https://en.wikipedia.org/wiki/Foundations_of_mathematics

Well... explained rather than resolved.

It has been known for over a hundred years that there is no foundation to mathematics (in particular axiomatic mathematics). While only a handful of axioms are required to define a given axiomatic system - it is impossible to define that handful of axioms in the first place.

This is a pretty large problem. It has generally been ignored because:

1. There didn't seem to be an alternative.

2. "The Empty Set" is pretty obvious even if we can't define it in a non-tautological way.

In response:

2. Yeah - a tautology isn't a good starting point.

1. We can directly measure relationships.