1.10. The “no-self-defeating object” argument, revisited 21 only shallowly underwater, one can still see it well enough to conclude that it is unbroken. But if it sinks too far into the depths of the counterfactual world, then it becomes suspicious. Finding a non-counterfactual formulation of an argument then resembles the action of lifting the rope up so that it is mostly above water (though, if either the hypothesis A or conclusion B are negative in nature (and thus underwater), one must still spend a tiny fraction of the argument at least in the counterfactual world also, the proof of the non-counterfactual statement may still occasionally use contradiction, so the rope may still dip below the water now and then). This can make the argument clearer, but it is also a bit tiring to lift the entire rope up this way if one’s objective is simply to connect A to B in the quickest way possible, letting the rope slide underwater is often the simplest solution. 1.10.1. Set theory. Now we revisit examples of the no-self-defeating ob- ject in set theory. Take, for instance, Cantor’s theorem: Proposition 1.10.6 (Cantor’s theorem). The reals are uncountable. One can easily recast this in a “non-counterfactual” or “all objects can be defeated” form: Proposition 1.10.7. Let x1,x2,x3,... be a countable sequence of real num- bers (possibly with repetition). Then there exists a real number y that is not equal to any of the x1,x2,x3,.... Proof. Set y equal to y = 0.a1a2a3 ..., where (1) a1 is the smallest digit in {0,..., 9} that is not equal to the first digit past the decimal point of any9 decimal representation of x1 (2) a2 is the smallest digit in {0,..., 9} that is not equal to the second digit past the decimal point of any decimal representation of x2 (3) etc. Note that any real number has at most two decimal representations, and there are ten digits available, so one can always find a1,a2,... with the desired properties. Then, by construction, the real number y cannot equal x1 (because it differs in the first digit from any of the decimal representations of x1), it cannot equal x2 (because it differs in the second digit), and so forth. 9Here we write “any” decimal representation rather than “the” decimal representation to deal with the annoying 0.999 . . . = 1.000 . . . issue mentioned earlier. As with the proof of Euclid’s theorem, there is nothing special about taking the smallest digit here this is just for the sake of concreteness.
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