Misunderstanding the Continuum Hypothesis
(sometimes) A few days ago I read this article and realized I'd misunderstood the Continuum Hypothesis. The Continuum Hypothesis is a statement in the language of set theory that says something like this: There's no set whose cardinality is between that of the real numbers $\mathbb{R}$ and the integers $\mathbb{Z}$. Set Theory Set theory is an axiomatic theory designed to give a rigorous foundation to our intuitive beliefs about sets. The axioms of set theory take for granted just two things: That there is a collection - also known as a class - of objects, which are known as sets That there is a binary relation between sets, represented by the symbol $\in$, and where $x \in A$ is read as "x is an element of A". Those two assumptions in themselves do not create any sort of parallel between the objects discussed and the naive concept of the "set". That's the purpose of the axioms. There's about 10 of these - known as the Zermelo-Fraenkel axioms ...