If all transitive models have the same height, are they all "simple"?

Question

Suppose that $\alpha$ is the unique ordinal for which $L_\alpha$ is a model of $\sf ZFC$. In other words, there is no transitive model of $\sf ZFC$ in which there is a transitive model of $\sf ZFC$.

We know, of course, that there are many different models of $\sf ZFC$ of height $\alpha$. Starting with $L_\alpha$ itself, it is a countable model, so we can do a lot of different forcings over it. In fact, also class forcings can be used to extend $L_\alpha$. So we get models which are class-generic extensions, which may not have any set which is set-generic over $L_\alpha$ (e.g. a minimal coding real).

Is it true/consistent that if all transitive models have the same height, then all transitive models are class-generic extensions of $L_\alpha$?

(Yes, I'm including here things like "hyperclass" generic extensions, it's just the question of whether there is some relatively "tame" operation that generates all models from the minimal model; relative constructibility is not tame.)

If the answer is somehow positive, how much can this be pushed up to include other heights of transitive models? Can it include "there are 2/3/infinitely many different heights of transitive models"? What about "every real is in a transitive model"? What about uncountable heights?

Answer 1

I think the following result is related:

Theorem 1(Mack Stanley). Let $L_α$ be a minimal countable standard transitive model of ZFC. There exists a real $x_{nwg}$ having the following three properties:

1. $x_{nwg}\notin L_α$.

2. $L_α[x_{nwg}] \models ZFC$.

3. $x_{nwg}$ is not definably generic over any outer model of $L_α$ that does not already contain $x_{nwg}$.

Indeed Stanley proves something stronger. See his paper

• Stanley, M. C., A non-generic real incompatible with $0^\#$, Ann. Pure Appl. Logic 85, No. 2, 157-192 (1997). ZBL0877.03025. (Also on Stanley's homepage.)

But on the other hand, we have partial positive answers as well. For example see Stanley's paper

• Stanley, M. C., Invisible genericity and $0^\#$, J. Symb. Log. 63, No. 4, 1297-1318 (1998). ZBL0924.03097. (Also on Stanley's homepage.)

What is proved in this paper is simply that some instances of any type of non-constructible object are class generic over $L$. See also the paper ''$^*$forcing'' by Garvin Melles.