absoluteness and and transitivity

Question

I'm early in my reading about absoluteness, but one thing has me stuck, so I thought I'd ask. One reason absoluteness seems to matter is that we feel confident that we know what we're talking about when we ascribe properties that are absolute. But the structures considered are almost always transitive (and often also models of ZF). Can't one find a non-transitive model which "thinks of itself" as transitive? If so, then why think that because a property is absolute in transitive models that we have a better grip on it? Aren't we "cheating"?

Answer 1

What does it mean for a model to "think of itself transitive"? It means that whenever $x$ is in the model, and the model thinks that $y\in x$, then $y$ is in the model.

But in order for the model to even know about $y$ it had to be in the model to begin with. So every model thinks it is transitive.

Now take a countable elementary submodel of any large enough portion of the universe, e.g. $H(\omega_2)$, in which $\omega_1$ is definable, and show that your submodel cannot be transitive.