Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen ( 1936 ) proved the consistency of arithmetic using a finitistic system together with a principle of transfinite induction . Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Gödel ( 1958 ) gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types.
Does that give you a warm glow? It gives me one, even though I know there is another side to this feel-good story. Make-A-Wish would not say how much it cost to fulfill Scott’s wish, but it did say that the average cost of making a child’s wish come true is $7,500. Effective altruists would, like anyone else, feel emotionally drawn toward making the wishes of sick children come true, but they would also know that $7,500 could, by protecting families from malaria, save the lives of at least three children and maybe many more. Saving a child’s life has to be better than fulfilling a child’s wish to be Batkid. If Scott’s parents had been offered that choice—Batkid for a day or a complete cure for their son’s leukemia—they surely would have chosen the cure. When more than one child’s life can be saved, the choice is even clearer. Why then do so many people give to Make-A-Wish, when they could do more good by donating to the Against Malaria Foundation, which is a highly effective provider of bed nets to families in malaria-prone regions? The answer lies in part in the emotional pull of knowing that you are helping this child, one whose face you can see on television, rather than the unknown and unknowable children who would have died from malaria if your donation had not provided the nets under which they sleep. It also lies in part in the fact that Make-A-Wish appeals to Americans, and Scott is an American child.
Overall, this is an excellent start to a compelling essay. Keep in mind that although the committee allows you to write up to five pages, three double-spaced pages is usually adequate. Feel free to add more detail, but make sure your text is concise and transparent. You might consider "writing" ideas and then "rewriting" them. Too often, authors put ideas onto the page, but do not render those ideas compellingly. Giving your sentences a bit of extra effort, and giving your experiences a bit of extra reflection, will result in a persuasive essay that compliments your accomplishments and character.