I think you can derive enough real analysis to define the Cantor function using only ZFC. You can read the axioms in [1], but they are very simple and “obvious” so to speak. What’s more interesting is where ZFC is known to be incomplete. Cantor’s continuum hypothesis is known to be unprovable in ZFC, which is fun the think about.

[1] https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory