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visits member for 4 years, 4 months
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Sep
13
comment Proof: Series converges $\implies $ the limit of the sequence is zero
You are mistakingly assuming that the sequence actually converges to something. The other proofs also show that the limit exist, AND it is 0.
Sep
11
comment Boy Born on a Tuesday - is it just a language trick?
@Gadi The formulation you give is indeed ambiguous: we do not know a probability model for who goes to open the door. The formulation in the original question is unambiguous and indeed is a simple problem of conditional probability.
Sep
11
comment Boy Born on a Tuesday - is it just a language trick?
@T. No, it is not. In the Monty Hall problem there is a random choice, and then the showman has to make a choice about which door to open (he certainly does not want to open the door with the goat). In this case, there is not a random choice, after which the parent has to reveal information (for instance a random question). It is just plain conditional probability: the fact that the parent reveals this piece of information has nothing to do with the fact that he/she could reveal another.
Sep
11
comment Boy Born on a Tuesday - is it just a language trick?
@Gadi: I do not agree. I think you are making confusion with similar sounding problems like the Monty Hall one. In this case there is a very precise information; there is no need to speculate why the parent gave this piece of information instead of another one. Of course he/she might have said other things, for instance "I have a boy which is not born on Sunday", and you would obtain a different question, but this does not affect our problem.
Sep
7
comment Induction on Real Numbers
It seems to me that ii) is flawed. You could have $[0, x] = [0, y)$ (think of the case of natural numbers where y = x+1). Hence in your argument you cannot deduce that $y \notin S$.
Aug
26
comment How many ways to define sine and cosine?
Proving their properties from this definition (especially the periodicity) is a wonderful exercise in a first course on (qualitative study of) differential equations.
Aug
22
comment Good books and lecture notes about category theory.
Are you implying that usually books are not written by experts? :-?
Aug
20
comment Why is compactness in logic called compactness?
Why the downvote?!? This is the same answer as Akhil, even if in a different language. I used as elementary language as possible since I did not know whether the reader knew about propositional calculus. I would like very much if the downvoter could explain what is wrong in this answer.
Aug
12
comment Bugs walking in a plane
This reminds me of Hamilton's flow... :-)
Aug
12
comment Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?
In this case the topic was perfectly clear to me, and basic topology has close to no prerequisites. Yet I did not see the bug in the proof. And I'm in good company, as you can see from the link to the NForum. This is the reason I am now less confident in mathematical proofs, even in the case where it is easy for me to check all the details.
Aug
12
comment Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?
My point is that I was not confused about basic topology! I could easily teach that without reviewing anything. Indeed I was completely convinced that I had a proof of this implication. Not that this was one thing I actually cared about: it was something I (mistakenly) knew to be true and was certain I could provide a proof in 5 minutes if needed. Yet, I did not spot the mistake. It is obvious that one can fail to see bugs in proofs in topics about which he is confused. Also one can fail to spot bugs in topics where one has a clear understanding, but which involve a lot of prerequisites (cont)
Aug
11
comment Toy sheaf cohomology computation
As many have other people have pointed out, there are other techniques for computation than the definition. If you want to stick to the definition, you are in bad luck: you have to write explicitly injective sheaves, and these will be quite bigger than the sheaves which appear naturally. Maybe in a finite topological space you can actually do such a computation.
Aug
11
comment A curious compactness confusion: space filling curves in the hilbert cube that contradict a bona fide theorem?
You also may want to recall Tychonoff theorem, stating that the product of compact spaces is compact; this is easy to remember and surely clears that what fails is your argument is that the Hilbert cube is compact. By the way, if you know that the Hilbert cube is the image of an interval, there is no need appeal to Hahn-Mazurkiewicz theorem, since trivially the image of a compact space has to be compact.
Aug
8
comment Sets. Classes. …?
For what it's worth it does not have "stability condition" either...
Aug
8
comment Sets. Classes. …?
Is it a new criterion for deciding what exists in mathematics? :-D
Aug
7
comment (undergraduate) Algebraic Geometry Textbook Recomendations
By the way "presupposes only some commutative algebra" seems no little requirement to me. David Eisenbud has written a textbook explicitly designed to give the background to study Hartshorne, and the result is that Eisenbud's book is thicker than Hartshorne's. You will not get through Hartshorne with Atiyah-MacDonald.
Aug
7
comment (undergraduate) Algebraic Geometry Textbook Recomendations
It depends on your background. In Italy it is much more common to have a basic understanding of differential geometry than commutative algebra. It seems to me that the prerequisites of GH are just some differential geometry, complex analysis in one variable and some topologu. The rest is developed in Chapter 0, including more complex variables.
Aug
7
comment Dividing a disk into 7 equal pieces with 3 line segments
I don't see how this answer says more that the original poster "I can see with some visual arguments that the answer should be no".
Aug
7
comment Meaning of closed points of a scheme
Just to be precise: F is not a scheme, or better, it has different scheme structures which are not determined by F alone. Anyway, what matters for the argument is that it has one, so one can choose for instance the reduced scheme structure.
Aug
6
comment Sets. Classes. …?
It is actually possible to talk of things larger than classes, as I explain in my answer.