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Statement

Setting (p. 6). A set $Y\subseteq\mathbb C$ with $|Y|<2^{\aleph_0}$ is universal (for entire functions) when for every $X\subseteq\mathbb C$ with $|X|<2^{\aleph_0}$ there is a non-constant entire function $f$ with $f(X)\subseteq Y$ (Definition 3.4, p. 6).

Theorem 7.1 (p. 21, quoted). "(CH) There is a proper forcing extension of $V$ preserving all cardinals and cofinalities in which $\mathbb{C}^{V}$ is universal and $2^{\aleph_0} = \aleph_2$. In particular, the existence of a universal set is consistent with $2^{\aleph_0} = \aleph_2$."

Here $\mathbb C^V$ is the set of complex numbers of the ground model $V$, which has cardinality $\aleph_1$ under CH. By Proposition 3.7 the extension also has a Wetzel family, and by Theorem 6.5 it fails MA. The paper asks whether a universal set is consistent with $2^{\aleph_0}=\aleph_3$, or with any successor value of the continuum (Question 8.3, p. 24).

Proof pointer

Section 7, pp. 21--23. For a set $Y$ of complex numbers the poset $\mathbb P(Y)$ (p. 22) adds an entire function by conditions from the poset of Definition 5.1, with finite chains of pairs of countable elementary submodels as side conditions; the points of a condition that first appear in one model of the chain are sent to mutually Cohen generic points of $Y$. It is proper (Lemma 7.2, p. 22), almost preserves Cohen reals (Lemma 7.4, p. 23), and adds a non-constant entire function mapping $\mathbb C^V$ into an everywhere non-meager $Y$ (Lemma 7.5, p. 23). The proof of Theorem 7.1 (p. 23) iterates $\mathbb P(\mathbb C^V)$ with countable support for $\omega_2$ steps.

Dependencies

Lemmas 7.2, 7.4 and 7.5 (pp. 22--23); the poset of Definition 5.1 (p. 11).

Read depth

Claims checked: the statement was read on the printed page. 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: with Proposition 3.7 the extension has a Wetzel family while $2^{\aleph_0}=\aleph_2$, the route Kumar and Shelah proposed (p. 3); such a family has more than $\aleph_1$ members and takes at most $\aleph_1$ values at each point, a negative instance of the problem's question for $\mathfrak m=\aleph_1$. The paper's own answer to Kumar and Shelah's question is Theorem 5.14.