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 Jan 8 awarded Yearling May 18 awarded Popular Question Mar 31 awarded Nice Answer Feb 18 awarded Nice Answer Feb 21 awarded Yearling Jan 14 awarded Nice Answer Feb 22 awarded Yearling Dec 6 answered Is it consistent with ZFC that for all ordinals $\alpha, \beta < \omega$ it holds that $2^{\aleph_\alpha} = 2^{\aleph_\beta}$? Oct 14 comment Most 'unintuitive' application of the Axiom of Choice? Resurrecting an old answer here: It should be noted that many of these can be ruled out by resorting to countable AC or dependent choice, which avoid many of the strange consequences of full AC. For example, "A set can be infinite, but have no countably infinite subset", is ruled out by countable AC. Oct 14 revised When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals) struck erroneous statements Aug 24 comment Mathematics as a hobby "Everything else, like getting the books and papers you need, is basically solved by knowing google and wikipedia :-)" How are you seeing the papers? One does need a good university library nearby, for that at the very least. Even MathSciNet isn't available to the general public. Aug 22 awarded Nice Answer May 8 awarded Civic Duty Apr 19 awarded Necromancer Mar 13 comment How do we know that P != LINSPACE without knowing if one is a subset of the other? That's not a problem. The whole argument is proof by contradiction; you've just provided a different contradiction. Mar 13 comment Is a space with no covering spaces simply connected? Nitpick: I assume you mean "connected covering space", because otherwise... Feb 22 awarded Yearling Feb 10 comment Is the “closedness of the image of operator” needed in the defintion of Fredholm operators? That "im T + C^n" is supposed to be at the end of the first paragraph. Not sure why it didn't show up there. Feb 10 answered Is the “closedness of the image of operator” needed in the defintion of Fredholm operators? Dec 14 comment Proofs that require fundamentally new ways of thinking I took the last set theory course that Cohen taught, and this isn't how he presented his insight at all (though his book takes this approach). The central problem is "how do I prove that non-constructible [sub]sets [of N] are possible without access to one?", and his solution is "don't use a set; use an adaptive oracle". Once that idea is present, the general method falls right into place. The oracle's set of states can be any partial order, generic filters fall right out, names are clearly necessary, everything else is technical. The hardest part is believing it will actually work.