There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that

there is no infinite cardinality lying strictly between the cardinality of the natural numbers and the cardinality of the real numbers.

But, imagining trying to explain Continuum Hypothesis in, say, a popular-mathematics context, I can imagine people thinking, "Well, all this 'looking for different sizes of infinity in terms of one-to-one mappings' stuff sounds really cool, but it's ultimately just mathematicians engaging in abstract word-logic games." And, as I understand it, a significant question in the philosophy of mathematics is whether such people would actually be right! But nevertheless, we could still at least ask: **what is the least 'abstract-word-games'-feeling, most concrete-feeling, equivalent statement to the Continuum Hypothesis?**

Here's my own best attempt:

(I take $\mathbb{N}$ not to include $0$.)

Recall

Zeno's paradox: the sequence $(a_n)_{n \in \mathbb{N}}$ defined recursively by $a_1=0$ and $a_{n+1}=\frac{1}{2}(a_n+1)$ is a strictly increasing infinite sequence, and yet still has a finite upper bound.A

Generalised Zeno Paradoxis a partition of $[0,1)$ of the form $$ \Big\{ [ q_n , q_n+2^{-n} ) \ \Big| \ n \in \mathbb{N} \Big\} $$ for some sequence of rational numbers $(q_n)_{n \in \mathbb{N}}$.We say that two Generalised Zeno Paradoxes $$ \Big\{ [ q_n , q_n+2^{-n} ) \ \Big| \ n \in \mathbb{N} \Big\} \quad \text{and} \quad \Big\{ [ \tilde{q}_n , \tilde{q}_n+2^{-n} ) \ \Big| \ n \in \mathbb{N} \Big\} $$ are

qualitatively equivalentif there exists a strictly increasing bijection between the set $\{q_n | n \in \mathbb{N}\}$ and the set $\{\tilde{q}_n | n \in \mathbb{N}\}$.A

Qualitatively Generalised Zeno Paradox(QGZP) is a qualitative-equivalence class of Generalised Zeno Paradoxes.An

infinite-bit stringis an $\mathbb{N}$-indexed family of $0$s and $1$s.

Continuum Hypothesis:There exists a set of pairings-of-a-QGZP-with-an-infinite-bit-string in which

- every possible infinite-bit string has
at least oneQGZP paired to it, and- every QGZP is paired to
at most oneinfinite-bit string.

Are there other equivalent formulations that feel at most as "abstract" as (or perhaps, even less "abstract" than) this one?

Hypothèse du Continu. eudml.org/doc/219323#content $\endgroup$"Well, all this 'looking for different sizes of infinity in terms of one-to-one mappings' stuff sounds really cool, but it's ultimately just mathematicians engaging in abstract word-logic games."This certainly describes a more positive reaction than I'd hope for. ;-) $\endgroup$1more comment