Provability of the continuum hypothesis and the incompleteness theorem

Question

Since Cantor's continuum hypothesis is not provable or disprovable using the axioms in Zermelo–Fraenkel set theory, can we say it is true due to Gödel's incompleteness theorem?

Answer 1

No. There are some propositions that must be true if they are not disprovable: typically those are of the form "there is no $x$ such that $A(x)$" where, if there were any $x$ such that $A(x)$, it would be possible to verify that fact. For example, "there is no odd perfect number": if there were an odd perfect number, it would be possible (in principle) to write it down, find its divisors, and thus check that it is odd and perfect. But for the continuum hypothesis, if $x$ were a set of cardinality strictly between $\aleph_0$ and $c$, there is no reason to think we could even specify $x$ mathematically, let alone prove anything about its cardinality.

Answer 2

No, we can't. The statements constructed in Gödel's theorem must be true, because they claim they are not provable. But that doesn't mean that all undecidable statements are true by construction. The contrary to the continuum hypothesis is consistent because there must be a model, if the theory with the continuum hypothesis is consistent (similar, but much more complicated than a model of the hyperbolic plane in the Euklidean plane, reinterpreting the metric).

Answer 3

No: there are models of ZF(C) where CH is true, and models where it is false.

• Gödel showed in 1940 that CH and AC hold in the constructible universe.

• Cohen in 1963 invented/discovered forcing and used it to show that there are models of ZF(C) in which CH is false.

The Stanford Encyclopedia of Philosophy and Wikipedia articles have plenty of references between them.

Answer 4

There is a problem that you're not seeing here.

Truth is relative to a structure. In the context of arithmetic, where the incompleteness theorems are originally proved, there is a canonical structure of interest: the natural numbers themselves. So in the context of arithmetic, "true" is just a shorthand for "true in the natural numbers".

In set theory we are not that lucky. There is no consensus about a canonical model for set theory. In fact, when going to lengths to ask about what goes on in different models of set theory, you might even find yourself with different arithmetical truths, which makes even the natural numbers themselves somewhat "less canonical" than what we might think about them.

So where the incompleteness theorem says "There is a true statement which cannot be proved", it generally speaks about a given model of Peano Arithmetic, or some other set of axioms. The full formulation, if so, should be "In any given model of $\sf PA$, there is a statement which is true in that model that cannot be proved from $\sf PA$ itself". But of course, we are mainly interested in $\Bbb N$ itself, so we can omit the redundancy and keep the statement in the first sentence of this paragraph.

Finally, since there is no "obvious notion for truth" in the context of set theory, we cannot quite determine which one is better than the other. There are both philosophical and mathematical arguments for both sides of the coin; as well as people who simply argue that the whole concept of "absolute truth" is flawed.