Can we describe any subsets of $\mathbb{N}$ occurring in a late layer of the Constructible Universe?

There is a certain large countable ordinal referred to in the literature as $$\beta_0$$. It was first discovered by Paul Cohen, and here are some equivalent characterizations of it:

• The smallest ordinal $$\beta$$ such that $$L_\beta$$ is a model of $$ZFC-P$$

• The smallest ordinal $$\beta$$ such that $$L_\beta\cap P(\mathbb{N})=L_{\beta+1}\cap P(\mathbb{N})$$

• The smallest $$\omega$$-admissible ordinal

My question is, is it possible to describe an actual example of a subset of $$\mathbb{N}$$ which is not an element of $$L_{\beta_0}\cap P(\mathbb{N})$$? Or are all such sets undefinable?

I’m even okay with something like “the set of Gödel numbers of all true statements in language $$X$$“. Or will even such descriptions not suffice?

Such an object is complicated to describe, but not too complicated. In general, the appearance of reals throughout $$L$$ is technical but not mysterious: we sort of keep using the same basic tricks over and over again. Standard go-to's include countability witnesses and first-order theories of countable levels of $$L$$ and related structures; common techniques include Lowenheim-Skolem, the condensation lemma (and the Mostowski collapse), and the use of the $$L$$-ordering to eliminate parameters.

First, there is a general approach that applies more-or-less to every countable ordinal. Whenever $$\alpha$$ is countable, so is $$L_\alpha$$, which means there is a (not unique of course) relation $$R\subseteq\omega^2$$ such that $$(\omega; R)\cong (L_\alpha;\in)$$ (I'm assuming $$\alpha$$ is infinite, here). However, it's easy to see that such an $$R$$ can never, itself, be in $$L_\alpha$$. That is, for every countable $$\alpha$$ there are reals which code bijections between $$L_{\beta_0}$$ and $$\omega$$, none of which are in $$L_\alpha$$, and particular this is true for $$\alpha=\beta_0$$.

We can further identify a specific such real (using $$\alpha$$ as a parameter): the least real with respect to the parameter-freely-definable well-ordering of $$L$$ which codes a bijection between $$\omega$$ and $$L_\alpha$$. In case $$\alpha$$ itself is parameter-freely definable - as $$\beta_0$$ is - this real is also parameter-freely definable. (We can also give a quick complexity analysis: for ordinals such as $$\beta_0$$ corresponding to the first level of $$L$$ satisfying a given first-order theory, the resulting definition is $$\Delta^1_2$$.)

A more specific argument would be to observe that - conflating a transitive set $$A$$ with the corresponding $$\{\in\}$$-structure $$(A; \in\upharpoonright A)$$ - the structure $$L_{\beta_0}$$ happens to be a pointwise definable structure; that is, each element in it is definable without parameters in it. This means that $$Th(L_{\beta_0})$$, the set of Godel numbers of all $$\{\in\}$$-sentences which are true in $$L_{\beta_0}$$, is not itself an element of $$L_{\beta_0}$$.

But this relies on particular properties of $$\beta_0$$; there are many countable ordinals $$\gamma$$ such that $$L_\gamma$$ is not pointwise-definable; indeed, most countable ordinals have this property, in the sense that the set of $$\gamma$$ such that $$L_\gamma$$ is not pointwise definable is club. Such an $$L_\gamma$$ can indeed contain its theory as an element, avoiding Tarski by way of that specific element not being parameter-freely definable. For example, $$L_{\omega_1}$$ contains every real in $$L$$, including (since $$L$$ computes first-order theories correctly) the theory of $$L_{\omega_1}$$ itself. And we can bring this down to the countable realm too, by applying Lowenheim-Skolem, Mostowski collapse, and condensation to get a countable $$\gamma$$ such that $$L_\gamma\equiv L_{\omega_1}$$ and $$Th(L_{\omega_1})\in L_\gamma$$ (hence $$Th(L_\gamma)\in L_\gamma$$ since $$L_\gamma\equiv L_{\omega_1}$$).

Incidentally, if you're not already familiar with it you'll probably be interested in the paper "Gaps in the constructible universe" by Marek and Srebrny.