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I’ve always been severely dissatisfied with this argument. It feels like a sleight of hand as opposed to something profound. Are there any other roads to “size
by thethimble 5y ago
I’ve always been severely dissatisfied with this argument. It feels like a sleight of hand as opposed to something profound.
Are there any other roads to “sizes of infinity” that are more palatable than the diagonalization argument?
- morelisp 5y agoDo you also see the First Incompleteness Theorem, or Halting Problem, as sleights of hand? Informally, the answer to your question is no - the Schröder-Bernstein theorem, which lets us order the size of sets, is sufficient to derive the law of the excluded middle. Therefore if you don't like the "trick" i.e. proof by contradiction (even given the contradiction is "actually constructed" in this case), and instead demand constructive mathematics, you will not be able to say much about relative cardinality.