An upper bound for the continuum hypothesis [duplicate]

Solution 1:

First, a quick comment: you've phrased your question in terms of $\vert\mathbb{R}\vert$ and sets of reals. However, it's much easier to talk instead in terms of powersets and cardinals $\le 2^{\aleph_0}$, and this gives the same result (since $\vert\mathbb{R}\vert=\vert\mathcal{P}(\mathbb{N})\vert$ and every cardinal $\le 2^{\aleph_0}$ is the cardinality of some set of reals and vice versa). That is, your question is just

How big can $2^{\aleph_0}$ be?

and this is how I treat it below.

It turns out that basically nothing can be said about $2^{\aleph_0}$ in ZFC alone. Specifically, while Konig's theorem tells us that $\mathfrak{c}$ must have uncountable cofinality, that's basically all that can be said. This was proved by Solovay, and later extended dramatically by Easton.

A very small case of Solovay's result is the following:

For any $n>0$, it is consistent that $2^{\aleph_0}=\aleph_n$.

For example, we can have $2^{\aleph_0}=\aleph_{17}$. Even more is possible however: we could have $2^{\aleph_0}=\aleph_{\omega+1}$, or $\aleph_{\omega_1}$, or even be weakly inaccessible.

The whole situation can best be summed up as follows:

Suppose $M$ is a countable (= so that we can actually force over it) model of ZFC, and $\kappa$ is an $M$-cardinal (= $\kappa\in M$ and $M\models$"$\kappa$ is a cardinal"). Then there is a forcing extension $N$ of $M$ such that $(i)$ the cardinals of $M$ are exactly the cardinals of $N$, and $(ii)$ $2^{\aleph_0}\ge\kappa$.

The proof is a straightforward forcing argument: let $G$ be generic over $M$ for the forcing $\mathbb{P}$ which adds $\kappa$-many Cohen reals (specifically, elements of $\mathbb{P}$ are finite partial maps from $\kappa\times\omega$ to $2$, and we order $\mathbb{P}$ by reverse extension). Let $N=M[G]$. We clearly have in $N$ an injection $\kappa\rightarrow\mathbb{R}^N$, and the fact that $Card^M=Card^N$ follows from the fact that $\mathbb{P}$ has the countable chain condition.

So in a precise sense, there is absolutely no ZFC-provable bound on the size of $2^{\aleph_0}$.

Finally, you ask about the possible impacts on analysis of a large continuum. To the best of my knowledge there are very few of these (and fewer still that aren't simply equivalents of $\neg$CH right off the bat). However, there are interesting set-theoretic propositions which imply a large continuum and which have meaningful analytic consequences: e.g. cardinal characteristic inequalities or forcing axioms. But in general, simply knowing that the continuum is large doesn't really tell you very much.