Suppose 1st-order arithmetic is inconsistent along with Voevodsky http://video.ias.edu/voevodsky-80th.

It nevertheless remains true that when you have 2 apples and 2 apples, you have 4 apples. Preforming an experiment gives you the result of an experiment, which cannot be inconsistent. So there is a subset of arithmetic that is "necessarily" consistent, given the notion (maps) that arithmetic models reality. The question is, what is this "physically consistent" proper subset of arithmetic?

The second question is, what happens if the physical theory is quantum field theory, where quanta loose their individual identity or "primitive thisness"?

applesto lose their individual identity: all you have to do is tweak the temperature or pressure of your surroundings... – Qiaochu Yuan Mar 18 '11 at 22:40