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I still struggle mighty with basic conceptions of truth and proof.

For example: The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds. Now, Goedel and Cohen proved that CH/~CH are independent from ZFC, so ZFC + CH and ZFC + ~CH are consistent (in case ZFC is consistent but mathematicians assume that anyway). But since we know that CH or ~CH must be false, how can that be? One of those axiom systems must be inconsistent since it has no models (because one of its axioms is false).

Another example is the parallel axiom (P) in euclidean geometry. P is true or false, i.e. P or ~P. That would mean that either the euclidean or non-euclidean geometry system has to be inconsistent (= no model).

Can somebody explain where I make a mistake?

3 Answers3

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Your "Example" statement misses an important point: CH is a sentence of set theory, and within any model of set theory CH is either true or false. There are some models of set theory in which CH is true, there are other models of set theory in which it is false, and in every model of set theory it is either true or false. Contradiction averted.

Here, perhaps, is a simpler example. Think about this sentence in group theory: $\forall a,b, \,\, a \cdot b = b \cdot a$. Would you say this sentence is either true or false? You shouldn't, but, what you might say instead is that in any model of group theory, that is, in any group, that sentence is either true or false. In words, that sentence says the group is abelian. Some groups are abelian, some aren't, and every group is either one or the other.

Your parallel axiom example is quite similar. In some models of geometry, all the axioms except the parallel axiom are true, and the parallel axiom is also true: this is the case, for example, in the Cartesian coordinate plane. But there are other models of geometry in which all the axioms except the parallel axiom are true, and the parallel axiom is false: for example, in the hyperbolic plane.

Keep in mind: "axioms" do not exist in a vacuum. We use them to study mathematical objects, sometimes called "models", and we evaluate their truth or falsity within those models. We also reason with axioms, using them to prove theorems of the form "these axioms imply those properties". From a theorem of that form, we can conclude that in any mathematical model where these axioms are true, those properties are also true.

Lee Mosher
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As in this other question of yours, the issue comes from conflating truth/falsity (which are relevant to structures) with provability/disprovability (which are relevant to theories).

More specifically, you're trying to apply bivalence - "everything is either correct or incorrect" - in a way it can't be applied. The satisfaction relation for individual structures is indeed bivalent: if $\mathcal{A}$ is a structure and $\varphi$ is a sentence (in the same language), then either $\mathcal{A}\models\varphi$ xor $\mathcal{A}\models\neg\varphi$. However, bivalence does not hold for provability: e.g. there is no sense in which "$\mathsf{CH}$ is either true in $\mathsf{ZFC}$ or false in $\mathsf{ZFC}$."

So when you invoke bivalence to claim

The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds,

that's correct iff we're tacitly referring to some specific structure $\mathcal{S}$. However, your later claim

One of those axiom systems must be inconsistent since it has no models (because one of its axioms is false)

is then mixing up provability and truth. We can see this clearly if we make explicit the tacit assumption above:

One of those axiom systems must be inconsistent since it has no models (because one of its axioms is false $\color{red}{\mbox{in the structure $\mathcal{S}$}}$).

It should be clear that this is unwarranted: just because a theory is not satisfied in some particular structure doesn't mean it's not satisfied in any structure.

To avoid exactly this sort of confusion, we should be careful - at least until the fundamentals are mastered - to only use true/false/etc. when talking about structures, and use provable/disprovable/etc. when talking about theories. This way we won't be tempted to improperly use our intuitions about truth/falsity when talking about provability/disprovability (or at least, we'll be less tempted).

Noah Schweber
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Before getting to the mathematics, here is some philosophy:

To a first approximation, there are three ways of approaching mathematics:

  • Platonism is the belief that mathematical statements have an intrinsic truth value, and that mathematical objects really exist in some ideal universe. To a platonist, CH is either true or false. The independence results in set theory just say that ZFC is insufficient to decide which one it is. More generally, by Gödel's incompleteness theorems, first order logic as a whole is insufficient to decide the truth value of every statement (in the sense that any first order theory which is consistent and has enough arithmetic cannot prove everything).

  • Formalism is sort of the agnostic approach. A formalist makes no judgments regarding to the inherent truth of mathematical statements, but rather views mathematics as playing around with symbols (subject to some rules). To a formalist, the independence results of Gödel and Cohen just say that there's no way to play around with strings of symbols so that the end result is a proof of CH (or ¬CH). Hence, the formalist will not waste time trying to come up with a proof, since they know that is not possible. Here, the notion of "is a proof" (together with other notions of first order logic) are defined in the metatheory, and are presumably finitistic statements on whose validity we can all agree. On that note:

  • Finitism is (more or less) the belief that infinity is a fiction. To a finitist, CH is a meaningless statement about non-existent objects. Still, a finitist can derive some value from the independence results, in a similar fashion to a formalist. Nevertheless,

Regarding your statement about consistency of different theories, the key is that CH is valid in some models, and false in some others. We're not looking at the truth value of CH in the entire universe, but rather in some specific model. Still, this tells you that no proof of CH could exist, because any such proof would still go through in any model of ZFC (this is just the Soundness Theorem). In particular, it would go through in a model of ¬CH, which would result in a contradiction.

If you look at, for example, ordered sets, the statement "There is a largest element" is true in the model $\{0,1\}$, but false in the model $\mathbb{N}$ (with the usual order). It would be foolish to discard the study of bounded or unbounded ordered sets just because we have a statement which has to "really" be true or false, whatever that means.

On another note, the notion of "truth" itself is rather problematic. This is the content of Tarski's (aptly named) theorem of the Undefinability of Truth. Roughly, it says that it is not possible to express "truth" as a first order property. More precisely, working in ZFC (much weaker theories suffice), if $\phi$ is a formula in $\{\in\}$, we can define a formal counterpart, a code $\lceil \phi\rceil$ (for example, code the metalinguistic first order symbols using natural numbers). Then, Tarski's theorem says that, unless ZFC is inconsistent, there is no first order formula (in one free variable) $T(v)$ such that, for every sentence $\sigma$, the following is provable: $$ T(\lceil \sigma\rceil)\leftrightarrow \sigma $$

Reveillark
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