It is shown that the 3-stratifiable sentences are equivalent in to truth-functional combinations of sentences about objects, sets of objects, sets of sets of objects, and sentences stating that there are at least urelements. This is then used to characterize the closed 3-stratifiable theorems of with an externally infinite number of urelements, as those that can be nearly proved in with an externally infinite number of urelements. As a byproduct we obtain a rather simple demonstration of the consistency of 3-stratifiable (...) extensions of. (shrink)

It is shown that the 3-stratifiable sentences are equivalent in $\mathit{NFU}$ to truth-functional combinations of sentences about objects, sets of objects, sets of sets of objects, and sentences stating that there are at least $n$ urelements. This is then used to characterize the closed 3-stratifiable theorems of $\mathit{NFU}$ with an externally infinite number of urelements, as those that can be nearly proved in $\mathit{TTU}$ with an externally infinite number of urelements. As a byproduct we obtain a rather simple demonstration of (...) the consistency of 3-stratifiable extensions of $\mathit{NFU}$. (shrink)