by the proof: Kunen's construction and his results on $\Phi$ (Saturated ideals, J. Symbolic Logic 43 (1978), 65--76, the paper's [8]). The saturation notion comes from Laver, A saturation property on ideals, Compositio Math. 36 (1978), 233--242 (the paper's [10]). Neither has a card in the library.

Bears on.

• #1172: the model has $2^{\aleph_0}=\aleph_1$, $2^{\aleph_1}=\aleph_2$ and a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$, so by the § 3 theorem it satisfies $\omega_2\to(\omega_1+\omega)^2_2$, a one-line deduction made here; the paper does not state that GCH holds in it, which the problem's consistency question requires.
• #1170: by the same theorem the model satisfies $\omega_2\to(\alpha)^2_2$ for every $\alpha\le\omega_1+\omega_1+1$, again a deduction made here; the paper says nothing about the larger $\alpha<\omega_2$ that the problem also asks about.