[daxfohl]

Why is it always zfc plus optionally something else? Is there anything other than zfc that creates an interesting starting point?

[triska]

These are excellent questions! And yes, there is!

For example, I highly recommend the paper Rethinking set theory by Tom Leinster:

https://arxiv.org/abs/1212.6543

It highlights Lawvere set theory. The paper won this year's Chauvenet Prize:

https://www.maa.org/programs-and-communities/member-communit...

[QuesnayJr]

Isn't Lawvere set theory equiconsistent with bounded Zermelo set theory?

ZFC, its subtheories, and its extensions, serve as a universal yardstick for consistency results. It's not really clear why that is the case, but it has so far proven to be so.

[andrepd]

https://en.m.wikipedia.org/wiki/New_Foundations

[empath75]

https://en.m.wikipedia.org/wiki/Univalent_foundations