How could we use the continuum hypothesis if we are able prove its truth?

Question

I'm not sure if this is an appropriate question to ask, but if we are able to agree on the truth of the continuum hypothesis, then what problems could we solve using this newfound knowledge? I'm having trouble finding sources on the internet that give examples as to what follows if the continuum hypothesis is solved. This suggests to me that the continuum hypothesis is not actually a very important problem compared to something like the P vs NP problem since its implications are clear and documented across the internet.

Could I please have some examples?

Answer 1

The following book is entirely devoted to equivalences and consequences of the continuum hypothesis. I'm giving this as an answer rather than a comment mainly for archival purposes, since over the years I've cited this book (and some of the reviews) several times in comments where the information tends to get lost, forgotten, not-googleable, and mostly never seen again.

Wacław Franciszek Sierpiński (1882-1969, Hypothèse du Continu, [Hypothesis of the Continuum], Monografie Matematyczne #4, 1934, vi + 192 pages. Zbl 9.30201; JFM 60.0035.01

The 2nd edition was published by Chelsea Publishing Company in 1956 (xvii + 274 pages; MR 19,829c; Zbl 75.00903).

Some published reviews of the book:

Adolphe [Alfred] Buhl (1878-1949), L'Enseignement Mathématique (1) 32 (1933), pp. 417-418 (in French). freely available online

Paul [Pál] Dienes (1882-1952), Mathematical Gazette 19 #233 (May 1935), pp. 146-147. online at JSTOR and p. 146 freely viewable online here

Herbert Busemann (1905-1994), Matematisk Tidsskrift B [after 1952: Mathematica Scandinavica], 1935 (1935), pp. 43-44 (in German). online at JSTOR

Hans Hornich (1906-1979), Monatshefte für Mathematik und Physik 42 (1935), Literaturberichte, p. 23 (in German; separately paged). freely available online

John Robert Kline (1891-1955), Bulletin of the American Mathematical Society 42 #5 (May 1936), pp. 301-303. freely available online

Alonzo Church (1903-1995), Journal of Symbolic Logic 23 #2 (June 1958), p. 215. online at JSTOR and freely viewable online here

Evert Willem Beth (1908-1964), British Journal for the Philosophy of Science 10 #39 (November 1959), pp. 249-250. online at JSTOR and p. 249 freely viewable online here

Answer 2

It could be, albeit highly unlikely, that tomorrow morning someone will come up with a particularly compelling argument to accept a set theoretical axiom such as $V=L$ or $V=K$ for some other suitable canonical core model (to accommodate some large cardinals, anyway), or even Woodin might succeed in his plan to convince the rest of us that $V=\textrm{Ultimate-}L$ is "the correct axiom" we were missing.

Yes, it might be that something in the heart of mankind will change tomorrow morning, and we will all see the light and accept some axiom that implies $\sf CH$, or even $\sf GCH$. What can we prove, in that case? Well, there is a lot of work in set theory where we prove that certain things follow from $\sf CH$ or perhaps imply its negation (which is to say, the negation of the statement follows from $\sf CH$).

All those things will immediately become "true" in the broader context of mathematics, since they follow from what we sort of agreed to be the base theory for mathematics. The body of work on this is so huge that it is hard to even put it into a boundary. Dave L. Renfro gave a good start. I'll add to this and suggest looking into much of the work about forcing axioms (e.g. Martin's Axiom) and the cardinal characteristics of the continuum. In some sense, both of these can be thought of as certain extensions of Sierpinski's work.

Many other things, such as $\rm P=NP, RH$, and many others, will not be affected, since, as Noah Schweber points out, these statements have a certain logical complexity which allows us to dispense of $\sf CH$, or even $\sf AC$, as far as their truth in the mathematical universe goes. To that extent, this is why we do not have "reasonable counterexamples to $\sf CH$" in the context of the real numbers. Any Borel set must be countable or have size continuum. So, if we accept the premise that a counterexample should be reasonable, and that reasonable sets are Borel, then to that end, $\sf CH$ is already "true".

But does that mean that the truth, or even "definiteness", of the Continuum Hypothesis is more or less important than $\rm P=NP$? In a way, you are correct, it has less effect on every day mathematics and certainly almost no effect on real life as we understand it today. But to that end, $\rm P=NP$ is a huge red herring as well. If someone were to prove tomorrow morning that $\rm P=NP$, but any polynomial algorithm to find a solution for a $3\rm SAT$ problem must have complexity of $O(n^{U!})$ where $U=2^{2^m}$, where $m$ is $((\text{number of hydrogen atoms in the universe})!)!$, then every kind of attempt to find an actual general solution to these problems is going to be so ridiculously impossible, that for all intents and purposes this is not at all solved.

Many of these other, so-called "important problems", suffer similar fates. Just because there is a solution, or even a solution that is technically considered "efficient", it does not mean that there is one that is feasible to us or to our distant ancestors. And we may very well end up relying on guesswork and approximate numerical calculations until the heat death of the universe.

On the other hand, mathematics, as a subject of its own, and set theory in particular, has certain elegance to it. Asking questions about this elegance is important, and we do learn quite a lot from it at the end of the day. Both in philosophical aspects of mathematics and set theory, about what it means to be true, provable, and the foundations of mathematics; as well as its advances in set theory which slowly trickle down to things like algebra, analysis, and so on, which may, one day, have "an actual impact" on the daily lives of your children's children's children.

I'll finish with a supposed quote of Faraday, "what good is a newborn baby?", when asked what this "electricity thing" is going to be good for. We don't know yet, we will never know. In a logical term, this is a $\Sigma_1$ problem: until you find a use for it, you don't know if it's useful. And I argue, that we already found uses for the study of the truth of the Continuum Hypothesis.