Is it true but unassertable that there are undefinable real numbers?

Question

I know of Joel David Hamkins's analysis of the so-called "math tea argument", namely that there are undefinable real numbers. Supposedly, he debunked this argument by constructing a countable model of set theory where all sets and all real numbers are definable. However, I don't think he really did, because he merely constructed a countable model. Even though the model thinks that it is not countable, it is countable in reality. I still think there are undefinable real numbers, but now I also think that this fact can't be asserted in the formal theory of ZFC. In much the same way that no formal theory can define its own truth predicate, so also no formal theory can define its own definability predicate. But I am interested to know, what have other mathematicians thought about this topic?

Answer 1

I suppose we need to consider some context here. ZFC is a first order theory over the language $\mathscr{L}=\{\in\}$. It interprets $\in$ as just some binary relational symbol (so if $M\models ZFC$ is a model of ZFC, then $\in$ will be interpreted as some relation $R\subseteq M^2$ (in other words $\in^M=R$)). Notice that firstly, the relation $\in$ in ZFC doesn't has to necceserily be interpreted as the "meta-$\in$" with a cut domain (to avoid ambiguities I will denote "meta-$\in$" as $\in'$). I.e it isn't necceserily the case that in a given model $M\models ZFC$ we have that the $\in^M=\{(m_1,m_2)\in M^2: m_1\in' m_2\}$.

Now I say it to make some clearness here. When you define things in ZFC you define everything in accordance to the relation $\in$ (which will be interpreted as $\in^M$ in some particular model $M$). In particular you define cardinality in accordance to $\in$, you define it with some first order formula over the language, and it doesn't have to correspond to the meta-cardinality. They are two distinct things.

(If we will assume that there's a transitive ($\in^M$ is the same as $\in'$ just cut to the $M$ and $\in^M$ is transitive on $M$) model of ZFC then there is transitive pointwise definiable model of $ZFC$ (as we see in this article by Hamkins https://arxiv.org/abs/1105.4597). So the analogy isn't perfect but still I think it might be helpful. Also still definition of cardinality etc. depends also on that what are you quantifying over so still it can be diffrent than the "meta version" ).

In a pointwise definiable model at the same time we can prove that based on it's internal definition of what "two sets has distinct cardinalities" that reals and natural numbers have diffrent cardinality, but also we can show that in a sense of "meta-cardinality" they are diffrent. It also shows why we can have definiable reals, because the concept of cardinality might differ.

We can't define definiability within a theory as you notice, and that's one of the reasons why we can have models where even every set is definiable. Formal theories oftenly have many limitations, the fact that all sets might be definiable (in some models) shows that we also here have some sort of limitations, that ZFC on it's own doesn't impede diffrences between what it thinks and what works in the metatheory (if all would work the same way then from the fact that there are countably many formulas, we couldn't have all reals definiable).