Is the standard model of ZFC a subset of Godel's constructible universe $L$? Since everything true in the standard model must be true in $L$, and there is no intermediate cardinal in $L$, there can't be one in the standard model. So Godel proved CH? What's wrong with my argument? If my reasoning is correct but conclusion wrong, what's the use of the standard model?

  • I am curious - why do you think that statements true in "the standard model" are true in $L$? – Noah Schweber May 2 at 15:01

The root of the mistake is the usage of "the". In set theory a standard model is a model of set theory whose membership relation is the same as the universe. Often we also require transitivity, but this is the same because of Mostowski's Collapse Lemma.

If there is a standard model, then there is one which satisfies the failure of CH, and anything you can do with forcing. This is Cohen's work on forcing, and all the subsequent work since then.

This is the difference between set theory and arithmetic. Set theory has no unique model to draw upon for the notion of "true". But in any case, let me also point out that even in arithmetic there is a difference between "true" and "provable".

  • Truth is with respect to a model, isn't it? While provability is a language feature. Isn't N the standard model of PA? I hope I have the basics right. BTW, where do you learn all these stuff? Any book? – Zirui Wang Apr 30 at 13:58
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    @ZiruiWang Just because we think of PA as having a standard model doesn't mean that we also think of ZFC as having a standard model (although some people do). Meanwhile, note that the phrase "standard model" actually means something different in set theory than it does in arithmetic, as per Asaf's first paragraph. Regardless, your claim that statements true in "the standard model" are true in $L$ is unwarranted. – Noah Schweber Apr 30 at 15:10
  • @Zirui: Yes, you are correct. And even if you do have a standard model, it's just one model. Provability, as the completeness theorem states, is about all models. – Asaf Karagila Apr 30 at 21:46

As Asaf has pointed out, the phrase "the standard model" is highly problematic in the set theory context (whereas it isn't really, or at least is much less so, in the context of arithmetic). However, that's not the only problem here. You write:

"Since everything true in the standard model must be true in $L$."

Even if we accept the use of the term "the standard model," this claim is completely false. We do know that certain statements can't change truth values between inner models of ZFC, but that's a very limited situation (and CH isn't such a statement).

To extend Asaf's good answer, I want to make a few comments on "the" standard model of set theory.

It is well known that many mathematicians have a so-called "platonist" attitude towards mathematical objects. This extends to set theory - many set theorists have had the viewpoint that "sets" are mathematical objects which exist apart from the axioms we use to study them, and apart from our ability to understand them.

One motto of such mathematicians might be that "set theory is the study of sets, rather than the study of models of set theoretic axioms". This viewpoint has been very common historically in set theory, and it has influenced the way that many set theory texts are written. This is why the phrase "the standard model" seems to arise so often.

For mathematicians who do believe in a collection of pre-existing sets, "the" standard model of set theory simply means "the model consisting of all the sets that exist in that sense". The use of the standard model, for these mathematicians, is as the motivational object of study in set theory.

Without getting into the philosophical question about whether that model is well defined, we can see that there are many things we do not know about it. The continuum hypothesis is one of those things. The axioms that are generally accepted for sets don't decide CH, and nobody has managed to find new axioms that both (1) are broadly seen as acceptable and (2) resolve CH.

This leads to several very active areas in the philosophy of mathematics:

  • To what extent is platonism with respect to set theory viable? Does our relatively recent experience with the method of forcing shed any light on this old question?

  • Are there new axioms that could decide CH and which could be broadly accepted? Does our experience with forcing suggest that any new axiom that does decide CH would be viewed with suspicion?

Each of these is deep enough to take a book to answer.

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory.

Indeed, if I were to encounter this phrase, I would think of two possible interpretations:

  1. The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is either a certain countable ordinal, or $\mathsf{ORD}$. Or
  2. The author actually meant "the true universe of sets".

Now, in the case at hand, 1. does not appear to be a viable interpretation, since trivially $L_\Omega$ is a (perhaps proper class) submodel of $L$. Even so, however, it is not necessarily true that everything true in the minimal model is true in $L$. For instance, if $\Omega$ is a countable ordinal, then $L_\Omega$ is a model of "there are no standard models of set theory" (in the traditional sense of the term "standard model", that is, transitive, as in Asaf's answer), but this fails in $L$ (as witnessed by $L_\Omega$).

The second interpretation is perhaps what most set theorists would think was meant. However, what we mean by "the true model" varies considerably.

For instance, some set theorists may simply say "the true model" to mean the universe of sets they are dealing with in the midst of a specific argument (a local notion). So, we can start with a ground model (which would then be the true model), and then pass to a forcing extension (and then that extension would be the true model), or we can start with a countable model $M$ (which does not even need to be well-founded or even an $\omega$-model), and argue about properties of that model, and for convenience phrase the arguments as if they take place inside $M$, and then refer to $M$ as the true model, even though, of course, $M$ is in a sense far from being the true model, being that it is countable.

Other set theorists would say that saying "the true model" is interchangeable with saying "$V$", and so if in some arguments they assume $\mathsf{CH}$, then in those arguments $\mathsf{CH}$ holds in the true model, while if in other arguments they assume, say $\mathsf{PFA}$, then in those arguments $\lnot\mathsf{CH}$ holds in the true model, and that's it. There is no ontological commitment in their part when using the phrase. They do not mean, in particular, that "in real life" (whatever that means) there is a universe of sets.

Yet other set theorists, of course, actually think of the universe of sets as a mathematical object just as precise as the natural numbers or the group of invertible $2\times 2$ matrices with complex entries. They would say that what we do as set theorists is to study this universe, and that the axioms are our attempts to capture some of the features of the universe that we seem to agree on. Under this interpretation (which Carl describes in his answer) $\mathsf{CH}$ is a precise question with a definite answer, regardless of whether we currently know what that answer is. Most set theorists in this camp also agree on the existence of significant large cardinals, and therefore they agree that $V\ne L$.

There are intermediate positions as well. For instance, some set theorists think of the axioms as statements we agree on, not necessarily because they are "true" (whatever that means) but because we find that they capture well certain practices and seem better suitable to our mathematical needs and our sense of aesthetics. In this sense, the axioms of $\mathsf{ZFC}$ hold in the true model (because we have reached consensus about them), and (most would say) so do many large cardinal axioms, but at the moment there is no consensus on $\mathsf{CH}$, so part of the goal of research in set theory (defenders of this view would say) is to reach a point where we decide one way or the other (and if this happens, then we can add the statements we end up agreeing on to our list of "axioms", and accordingly we will now say that $\mathsf{CH}$ is true in the true model---or it is false, whichever ends up being the decision).

As you can imagine, these views tend to be controversial and there are disagreements. But these are philosophical positions and are in a sense irrelevant when proving theorems (they do not appear in the assumptions we write down, nor in the proofs we detail). Almost no practicing set theorist, though, would say that "the true model" is $L$ or a subclass of $L$.

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