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 Jul24 asked Why is the volume of a cone one third of the volume of a cylinder? Jul24 comment Is $0$ a natural number? @Justin, I know that there are mixed views (as indicated in the second paragraph of my question). But for the case of 1 being classified as a prime number, it seems the consensus view of the Mathematical community is that it should not count as a prime number. My actual question is 'Is there a consensus on whether zero is a natural number?' (although the question's title is simpler), so a suitable answer would be 'No, there is no consensus' combined with a quick demonstration from a few Mathematical dictionaries or articles that there are conflicting definitions. Jul23 comment Explanation of method for showing that 0 / 0 is undefined Neither 0/x = 0 nor x/x = 1 would be considered axioms though -- they're just observations that can be proved for any x other than zero, so I don't think it makes sense to pit them off against each other. Jul23 comment Explanation of method for showing that 0 / 0 is undefined Except that the 0^-1 is undefined (the original question didn't ask why 1/0 was undefined, but that seems acceptable), so there is no need for us to be disturbed. Jul23 answered Explanation of method for showing that 0 / 0 is undefined Jul23 comment How would you describe calculus in simple terms? Fair enough. I think I choose them in this order because I the concept of area seems more basic to understand, whereas the concept of tangents/gradients at points seems a bit more complex. But I would never teach integration before differentiation! Jul22 comment Why isn't reflexivity redundant in the definition of equivalence relation? At the risk of making another embarrassing mistake, I think I've realised why I got the last one wrong -- I was using the definition of Euclidean that I got from your link to the wikipedia article, rather than the one you gave above. I think my counterexample works with that definition of Euclideanity. Jul22 comment What is the meaning of the double turnstile symbol ($\models$)? @Noldorin, perhaps the simplest example of the distinction could be seen in classical propositional logic. A statement like |= p -> (q -> p) means that the no matter the truth values of p and q, the value of p -> (q -> p) is always truth (which can be seen by looking at the truth table). A statement like |- p -> (q -> p) means that we can prove this formula purely by using axioms/rules of deduction, with no mention of truth or meaning. Incidentally, I think you could ask this as a separate question on the site, since we don't want the comment strand to be where one has to look for answers. Jul22 comment Why isn't reflexivity redundant in the definition of equivalence relation? @Charles: ouch: I typed it into a program that does a brute-force search for counterexamples, but I misrepresented the Euclideanity as 'Rxy & Rxz -> Ryz'. Perhaps I should have read the definition you gave! Sorry. Jul22 comment Why isn't reflexivity redundant in the definition of equivalence relation? Not to sound as though I'm picking on you, but I think this revised claim is also false. My first clue was the fact that the wikipedia article you referenced said that Euclidean+Reflexive => Equivalence. Since being reflexive is a stronger requirement than being serial I sought a counterexample, the smallest of which I can find is on the set {1,2} and is defined as Rxy <-> y = 2. Jul22 awarded Nice Question Jul22 comment Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Thank you for adding this answer. I was hoping to avoid an answer that involved integration, so I also prefer AgCl's answer. But I am happy to see more than one demonstration/proof. Jul22 comment Why is $x^0 = 1$ except when $x = 0$? Unfortunately there's also a debate about whether 0 is a natural number (see math.stackexchange.com/questions/283/is-0-a-natural-number). But I'll take your usage to mean 'nonnegative integer', and that you're voting for $0^0 = 1$. Jul22 comment What is the meaning of the double turnstile symbol ($\models$)? Sorry, I realise that I haven't actually answered your question (about the meaning). I've seen it used in logic to stand for semantic entailment, but I imagine it has other uses. Jul22 answered What is the meaning of the double turnstile symbol ($\models$)? Jul22 awarded Commentator Jul22 comment Why is $x^0 = 1$ except when $x = 0$? @Neil, except that 0^x = 0 is not true for all non-zero x (e.g. x = -1). Jul22 comment Why isn't reflexivity redundant in the definition of equivalence relation? Perhaps I made up the noun form. If you google 'serial binary relation' it will come up. Although if you google 'seriality of binary relations' it will come up there too. Jul22 comment Why isn't reflexivity redundant in the definition of equivalence relation? This seems to fail though because (as Akhil mentioned) the empty relation would satisfy this but it's not reflexive. Jul22 comment Are there any interesting semigroups that aren't monoids? I'll try my luck. If the community decides to close the question on grounds of arcaneness then so be it.