New Foundations and weaker forms of choice New Foundations (introduced by Quine) proves that $AC$ is false. Out of curiosity, is $NF$ consistent with countable choice or dependent choice? What's the strongest consequence of choice still consistent with $NF$ and has it been localized at what point consequences of $AC$ become inconsistent with $NF$? Hope my question is clear. Thx. 
 A: Perhaps I am not qualified to answer this question, but there are some experts claiming in email newsletters that NF + PP[1] is consistent. (no proof)
In a general way, the NF's choiceless set is very "close" to V.
Not all weaker forms of choice axiom are consistent with NF. For example,
Thomas Forster claimed[2] a proof that $NF + DC_α$[3] is inconsistent, where α is a non-Cantorian ordinal. No one has studied the specific upper / lower boundaries, But $\Omega$: "The order type of all the ordinals" satisfies the requirement.

Consider the family of all wellorderings under the binary relation of end-extension and consider a maximal chain, using (some form of) DC. The union of this chain will be a wellordering that all wellorderings embed into. But this is impossible.


[1] Partition Principle: if A is nonempty and surjects onto B, B injects into A
[2] http://www.dspace.cam.ac.uk/handle/1810/223940
[3] $DC_α$: Let $S$ be a nonempty set and let $R$ be a binary relation such that for every $α$-sequence $s = [ x_γ : γ < α ]$ of elements of $S$ there exists $y ∈ S$ such that $sRy$.Then there is a function f such that for every $γ < α, (f | γ) R f(γ)$.
