[dwheeler]

If you believe the big advantage of Principia Mathematica was that it starts with a very few axioms and then manages to formally and exactly prove many things, then MPE is a worthy successor. I'm in that camp.

However, if you think the main point of Principia Mathematica was the very specific set of axioms that they chose, then that's different. The PM authors chose to use a "ramified" theory of types, which is complex. It does have sets, it's just not ZFC sets. Few like its complexities. Later on Quine found a simplification of their approach and explained it in "New Foundations for Mathematical Logic". There's a Metamath database for that "New Foundations" axiom set as well (it's not as popular, but it certainly exists): https://us.metamath.org/index.html

More Metamath databases are listed here, along with some other info: https://us.metamath.org/index.html

[dmvdoug]

Although N.B.: it’s still an open question whether NF is consistent.

[dwheeler]

Strictly speaking that's true for any system that can handle arithmetic (as proved by Goedel). You can show an inconsistency, but you can't prove consistency. No one's found an inconsistency.

There are pros and cons for both systems.

[dmvdoug]

Yes, I was too loose with me words again. I was gesturing at relative consistency (especially, at least, when compared to the lack of serious doubt about ZF/C’s consistency), as well as the work of folks like RB Jensen and Randall Holmes.

The biggest problem NF has is, as usual, a social one: there just ain’t a lot of people working on, or interested in working on, NF compared to other set theories.