5 ms·

Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardina

• j2kun 12y ago
  Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.
  • dbecker 12y ago
    Though, if we apply your requirement, the set of problems without a unique answer must be countable as well. So, the cardinality of problems with a unique answer is still equal to the cardinality of problems without one.
    • j2kun 12y ago
      You're just using the terminology imprecisely is all. (Indeed, two things being countable does not mean they have the same cardinality; finite sets are also countable)
      • dbecker 12y ago
        I think it's trivially obvious that both sets are infinite. There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.
        • j2kun 12y ago
          I'm not making claims about the truth or falsity of your statement, just critiquing the logic and terminology you're using (the lack of a definition for a "problem," the minor misuse of the term "countable"). Regardless, I really hope you saw the article as something besides a mathematical claim, because it clearly did not pretend to be one.
          • dbecker 12y ago
            Indeed. My original comment was meant as a joke, not as a serious mathematical claim.
  • JadeNB 12y ago
    This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)
    • j2kun 12y ago
      The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of something that I don't agree is a problem (if you're saying what I think you're saying).
      • JadeNB 12y ago
        My weak, but (I think) practical, implicit definition of 'problem' was: > a question that a mathematician might ask I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.) By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)
        • j2kun 12y ago
          You're falling into the interesting number paradox here, regardless of not using the word interesting. You can't make a statement about the cardinality of a set if your definition of that set isn't expressed using mathematics. In other words, you're saying "I'm not going to give a mathematical definition of this thing, but I can nevertheless give a mathematical proof of a mathematical statement about that thing." For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!