Unnecessary uses of the Continuum Hypothesis

Question

This question was inspired by the MathOverflow question "Unnecessary uses of the axiom of choice". I want to know of statements in ZFC that can be proven by assuming the Continuum Hypothesis, but can also be proven by a more elaborate (perhaps even significantly more elaborate) proof without assuming the Continuum Hypothesis.

Answer 1

Theorem: The space $\mathbb N^*$ of non-principal ultrafilters on $\mathbb N$ is not homogeneous.

Using CH, it is fairly straightforward to prove there is a special kind of ultrafilter called a $P$-point. A point $u$ of $\mathbb N^*$ is a $P$-point if any countable intersection of open neighborhoods of $u$ is again a neighborhood of $u$. Not all points of $\mathbb N^*$ are $P$-points (regardless of CH). Walter Rudin proved in 1956 that CH implies that $\mathbb N^*$ contains $P$-points, so this shows the space is non-homogeneous.

But the non-homogeneity of $\mathbb N^*$ is a theorem of ZFC. This was proved years later in 1967 in Frolík - Sums of ultrafilters (building on some unpublished work of Kunen). As I understand it, the non-homogeneity of $\mathbb N^*$ was a hot-topic open question during the intervening years, which demonstrates how much tougher the non-CH proof is.

Answer 2

A Dowker space is a normal Hausdorff space whose product with the closed unit interval $I$ is not normal. In 1971, Mary Ellen Rudin constructed the first ZFC Dowker space, which had cardinality $\aleph_\omega^\omega$. This space was considered "large" and for a long time it was an open problem to construct a "small" Dowker space. Various people constructed small Dowker spaces by assuming extra hypotheses. Most relevant to the current question is the construction, by I. Juhász, K. Kunen and M. E. Rudin (Two more hereditarily separable non-Lindelöf spaces), of a small Dowker space assuming CH. Finally, in 1996, Zoltán Balogh constructed A small Dowker space in ZFC.

There might be other examples stemming from the work of M. E. Rudin. Initially, I thought that the normality of ${\mathbb R}^\omega$ in the box topology might be an answer to this question, but apparently it is still open whether it can be proved without CH.

Answer 3

Let $G_1,G_2,H_1,H_2$ be countable groups. If $G_i$ is elementary equivalent to $H_i$ for $i=1,2$, then $G_1\times G_2$ is elementary equivalent to $H_1\times H_2$.

Proof: (a) assume CH. Fix a nonprincipal ultrafilter $\eta$ on the set of integers. Then the ultrapowers $G_i^\eta$ and $H_i^\eta$ are elementary equivalent, have cardinality $\aleph_1$ (by CH) and are $\aleph_1$-complete. So they are isomorphic. Hence, denoting by $\equiv$ elementary equivalence and $\simeq$ isomorphism, we have $$G_1\times G_2\equiv(G_1\times G_2)^\eta\simeq G_1^\eta\times G_2^\eta \simeq H_1^\eta\times H_2^\eta\simeq (H_1\times H_2)^\eta\equiv H_1\times H_2.$$

(b) So the above result is a theorem of ZFC+CH. By Schoenfield absoluteness, it is a theorem of ZF.

(Note: group axioms play no role: this works with arbitrary countable algebras with countable signature, in the sense of universal algebras. Little further effort should remove the countability assumptions.)