Define: $n$-skew pair of $x,y$, denoted by $\langle x,y \rangle^n$, as: $(singleton^n(x), y)$


Define: $(-n)$-skew pair of $x,y$, denoted by $\langle x,y \rangle^{-n}$, as: $(x, singleton^n(y))$


Where $(-,-)$ is the Kuratwoski ordered pair implementation, and $n$ is a natural 


where: $singleton^0(x) = x$


$singleton^{i+1}(x) = \{singleton^i(x)\}$


so $\langle x,y \rangle ^0$ and $\langle x,y\rangle^{-0}$ are both level pairs.


Define: $f \text { is }n \text{-skew injection } \equiv\_{df} f \text { is injection} \land \forall p \in f (p \text { is n-skew pair})$


Define: $x \leq^\* y \equiv\_{df} \exists n \in \mathbb Z, \exists f (f:x \to y, f \text{ is n-skew injection)}$


Define: $x =^\* y \equiv\_{df} \exists n \in \mathbb Z, \exists f (f:x \to y, f \text{ is n-skew bijection)}$


Where $\mathbb Z$ is the set of Integers.


Write Cantor-Bernstein-Schroeder theorem in terms of $\leq^\*,=^\*$ , denoted as "skew-CBS", as:


$\forall x,y [(x \leq^\* y \land y \leq^\* x) \to x=^\* y]$



> 
> Question: Is skew-CBS consistent with NFU?
> 
> 
> 


Of course both $\leq^\*$ and $=^\*$ are non-stratified relations, so they work externally. 



Is Cantor-Bernstein-Schroeder theorem for skew cardinality, consistent with NF?
No. A counterexample to skew CBS follows from the existence of any cardinals
T(kappa) < lambda < kappa [cardinals in NF(U) are Frege cardinals; T(kappa) is
the cardinality of elementwise images of elements of kappa under the singleton map.]


This is a counterexample because an element of lambda will have an ordinary injective
embedding into an element of kappa, then an element of kappa will have a skew embedding to a subset of size T(kappa) included in an element of lambda.


And there cannot be any skew bijection from a set of size kappa to a set of size
lambda because the image of a skew bijection with domain in kappa must have
range in some T^n(kappa), and lambda is clearly not any T^n(kappa) [n might be negative here, which does not affect the conclusion].


In NFU+Choice there are cardinals strictly between |V| and T(|V|).


In NF finding cardinals T(kappa) < lambda < kappa is technically tricky but I know how to do it.


