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I'm not sure I agree with that characterization of the exceptions. What exactly did you mean by spending infinitely long writing the number down? If "infinitel
by hansvm 5y ago
I'm not sure I agree with that characterization of the exceptions. What exactly did you mean by spending infinitely long writing the number down?
If "infinitely long" means some unbounded, finite time then that gives a subset of the rationals, and the other rationals would be exceptions that we missed. If it means that there exists a program capable of spitting out successive digits then those are the computable numbers, but some of those (like pi) are transcendental. If it means that they're represented in at least one way by an infinite string of digits then our list of exceptions is now all the reals.
- ncmncm 5y agoInfinitely long means, literally, exactly the opposite of "finite time". In infinite time you can write, exactly, countably many digits, countably many fractions, and countably many roots. I.e., the number of exceptions is countably-infinite. But "countable infinity" is a more sophisticated concept than the problem statement called for.
- hansvm 5y agoI just missed the obvious context -- finite time writing <<what>> exactly. The exceptions can be written in finite time using integers and a specific set of operations, whereas the question I posed looked at decimal representations.
- ncmncm 5y agoNo. In infinite time you can write a countably-infinite number of symbols, identifying a countably-infinite number of exceptions. You might choose to write some random set other than the rationals and roots, but what you could write (absent infinite repetition) would have exactly as many members as those sets, and thus map to them one-to-one; so, just identify them possibly more abstractly. But the set will in any case remain negligible vs. the uncountably-infinite real numbers.