3 ms·

This has a similar feel to Russell's paradox: Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definit

by n4r9 4y ago

This has a similar feel to Russell's paradox: Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. https://en.m.wikipedia.org/wiki/Russell's_paradox https://en.m.wikipedia.org/wiki/Russell's_paradox Also the Lowest Uninteresting Number paradox: If there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, https://en.m.wikipedia.org/wiki/Interesting_number_paradox https://en.m.wikipedia.org/wiki/Interesting_number_paradox I suspect that the resolution boils down to what it means to "describe" a number.

pavlov 4y ago
I’m not a mathematician but I wonder why the Lowest Uninteresting Number paradox can’t be resolved by simply adjusting the definition of “interesting”? After all that’s a purely human concept. If we say a number isn’t interesting simply due to its proximity to other interesting numbers, then we can draw the boundary. It seems a bit like making a list of important cities. Borders are very important to countries, but not every border city is economically or historically important. It’s not unreasonable to say that a city must meet other criteria than simply being close to a border.
- mannykannot 4y ago
  Yes... when I first saw this paradox, the author of the article I was reading (maybe Martin Gardner?) pointed out that 'uninteresting' is itself not well-defined.
  - tuatoru 4y ago
    The set of uninteresting numbers is probably empty.
    - hgsgm 4y ago
      Please define "uninteresting" and "probably".
      - tuatoru 4y ago
        The standard mathematics teacher's response: I'll leave that as an exercise for the student.
  - kybernetikos 4y ago
    I think the uninteresting number one can be described without worrying about 'interestingness'. It works out as the same as saying something like the set of numbers that are not the smallest number in any set in some arbitrary collection of sets including this one. It's the 'including this one' part that is causing the trouble, not which other sets are included in the arbitrary collection.

by n4r9 4y ago

pavlov 4y ago

I’m not a mathematician but I wonder why the Lowest Uninteresting Number paradox can’t be resolved by simply adjusting the definition of “interesting”? After all that’s a purely human concept. If we say a number isn’t interesting simply due to its proximity to other interesting numbers, then we can draw the boundary. It seems a bit like making a list of important cities. Borders are very important to countries, but not every border city is economically or historically important. It’s not unreasonable to say that a city must meet other criteria than simply being close to a border.

mannykannot 4y ago
Yes... when I first saw this paradox, the author of the article I was reading (maybe Martin Gardner?) pointed out that 'uninteresting' is itself not well-defined.
- tuatoru 4y ago
  The set of uninteresting numbers is probably empty.
  - hgsgm 4y ago
    Please define "uninteresting" and "probably".
    - tuatoru 4y ago
      The standard mathematics teacher's response: I'll leave that as an exercise for the student.
- kybernetikos 4y ago
  I think the uninteresting number one can be described without worrying about 'interestingness'. It works out as the same as saying something like the set of numbers that are not the smallest number in any set in some arbitrary collection of sets including this one. It's the 'including this one' part that is causing the trouble, not which other sets are included in the arbitrary collection.