1.12. A computational perspective on set theory 35 (2) If x is an element of C, then there is another element y of C such that y x. (3) If (xβ)β∈B is a totally ordered set in C, indexed by another set B, then there exists an element y ∈ C such that y ≥ xβ for all β ∈ B. Then C is not a set. This lemma (which, of course, requires the axiom of choice to prove, and is in fact equivalent to this axiom) is usually phrased in the contrapositive form: any non-empty set C for which every totally ordered set has an upper bound, has a maximal element. However, when phrased in the above form, we see the close similarity between Zorn’s lemma and Propositions 1.11.1– 1.11.3. In this form, it can be used to demonstrate that many standard classes (e.g., the class of vector spaces, the class of groups, the class of ordinals, etc.) are not sets, despite the fact that each of the hypotheses in the lemma do not directly seem to take one from being a set to not being a set. This is only an apparent contradiction if one’s notion of sets is vague enough to accommodate sorites-type paradoxes. More generally, many of the objects demonstrated to be impossible in the previous articles in this series can appear possible as long as there is enough vagueness. For instance, one can certainly imagine an omnipotent being provided that there is enough vagueness in the concept of what “om- nipotence” means but if one tries to nail this concept down precisely, one gets hit by the omnipotence paradox. Similarly, one can imagine a foolproof strategy for beating the stock market (or some other zero sum game), as long as the strategy is vague enough that one cannot analyse what happens when that strategy ends up being used against itself. Or, one can imagine the pos- sibility of time travel as long as it is left vague what would happen if one tried to trigger the grandfather paradox, and so forth. The “self-defeating” aspect of these impossibility results relies heavily on precision and definite- ness, which is why they can seem so strange from the perspective of vague intuition. 1.12. A computational perspective on set theory The standard modern foundation of mathematics is constructed using set theory. With these foundations, the mathematical universe of objects one studies contains not only18 the “primitive” mathematical objects such as numbers and points, but also sets of these objects, sets of sets of objects, and so forth. One has to carefully impose a suitable collection of axioms 18In a pure set theory, the primitive objects would themselves be sets as well this is useful for studying the foundations of mathematics, but for most mathematical purposes it is more convenient, and less conceptually confusing, to refrain from modeling primitive objects as sets.

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