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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?

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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).

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I don't like defining a map as a formula. This suppresses maps which are not expressible as a finite formula. Well, in my particular case I can deal with this special case when we consider only maps expressible with a finite formula, because in my example I can write explicit formulas down. But I just don't like it. – porton Jun 14 '11 at 21:17
How do you define infinite formulas? – Martin Brandenburg Jun 14 '11 at 21:56
@Martin: I don't think that the OP is positing infinite formulas. But only countably many of the maps from $\omega$ to $\omega$ (for example) are defined by formulas. – John Bentin Jun 15 '11 at 11:06
@John: Perhaps we should allow parameters. – Martin Brandenburg Jun 16 '11 at 8:27
How does this prove $\kappa^2=\kappa$ for every $\kappa \ge \aleph_0$? It seems to only work for alephs. – Chris Eagle Jun 18 '11 at 10:18

ARC, an extension of Ackermann set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, PDF) proves the existence of the n-th powerclass of the set universe V for any $n \in \mathbb{N}$. That should make it possible to define pairs/triples of classes and thus functions and bijections the usual way.

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Many set theories with classes have the limitation of size principle. This says all proper classes have the same cardinality. The only interesting bijections are the ones that are definable by a formula (because formulas give extra information that anonymous bijections hide). There probably are alternative foundations where the limitation of size principle does not hold or models of ZFC where the definable classes have different sizes. You might find an interesting theory of functions (and bijections) of proper classes there.

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It is consistent with ZFC that there is no global well-ordering, which is equivalent to what you call the limitation of size (since bijections with Ord provide a global well-ordering and conversely). In Goedel-Bernays set theory, one has global well-orderings and hence limitation of size, but not every class need be definable. For example, the classes added by class forcing (which are definitely interesting) are usually not definable. – Joel David Hamkins Jun 18 '11 at 10:30
Is there an example of a set model of set theory with classes, in which the classes are, externally, not all of the same size? – Norman Lewis Perlmutter Aug 20 '11 at 17:17

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