in MSE we find that Axel Harnack in 1885 proved that the interval [0,1] can be covered by a countable number of small intervals such that a countable number of intervals remains in the complement. Nobody thought that Harnack's proof was flawed till Emile Borel appeared on scene. He proved that the complement of a countable union of intervals can be an uncountable union.

Harnack, a pioneer of measure theory, could not react because he passed away in 1888. But he became professor in 1876 and may have had some students.

My question: Has any mathematician defended his proof?

  • 2
    $\begingroup$ Mathematics Genealogy Project says "No students known". This means doctoral students, but it is unlikely that regular ones could weigh in on an active topic of research. Since [0,1] having measure 0 was highly undesirable, people were looking for a way around Harnack's result even before Borel (e.g. Jordan in 1892), so there was little motivation to defend it after 1895. Bressoud gives details on Harnack's work in A Radical Approach to Lebesgue's Theory of Integration. $\endgroup$ – Conifold Jan 6 at 11:35
  • $\begingroup$ There is even strong motivation today to defend Harnack's result because it is obviously correct. See mathoverflow.net/questions/320136/…. Further every mathematician should know that the real line is continuous and therefore every point on the real axis has two adjacent intervals. Therefore points of "scattered space" are incompatible with mathematics. $\endgroup$ – Ibrahim Abd el Faruk-Shaik Jan 6 at 14:22
  • $\begingroup$ Current debates are outside our scope, you can try Philosophy SE. $\endgroup$ – Conifold Jan 7 at 4:52

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