The question of consistency of $\sf NF$ can be seen to be equivalent to the question of whether the theory "Stratified $\sf Z$ - Regularity - Infinity + There exists a set as big as its powerset" is consistent?
Stratified $\sf Z$ is exactly the theory "$\sf ZF$-Replacement" but with Separation restricted to stratified formulas.
"As big as" means "is equinumerous to" defined in the customary manner after existence of a bijection.
Now, the question which presents itself is:
If we assume the consistency of $\sf NF$, can we infer that the above mentioned theory is consistent with the axiom of Regularity?
