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Claimed Proof of Riemann Hypothesis
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
The Continuum Hypothesis is known to be neither provable nor disprovable in the standard axiomatic set theory ZF, enriched with the axiom of choice (ZFC). So I wouldn't really count on someone settling that one either way any time soon. Of course one could come up with a new set of axioms for the set theory and *then* prove or disprove CH but you would be hardpressed to find anyone showing interest in that result. After all, I could just add CH or not(CH) to ZFC and trivially prove or disprove it. So anything in that line first needs to even define what a sensible problem is.
For those who have no clue what I said above:
Continuum hypothesis: There is no set strictly larger than the set of natural numbers and at the same time strictly smaller than the set of real numbers. The size of a set in relation to other is defined in terms of mapping. Positive integers are the same number as even numbers because you can define a bijection between the two. Reals are strictly more than naturals.
ZF: Set theory made axiomatic. Few axioms (like empty set exists, supersets are larger than original sets etc) that you need to believe and most of the set theory believed to follow.
Axiom of Choice: Given a set of sets, one can make a set containing one element from each set. Looks obviously true but in some equivalent but different sounding formulations looks obviously false. Known to be independent to ZF.
Y Independent to axioms X: Believing that Y is true does not yield contradiction together with X unless X itself yield contradictions. Same holds for believing that Y is false.
PS: Apologies for not including links. I am feeling lazy. Wikipedia has nice articles about all of the above. Articles on ZF, CH or Axiom of Choice are the place to start for a fun reading.