2 added 30 characters in body edited Feb 21 at 22:50 gowers 19.9k2525 gold badges125125 silver badges171171 bronze badges This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $$f$$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to believeprove that $$f$$ existsis a well-defined bijection. So for example if you order the elements of two sets $$A$$ and $$B$$ in some way that has nothing to do with why $$|A|=|B|$$, then you need to know that $$|A|=|B|$$ in order to conclude that the bijection that maps the $$k$$th element of $$A$$ to the $$k$$th element of $$B$$ is indeed a well-defined bijection. I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy). This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $$f$$ to count as explicit, one shouldn't need to know that there exists a bijection in order to believe that $$f$$ exists. So for example if you order the elements of two sets $$A$$ and $$B$$ in some way that has nothing to do with why $$|A|=|B|$$, then you need to know that $$|A|=|B|$$ in order to conclude that the bijection that maps the $$k$$th element of $$A$$ to the $$k$$th element of $$B$$ is indeed a well-defined bijection. I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy). This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $$f$$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to prove that $$f$$ is a well-defined bijection. So for example if you order the elements of two sets $$A$$ and $$B$$ in some way that has nothing to do with why $$|A|=|B|$$, then you need to know that $$|A|=|B|$$ in order to conclude that the bijection that maps the $$k$$th element of $$A$$ to the $$k$$th element of $$B$$ is indeed a well-defined bijection. I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy). 1 answered Feb 21 at 22:18 gowers 19.9k2525 gold badges125125 silver badges171171 bronze badges This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $$f$$ to count as explicit, one shouldn't need to know that there exists a bijection in order to believe that $$f$$ exists. So for example if you order the elements of two sets $$A$$ and $$B$$ in some way that has nothing to do with why $$|A|=|B|$$, then you need to know that $$|A|=|B|$$ in order to conclude that the bijection that maps the $$k$$th element of $$A$$ to the $$k$$th element of $$B$$ is indeed a well-defined bijection. I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy).