Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is?

Question

By what methods can we identify sentences that are true in the standard model of set theory, but not in other models? In particular, how do we prove that Gödel sentences are true in the standard model? Many answers on Math SE suggest to think of them as simply undecidable, but I'd like to understand what makes them different from the continuum hypothesis in this regard. It seems that the set theory axioms pretty much exhaust our intuition about sets, yet there are truths about sets that are not provable from them.

One explanation I read is that Gödel's proof in addition assumes consistency, which we have to believe if we believe in the standard model. But why would the consistency assumption single out the standard model as opposed to any other? Also, sentences of set theory are about sets, they can express something like consistency only when reinterpreted using some external numbering scheme for formulas (Gödel numbering). Even if Gödel sentences are "intuitively" true under such reinterpretation (because they "say" I am unprovable, and they are), how do we prove that they are also "internally" true, in their original meaning as statements about sets?

Answer 1

As Noah points out, in the context of $\sf PA$ we have a unique "very nice" model which has very nice properties. $\Bbb N$, we can show that any well-founded model of $\sf PA$ is isomorphic to it, and we know that this model exists, if we assume a sufficient meta-theory. So in the context of arithmetic, we can say "the standard model" and confuse between "true" and "true in the standard model".

In the case of set theory, there is no such thing. For several reasons:

1. While many people would argue that $\sf ZFC$ is self-evident, some people might disagree. It is much harder to argue against the natural numbers, though.

2. Even if $\sf ZFC$ is in fact self-evident, what sort of uniqueness can we expect from a canonical model? In comparison to $\sf PA$, when moving to a second-order theory (i.e., taking the second-order axiom of replacement instead of a schema), we can prove that any model is necessarily $V_\kappa$ for an inaccessible $\kappa$. But without adding more assumptions about what sort of large cardinals are in the universe, or what large cardinals are in the model, we cannot guarantee uniqueness.

3. The term "standard model" in set theory, means that the model, which is a pair $(M,E)$ where $E$ is a binary relation over $M$, is such that $E=\in$. So $M$ agrees with the background universe about the membership relation; and that the model is transitive, namely if $x\in M$ and $y\in x$, then $y\in M$.

4. The crux is that if there is a standard model, then there are many of them. Forcing takes a countable transitive model, and constructs a different countable transitive model.

   And depending on your meta-theory, there might be many many many different countable transitive models (e.g. what sort of large cardinal assumptions are true). In particular, there will be transitive models where $\sf CH$ is true and others where it is not.

And now we can turn our attention to your question about the Gödel sentences. This is really a statement about the natural numbers [of the model]. But as luck would have it, if $M$ is a transitive model, then it agrees with the universe about $\omega$, and about its first-order theory of $\sf PA$.

In particular it agrees with the universe about whether or not Gödel sentences ares true or false. And in particular, any standard model agrees with any other standard model, and with any inner model and with the entire universe, about these sort of questions.

And this is what makes it so different from $\sf CH$. While statements about sets can sometimes be changed by forcing, statements about the natural numbers are robust. They can be changed by considering other models, but not standard models, not models which are isomorphic or otherwise elementarily equivalent to standard models, or any model which just happened to agree with the universe about the set $\omega$ (these are called $\omega$-models).

That means that the completeness theorem ensures us that if something is not provable from $\sf ZFC$ we can find a model where it is false; but nothing tells you how that model looks like. More specifically, from a set theoretic vantage point, this model will not be particularly nice. It would be ill-founded, and will have non-standard integers.

Answer 2

The crux of the matter is that (people claim) we have evidence that PA is consistent, but we do not have similar evidence that CH is true. Note that "true in the standard model of set theory" is (basically) synonymous with "true."

Why is this? Well, let's begin with: why do we believe PA is consistent? Usually, actually, a stronger assertion is made: PA is true. The reasoning behind each claim is often, "We have intuitive access to the natural numbers, and this includes the knowledge that they satisfy PA." (If this sounds circular to you, don't worry, you're in good company.)

Now let's leave aside the issue of how convincing or not our ability to visualize the natural numbers is as an argument for the consistency of PA, and look at CH. First, note that we arguably know how to convince ourselves that PA is consistent: all it takes is the ability to find a single ordered semiring in which PA is true, and what goes on in other ordered semirings doesn't matter. By contrast, if we had a model of ZFC+CH, this would only be evidence for the consistency, not truth, of CH; in order for the existence of a model $M$ of ZFC+CH to count as evidence for the truth of CH, we would need

• a reason to believe that $M$ is the standard model of set theory, or

• a reason to believe that $M$ is similar to the standard model of set theory, at least as far as CH is concerned.

This difficulty is compounded by forcing, which lets us explicitly build a model of ZFC+CH from a model in which CH fails, and vice versa, while preserving many nice properties (such as well-foundedness). This (in my mind) kills off, for example, the hope of arguing that there is a single model of set theory which is somehow "within reach": simple models have simple forcing extensions.

So now on to your first sentence:

By what methods can we identify sentences that are true in the standard model of set theory?

Here's one approach: identify mathematical properties which, according to some philosophy, the standard model of set theory must have; then, show that these mathematical properties imply/disprove the statement in question.

For example . . .

• There are arguments that the standard model of set theory satisfies "V=L"; insofar as you buy the philosophy behind these arguments, these are also arguments for CH being true. However, they tend to be unpopular.

• Large cardinals are very "in" these days (:P), but they don't settle CH (although they do imply, for instance, projective determinacy, and so many set theorists believe that projective determinacy is "true in the standard model").

• Forcing axioms - such as PFA - imply that $2^{\aleph_0}=\aleph_2$; on those rare days when I believe in the standard universe of set theory, I tend to believe in this direction, but I think that might be rarer (more common is the belief that forcing axioms hold in inner models; this is basically large cardinals round two).

• Woodin has examined some other means of settling CH, but I know less in this direction; basically, one of his approaches ("Ultimate L") is to argue that, assuming large cardinals hold in the "real" universe $V$, there is an inner model $N$ which is "large" (i.e., has the same large cardinals as $V$) and has many nice canonical features, including CH. One can then make arguments that $V$ "ought to" be equal to its own $N$.

For more and better information on these and other arguments, see https://mathoverflow.net/questions/23829/solutions-to-the-continuum-hypothesis.

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EDIT: The lectures etc. at http://logic.harvard.edu/efi.php#multimedia might be of interest to you; they discuss the nature of mathematical truth, the definiteness of mathematical statements, and whether CH has a truth value.