20 1. Logic and foundations and it will actually exhibit a prime not in that list (in this case, 31). The same cannot be said of Proposition 1.10.3, despite the logical equivalence of the two statements. Remark 1.10.5. It is best to avoid long proofs by contradiction which consist of many parts, each of which is different to convert to a non-counter- factual form. One sees this sometimes in attempts by amateurs to prove, say, the Riemann hypothesis one starts with assuming a zero off of the critical line, and then derives a large number of random statements both using this fact, and not using this fact. At some point, the author makes an error (e.g., division by zero), but does not notice it until several pages later, when two of the equations derived disagree with each other. At this point, the author triumphantly declares victory. On the other hand, there are many valid long proofs by contradiction in the literature. For instance, in PDE, a common and powerful way to show that a solution to an evolution equation exists for all time is to assume that it doesn’t, and deduce that it must develop a singularity somewhere. One then applies an increasingly sophisticated sequence of analytical tools to control and understand this singularity, until one eventually is able to show that the singularity has an impossible nature (e.g., some limiting asymptotic profile of this singularity has a positive norm in one function space, and a zero norm in another). In many cases this is the only known way to obtain a global regularity result, but most of the proof is left in the counterfactual world where singularities exist. But, the use of contradiction is often “shallow”, in that large parts of the proof can be converted into non-counterfactual form (and indeed, if one looks at the actual proof, it is usually the case that most of the key lemmas and sub-propositions in the proof are stated non- counterfactually). In fact, there are two closely related arguments in PDE, known as the “no minimal energy blowup solution” argument, and the “in- duction on energy” argument, which are related to each other in much the same way as the well-ordering principle is to the principle of mathematical induction (or the way that the “no-self-defeating object” argument is related to the “every object can be defeated” argument) the former is counterfac- tual and significantly simpler, the latter is not but requires much lengthier and messier arguments. But it is generally accepted that the two methods are, on some level, equivalent. (See [KiVa2008] for further discussion of these arguments.) As an analogy, one can think of a long proof as a long rope connecting point A (the hypotheses) to point B (the conclusion). This rope may be submerged in murky water (the counterfactual world) or held up above it (the non-counterfactual world). A proof by contradiction thus is like a rope that is almost completely submerged underwater, but as long as the rope is

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