I'm not sure if my question make sense, but it would also be interesting to know if it didn't, so I will ask anyway:


There exist a countable model of ZF. This means (if I understand it correctly) that we can find a model of ZF in ZF, such that the set of elements in the model is countable seen from the outer ZF. 


**My question is:** Can every model of ZF be "seen from the outside" in a way that makes it countable? 


It seems to me, that if we have a model A of ZF, the model will be a element in a countable model B of ZF. Now, if you look at B from "the outside" A is countable.



Is every model of ZF countable "seen from the outside"?
Technically, a model of ZF consists of a set with some relation, representing "being an element of". So the literal answer is no. An uncountable model is simply uncountable. Your argument that one can see every model as a model in a model which is countable doesn't work. Cardinality in a model depends on whether there exists certain bijections in the model. This has nothing to do with "outside cardinality". 


However, the first-order statements true in some model of ZF are true in some countable model by the [Löwenheim-Skolem theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem). Since we are usually not interested in differences between models that cannot be formulated in the first order theory of sets, we can say that every model of ZF has an, for practical purposes, equivalent countable model.


