Given a set $X$ let $\mathsf{PP}(X)$ be the following statement:
For every equivalence relation $\sim$ on $X$, there is a function $f:X\rightarrow X$ such that $$\forall x,y\in X, \ x \sim y \iff f(x)=f(y)$$
This is the Partition Principle restricted to a set $X$. In particular $\mathsf{PP}(X)$ is equivalent to saying that for every surjective $g: X\rightarrow Y$ there is a injective $f: Y \rightarrow X$.
Now consider the first Cohen model $\mathcal{N}$, the one in which there is an infinite Dedekind-finite set $A\subset\mathbb{R}$:
- Does $\mathsf{PP}(A)$ hold in $\mathcal{N}$?
Ideas?
Thanks!
No. Every infinite set of reals maps onto $\omega$, but Dedekind finite sets have no countably infinite subsets.