If NF is consistent, then yes Con(NF) would be one of these statements. NF can interpret finite order arithmetic, so by that it would be subject to Godel incompleteness theorems. If Randall Holmes's proof of NF is correct, then NF is slightly stronger than finite order arithmetic, this means that all [strong axioms of infinity][1] are independent of it.

 


  [1]: https://en.wikipedia.org/wiki/New_Foundations#Strong_axioms_of_infinity