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Statement

Setting (pp. 4--5). $\mathcal H(\mathbb C)$ is the set of entire functions. A family $\mathcal F\subseteq\mathcal H(\mathbb C)$ is a Wetzel family when, for every $z\in\mathbb C$, the set $\{f(z):f\in\mathcal F\}$ has cardinality less than $|\mathcal F|$ (Definition 3.1, p. 5).

Theorem 5.14 (p. 17). Assume GCH, and let $\kappa$ be an infinite cardinal of uncountable cofinality. Then some forcing extension preserving cardinals and cofinalities satisfies all of the following:

(1) $2^{\aleph_0}=\kappa$;

(2) a Wetzel family exists;

(3) when $\kappa$ is regular, Martin's Axiom (MA) holds.

By Lemma 3.2 (p. 5) a Wetzel family in the extension has cardinality $2^{\aleph_0}=\kappa$; the introduction states the theorem as giving a Wetzel family of size $\kappa$ (p. 3). Uncountable cofinality is no restriction on the value: by König's theorem $2^{\aleph_0}$ always has uncountable cofinality, and the abstract reads the theorem as the consistency of a Wetzel family with every possible value of the continuum (p. 1).

Context in the paper

Pp. 2--3. Erdős proved that under CH there is a Wetzel family and asked whether its existence is provable without CH. Kumar and Shelah showed that there is no Wetzel family in the side-by-side Cohen model and built a model with a Wetzel family and continuum $\aleph_{\omega_1}$, asking whether a Wetzel family is consistent with $2^{\aleph_0}=\aleph_2$. The paper states that Theorem 5.14 settles that question completely, Wetzel families putting no further restriction on the size of the continuum, and notes that the Kumar and Shelah model necessarily fails MA.

Proof pointer

Pp. 17--19, using the tools of Sections 5.1--5.3 (pp. 11--17). The ground model is the extension of Proposition 4.1 (p. 7; see the Corollary 4.5 page), which has $2^{\aleph_0}=\kappa$ and a sequence of $\kappa$ functions with pairwise finite intersections. Over it a ccc finite support iteration of length $\kappa$ adds, one at a time, entire functions whose values at the complex numbers already listed fall into prescribed countable dense sets of Cohen generic numbers, steered by the almost disjoint sequence, while a bookkeeping function supplies the ccc posets needed for MA when $\kappa$ is regular. The work is to keep the products of the function-adding posets ccc at successor and limit stages.

Dependencies

Proposition 4.1 (p. 7); Lemma 3.2 (p. 5) for the size of the family.

Read depth

Claims checked: the statement was read clause by clause on the printed page, and the context on pp. 1--3. The proof was not checked step by step. Nothing here is independently reviewed.

Source. Jonathan Schilhan and Thilo Weinert, Wetzel families and the continuum, J. Lond. Math. Soc. (2) 109 (2024), no. 6, Paper No. e12918, doi:10.1112/jlms.12918; arXiv:2310.19473. Labels and pages here are those of arXiv:2310.19473v3, the edition read, named on the source card.

Bears on

• Problem 1119: the problem asks, for an infinite cardinal $\mathfrak m$ with $\aleph_0<\mathfrak m<2^{\aleph_0}$, whether every family of entire functions taking at most $\mathfrak m$ values at each point has at most $\mathfrak m$ members. Take $\kappa=\aleph_2$ in Theorem 5.14. In the extension $2^{\aleph_0}=\aleph_2$ and there is a Wetzel family, which has $\aleph_2$ members and fewer than $\aleph_2$, so at most $\aleph_1$, values at each point; with $\mathfrak m=\aleph_1$ it is a family of more than $\mathfrak m$ functions taking at most $\mathfrak m$ values at each point. So the problem's answer is no in that extension, for the case $\mathfrak m^+=2^{\aleph_0}$; the same reading with $\kappa=\mathfrak m^+$ applies to every uncountable $\mathfrak m$. This is the $\aleph_2$ case the paper says answers Kumar and Shelah's question (p. 1). The theorem gives consistency over a model of GCH; it does not address the case $\mathfrak m^+<2^{\aleph_0}$.