Franklin's Notes


Continuum Hypothesis

The Continuum Hypothesis, often abbreviated $\mathrm{CH}$, is a statement about the size of the cardinality of the continuum $\mathfrak{c}=2^{\aleph_0}$ relative to other cardinals in ZF set theory . In particular, it states that $\aleph_1=2^{\aleph_0}$, or that the cardinality of the real numbers is the smallest cardinality strictly greater than the cardinality of countable sets.

It happens that $\mathrm{CH}$ is independent from the axioms of ZFC . Proving this is very difficult and requires a lot of model-theoretic machinery, such as a technique called forcing, but the following is a rough sketch of the "game plan" for proving this:

set-theory

model-theory

nonstandard-model

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