I have two proper classes which intuitively are like bijectively equivalent, i.e. for every element of one of these two classes we can define an expression for the corresponding element of the other class, and these behave nicely (like a bijection).
I wonder, is the notion of bijection extended for proper classes? Where could I read about such generalized bijections?
Best Answer
If $X,Y$ are classes defined by formulas $\phi(x), \psi(y)$, then a map $X \to Y$ is just a formula $\alpha(x,y)$ such that $\forall x (\phi(x) \Rightarrow \exists^1 y (\psi(y) \wedge \alpha(x,y)))$. Here $\exists^1$ abbreviates "there exists exactly one ...". This defines the (meta)category of classes and maps of classes. The isomorphisms are exactly the bijections, i.e. with the above notation the maps $\alpha : X \to Y$ such that $\forall y (\psi(y) \Rightarrow \exists^1 x (\phi(x) \wedge \alpha(x,y)))$. In this MO thread it was shown that Schröder Bernstein holds in this setting.
I expect that you can find this notion of bijection in almost every introduction to set theory. A very basic example is the following: Define a (class) well ordering on $\text{On} \times \text{On}$ by
$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$ $(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.
Its type can be used to define a bijection of classes $\text{On} \cong \text{On} \times \text{On}$, but also it yields the equality $\kappa^2=\kappa$ for every cardinal number $\kappa \geq \aleph_0$ (even without AC).