Return to Question

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms to ZFC?

2. Does an axiom like this make sense (ie is neither a theorem of ZFC, nor blatantly inconsistent):

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ such that $M=V_{\eta_0}^N$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms to ZFC?

2. Does an axiom like this make sense (ie is neither a theorem of ZFC, nor blatantly inconsistent):

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ such that $M=V_{\eta_0}^N$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms?

2. Does an axiom like this make sense:

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ such that $M=V_{\eta_0}^N$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms to ZFC?

2. Does an axiom like this make sense (ie is neither a theorem of ZFC, nor blatantly inconsistent):

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ such that $M=V_{\eta_0}^N$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms?

2. Does an axiom like this make sense:

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ containing $M$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.

Using forcing techniques, at least the ones I know of, one starts from a ground model and then enlarge it, by adding a generic set and closing up with respect to relative constructibility.

The new model shares the same ordinals and the same height of the original one.

On the other hand, if one postulates some suitably large cardinal axiom, one gets as a side effect new transitive models of ZFC, shorter than the one corresponding to the large cardinal's height.

Looks like one can fatten a model, keeping the same height, or make it shorter, but what about creating taller ones?

Question(s) :

1. Are there standard techniques to make a transitive model taller without adding new axioms?

2. Does an axiom like this make sense:

For any transitive model of ZFC $M$ of height $\eta_0=\text{ht}(M)$, there is always another transitive model $N$ such that $M=V_{\eta_0}^N$ of height $\eta_1=\text{ht}(N)$, where $\eta_0 < \eta_1$

NOTE: in the "axiom" above I have simply asked for the second ordinal to be higher than the height of the starting model, but I am also curious as to which extent one could strengthen that, by asking for the second height to be much higher (add your own brand of -much- ) than the first one