4 ms·

Well done. Has anyone named a set in between the rationals and the reals?

• daxfohl 7y ago
  The canonical example is https://en.wikipedia.org/wiki/Vitali_set https://en.wikipedia.org/wiki/Vitali_set You can kinda think of it as "the set of real numbers MOD the set of rationals". Kinda. And they're dorked up because the length is infinitesimal, but a countably infinite number of them add up to length 1. For a more precise explanation see here: https://math.stackexchange.com/a/137959/287133 https://math.stackexchange.com/a/137959/287133
  • daxfohl 7y ago
    Also note that the construction of the above set requires the axiom of choice. And, as we all know, the axiom of choice is equivalent to the continuum hypothesis in ZFC. So that's how it all fits.
• jfarmer 7y ago
  Whether a set of cardinality strictly between the rationals and reals exists is independent of ZFC. https://en.m.wikipedia.org/wiki/Continuum_hypothesis https://en.m.wikipedia.org/wiki/Continuum_hypothesis There are many sets which are strict supersets of the rationals and strictly sheets of the reals, of course.
  • jfarmer 7y ago
    "strictly subsets of the reals", thanks autocorrect. For example, if ℚ is the rationals and ℝ is the reals then ℚ∪{√2} is a strict superset of the rationals but a strict subset of the reals. However, it still has the same cardinality as the rationals (cf. https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...)
• ginnungagap 7y ago
  If the continuum hypothesis (CH) holds then there is none. Assume that CH fails and well order the reals. An initial segment of length omega_1 is an example of such a set.