Has a counterexample to the continuum hypothesis ever turned out to be useful?

Question

If I understand the continuum hypothesis (and its undecidability) correctly, then one can safely assert the existence of some set $A$ whose cardinality is between that of the integers and that of the real numbers, and this assertion will not cause a contradiction within ZFC set theory.

Have such sets ever been useful within mathematics or physics? I'll leave "useful" up to your own interpretation, but here are some examples of what I mean:

1. The assumption that there exists a type of number whose square is $-1$ led to complex analysis, which has been useful in both pure math and applied physics.

2. The rejection of Euclid's 5th postulate led to non-Euclidean geometry, which was also useful in both math and physics.

Some starting criteria for "useful" might be:

• Launching a new field of math
• Shedding new light on existing fields of math
• Helping prove something that was already an open question
• Playing any role in theoretical physics or some other field

Answer 1

The assumption that there exists a type of number whose square is −1 led to complex analysis, which has been useful in both pure math and applied physics.

The rejection of Euclid's 5th postulate led to non-Euclidean geometry, which was also useful in both math and physics.

In both cases this was not an assumption, but a definition.

• You define $\Bbb C$ as the quotient ring $\Bbb R[X]/{\left<X^2+1\right>}$ and observe that in this structure, $X$ is an element with square $-1$.

• Similarly, non-Euclidean geometry is not assumed, it is observed in frameworks defined within the ZFC axioms.

An example of a set with cardinality between $\omega$ and $\mathfrak{c}$, by definition, cannot be constructed within ZFC. You can always assume its existence in a non-constructive manner, but then there is very little you can actually do with it. Or you could possibly construct it if you forgot about ZFC and worked in a suitable set of axioms, but then again it would not be useful to the ZFC community.

Answer 2

Let $P(x)$ denote the power-set of a set $x.$

Let $L[x]$ be the class of sets that are constructible from $x.$ Then $L[x] $ satisfies ZF.

An exercise from K. Kunen's text:

Let $M$ be a countable transitive model for (enough of) ZFC. We may obtain two Forcing extensions $M(1$) and $M(2)$ with a set $x\in M(1)\cap M(2)$ such that $$(i)\quad x=(P(\omega))^{M(1)}=(P(\omega))^{M(2)}$$ $$(ii)\quad (|x|^+)^{M(1)}<(|x|)^{M(2)}. $$

Now $(L[x])^{M(1)}=(L[x])^{M(2)}.$ Denote this set by $S.$ Then $S$ satisfies ZF. But we obtain a contradiction from (i) and (ii) if we assume there exists some $W\in S$ such that $S$ satisfies "$W$ is a well-order of $x$ "

So from the use of to different models for ZFC + ($\neg$ CH) we obtain a model for ZF+ ($\Bbb R$ cannot be well-ordered) and hence show that the Axiom of Choice is not a theorem of ZF (unless $1=0$ is).

Answer 3

If the continuum hypothesis is false, there is a canonical "counterexample": the set of all countable ordinals, commonly denoted nowadays by the symbon $\omega_1.$

Note that it's not the existence or the uncountability of $\omega_1$ that's undecidable; those are not in question. What's undecidable is whether $\omega_1$ is smaller than the continuum. If the continuum hypothesis is false (and the axiom of choice holds) then $\omega_1$ is smaller than the continuum.

And $\omega_1,$ the set of countable ordinals, is definitely useful in mathematics, regardless of the status of the continuum hypothesis (although in a sense it is even more useful if the continuum hypothesis is true). If I remember right, Cantor's interest in transfinite cardinals was motivated by his study of countable ordinals, which he found useful in his study of trigonometric series.