Question

Have a good week!
Quine argued against the principle of falsification that we never really know which axiom should be removed, since even if, for example, we did not find the bacterium under the microscope, perhaps it wasn’t standing properly, and so on.
So I would like to ask: once we know that every logical system is based on assumptions acquired intuitively, etc., and that the only thing analytic analysis does is show that we are not consistent within the system of arguments that we derive from the axioms—then it is difficult for me to understand how one can show that we are inconsistent. After all, according to Quine, we will never know which axiom we ought to give up.
Thank you very much!