 It should be emphasised again that these structures do not in any way challenge or repudiate existing mathematics, but rather extend our conception of what is mathematically possible. This, in turn, sharpens the issue of Mathematical Pluralism; see e.g., Davies (2005), Hellman and Bell (2006), or Priest (2013). Various authors have different versions of mathematical pluralism, but it is something along the lines that incompatible mathematical theories can be equally true. The case for mathematical pluralism rests on the observation that there are different mathematical “universes” in which different, indeed incompatible, mathematical theorems or laws hold. Well-known examples are the incompatibility between classical mathematics and intuitionist mathematics, and the incompatibility between ZF-like universes of sets respectively with, and without, the Axiom of Choice. It seems absurd to say that ZF with Choice is true mathematics and ZF without Choice is false mathematics, if they are both legitimate examples of mathematically well-behaved theories.