3 edited tags
2 removed confusing informal use of "class"

In the paper "Complete topoi representing models of set theory" by Blass and Scedrov, they consider a general class notion of Boolean-valued models model of set theory, and one of the conditions they impose is that the model contain "no extra ordinals after those of V", i.e. that for all z in the model we have

$$\Vert z \text{ is an ordinal} \Vert = \bigvee_{\alpha \text{ is an ordinal of } V} \Vert z=\check{\alpha}\Vert$$

where $\Vert-\Vert$ denotes the truth function of the model valued in some complete Boolean algebra.

My question is: do there exist models which do contain "extra ordinals" in this sense? I presume so, or they wouldn't have needed to impose this condition. What do such models look like?

(By way of clarification, certainly if the starting model V is a set model in some larger universe, then one can find other set models in that larger universe which contain more ordinals. But I'm interested in just starting with a single model V and building models from it, which can be sets or proper classes.)

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# Can models of set theory contain extra ordinals?

In the paper "Complete topoi representing models of set theory" by Blass and Scedrov, they consider a general class of Boolean-valued models of set theory, and one of the conditions they impose is that the model contain "no extra ordinals after those of V", i.e. that for all z in the model we have

$$\Vert z \text{ is an ordinal} \Vert = \bigvee_{\alpha \text{ is an ordinal of } V} \Vert z=\check{\alpha}\Vert$$

where $\Vert-\Vert$ denotes the truth function of the model valued in some complete Boolean algebra.

My question is: do there exist models which do contain "extra ordinals" in this sense? I presume so, or they wouldn't have needed to impose this condition. What do such models look like?

(By way of clarification, certainly if the starting model V is a set model in some larger universe, then one can find other set models in that larger universe which contain more ordinals. But I'm interested in just starting with a single model V and building models from it, which can be sets or proper classes.)