Please help me understand even the most basic ideas in model theory:
When in model theory we speak of the cardinality of a model, what exactly is meant by that? I assume that when we say that the model $M$ has cardinality $\kappa$, then $\kappa$ is a cardinal number, and there is a bijection from $\kappa$ ot $M$. But a bijection where? In what system? In an extension model? How can these terms make any sense except inside a model?
Also, is it necessarily the case that $M$ is even in one-to-one correspondence with any cardinal number? Because at least inside a model $M$ for ZFC, the number of elements in $M$ exceeds any cardinal number.