Bijection of proper classes - MathOverflow

Bijection of proper classes

2011-06-14T17:55:40Z

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?

Answer by Martin Brandenburg for Bijection of proper classes

2011-06-14T18:15:13Z

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).

Answer by Wouter Stekelenburg for Bijection of proper classes

2011-06-18T08:06:37Z

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.

Answer by Bernhard Stadler for Bijection of proper classes

2012-05-30T09:59:36Z

ARC, an extension of Ackermann set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, http://en.scientificcommons.org/49425946 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.