MathOverflow

Are there any well-known statements independent of NF?And also, are there prerequisites suggesting that NF in any way, to one extent or another, are not covered by the incompleteness theorem?

Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an undirected graph. Call this $G$.
Let $B \subset G$ be a ball around some vertex in $G$ of large radius. Since $G$ is vertex-transitive, the choice of vertex is irrelevant. The radius must be large enough that $B$ includes the cycles corresponding to the relations in the presentation...

Suppose we have $n$ lines in general position in the plane. Prove that there are at least $n-2$ ''small'' triangles. Here a "small" triangle is a triangle that is not contained in any larger triangle.

Let $R$ be an integral domain with Krull dimension $1$. If $0\ne a \in R$ is such that for every $b \in R$ , the ideal $Ra \cap Rb$ is principal , then is it true that for every $b\in R$, the ideal $Ra+Rb$ is also principal ?
It is clear that if $0\ne a,b$ and $Ra+Rb=Rd$ , then $Ra \cap Rb=R(ab/d)$. So if we know $Ra \cap Rb =Rc$ and want to prove $Ra+Rb$ is principal, then the unique (upto aasociates) possible candidate for the generator is $ab/c$. I'm unable to say anything...