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Asaf Karagila ♦
Asaf Karagila ♦
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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?

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 $L_\alpha$?

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?

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?

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Asaf Karagila ♦
Asaf Karagila ♦
  • 39.7k
  • 8
  • 134
  • 282

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

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 $L_\alpha$?

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?