I'm talking about topics such as model theory, proof theory, predicate calculus, Godel's incompleteness, ordinal numbers, large cardinals, etc. How much of these do I need to know if I want to specialize in some other areas of mathematics such as algebra and topology?
I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good to learn such advanced topics, I may not have enough time to do so, since I will probably put my emphasis on algebra, analysis or geometry as well as mathematical physics.   Although how much set theory and logic I should learn is totally up to what sort of analysis, algebra or geometry I will study, I want to know how much of them typical competitive algebraists/analysts/geometers know. If you are one of them, how much do you know?
How to get status start_date and end_date from status_track table? I have two tables:   things:  +------+----------------+------------+ | id   | current_status | created_at | +------+----------------+------------+ | 7770 | active         | 2016-08-09 | +------+----------------+------------+  thing_status_tract:  +----------+-------------+-----------+---------------------+ | thing_id | status_from | status_to | changed_at          | +----------+-------------+-----------+---------------------+ |     7770 | incomplete  | inactive  | 2016-08-09 16:26:22 | |     7770 | inactive    | active    | 2016-08-10 12:31:04 | +----------+-------------+-----------+---------------------+    I need data in the following form. This table has a status and its corresponding start and end timestamp for a particular thing_id:  +----------+-------------+---------------------+---------------------+ | thing_id | status      | status_start_date   | status_end_date     | +----------+-------------+---------------------+---------------------+ |     7770 | incomplete  | 2016-08-09 00:00:00 | 2016-08-09 16:26:22 | |     7770 | inactive    | 2016-08-09 16:26:22 | 2016-08-10 12:31:04 | |     7770 | active      | 2016-08-10 12:31:04 | now()               | +----------+-------------+---------------------+---------------------+   How to do that with a SQL query? How to distinguish gift from payment for the service? If somebody gives me reasonable advise or share important to me information.   Being attentive I would like to make a little gift or give some little (&lt;100$) amount of money (cash/wire/online) for that   How it would be classified by IRS?   Which obligations originate with this matter for me and my party?  How to do this in a most comfortable way to my party? How to set up beamer to grey out past items but hide future items? I know about beamer's  feature \setbeamercovered{invisible} and \setbeamercovered{transparent} to hide or grey out upcoming item's . But is there some way to have past items greyed (transparent) and future items hidden (invisible), so that only the current item is black/highlighted? And the best would be if on or 1 slide after the last all the item's could be visible highlighted. Something like this:   [grey] past 1 [grey] past 2 current [hidden] future 1 [hidden] future 2 [hidden] future 3   Is there any easy way of doing this? How to restore file copy / transfer status window? I think some good words would be conflicted, or confused. As an example of context, think of someone who believes in freedom of speech, but, at the same time, they would support banning comedians who make jokes about certain issues that person cares about. Volume of the torus with an integral Recursively delete empty directories in Windows Over the years I've written some libraries. These libraries sometimes depend on each other.  As a result, the "tex" directory is structured as follows:  tex ├── library |   ├── library1.sty |   ├── library2.sty |   └── library3.sty ├── project1 |   ├── master.tex |   ├── chapter1.tex |   └── chapter2.tex ├── project2 |   └── article.tex └── project3     └── paper.tex   The libraries depend on each other. As a result, in every library that depends on another library, this is part of the header:  %library2.sty \ProvidesPackage{library2} \RequirePackage{../library/library1}   In other words, the library is imported from the "root of every project". This is not a good design choice. What if someone copies the libraries in the folder of the root of a project, or what if the folder "library" is renamed...  It is however reasonable the "active directory" still remains in the project root.  What can be done to resolve such library dependencies? How can I get barcode-scan information into my photo metadata, onsite, untethered? I need a DSLR camera with a barcode scanner for on-site tours photography, which means it needs to be untethered! The image would have to contain the barcode scanned information in the image metadata or in the name itself.  I've seen solutions like, Foolography and a $2600 Canon Studio Edition launched in 2010. But I'm not sure of either those products!  So I'm asking the gurus here how I can get this working, which camera would I need and any accessory necessary?  I'm trying to recreate something like I saw the on-site volume photographers do in Disney's parks. Every finite set contains its supremum: proof improvement. Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance. connectedness of uniform topology Every natural number can be written as the sum of distinct Fibonacci numbers? I'm on 10.9, and I had been using internet sharing successfully again for the first time since Lion. Today though, it just decided to stop working again.  Network layout:  Modem -ethernet-&gt; Router -ethernet-&gt; Mac Pro -ethernet-&gt; MacBook Pro Retina   The internet works fine on the Mac Pro, but it is no longer forwarding the proper DHCP settings to the MacBook.  Previously, the MacBook would get the IP address of 192.168.2.12, and was able to surf the web fine. (Though it wasn't able to connect to any AirPlay/AirPrint devices while on the ethernet interface)  Sometime today, a setting got messed up, and I'm not sure how to fix it or what setting it is.  The MacBook now has a "Self-assigned" IP of 169.254.121.24, even though it is set to use DHCP. The ethernet port on the Mac Pro that the MacBook Pro is connected to is showing a "Self-assigned" IP of 169.254.36.53, even though it is set up to use DHCP as well.  The internet sharing options appear to be correct: share connection from Ethernet 1 to Ethernet 2.  But alas, for no apparent reason, I'm not getting internet access. I am happy to always not understand how does a transformer works ... One answer give me 10 questions ... (So I suggest you to not answer to this question, if you do not want to have more questions :D)  Here is the circuit :     So I put the "coupling" factor to 1, in order to have no leakage inductance and I set the value of the inductors very high in order to have a very low magnetizing current. So the primary current is approximately equal to the secondary current. My question is the following : How the primary current can reach an equilibrium to be equal to the secondary load current ? I will not accept answer which tells Pout is equal to ... So Pin is equal to Pout... There is something happenning into the transformer ...   Thank you very much and have a nice day ! What uses are there for "placement new"? How to prove the convergence of $\sum_{n\geq 0}\frac{n!^d}{(dn)!}$ for $d\geq 2$? How can we find the general formula about the convergence of the serie $\sum_{n=0}^\infty \frac{(n!)^d}{(dn)!}$ for all $d\geq2$ ?  I tried using the d'Alembert Criteria but it doesn't simplify itself enough to prove the convergence. Arguments for/against same birthday for siblings? My son is born on October 1st three years ago. My wife is pregnant again and the calculated delivery date is October 22nd. Both children are boys, 3 years apart. We might end in a situation where the delivery is earlier, for reasons irrelevant to this question.   I believe that around their birthdays is the only time in the whole year that the actual date matters. It really does boil down to practical considerations around patience (gifts) and planning. I think that if there is, say, a week or more between the birthdays then it's easy to separate but if the difference is only a very few days then it gets complicated, and that's the situation I'd like to avoid.  I know from friends that having the same birthday makes events easier to plan, from the parents' perspective. It would also remove the unhappiness of a small child saying "why does he get presents today and I don't." Being a twin, and married to one, I know that there's nothing problematic in sharing a birthday as long as both get their fair share of attention.    What are practical arguments for, and against, choosing the same birthday for the second child? What considerations apply when the second child's birthday is only a very few days after the elder sibling's? Best way to save a ordered List to the Database while keeping the ordering I was wondering if anyone has a good solution to a problem I've encountered numerous times during the last years.  I have a shopping cart and my customer explicitly requests that it's order is significant. So I need to persist the order to the DB.  The obvious way would be to simply insert some OrderField where I would assign the number 0 to N and sort it that way.  But doing so would make reordering harder and I somehow feel that this solution is kinda fragile and will come back at me some day.  (I use C# 3,5 with NHibernate and SQL Server 2005)  Thank you A Question On Euler's Proof Of the Basel Problem I've studied the proof that Euler gave for the famous , and it would seem that while it is technically correct, he does not justify all of his steps properly. Namely, he assumes that   $$\frac{\sin(x)}{x}=\left(1-\frac{x}{\pi}\right)\left(1+\frac{x}{\pi}\right)\left(1-\frac{x}{2\pi}\right)\left(1+\frac{x}{2\pi}\right)\dots$$  simply because they have the same roots, which is really not a strong enough condition. How do you really show that the equality holds?  He then notices that if you use the above equality, and consider it against the Taylor expansion for $\frac{\sin(x)}{x}$, then you can equate the coefficients of the two infinite expansions at each order, and the result of the Basel problem follows. But how do you know that if you have two different expansions for a function, then their coefficients at each order must be equal?  I would really appreciate if someone could show me how to make these two intuitive, yet informal, steps rigorous. how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$? Let $f(x) = \frac4\pi \cdot (\sin x + \frac13 \sin (3x) + \frac15 \sin (5x) + \dots)$. If for $x=\frac\pi2$, we have $$f(x) = \frac{4}{\pi} ( 1 - \frac13 +\frac15 - \frac17 + \dots) = 1$$ then obviously :  $$ 1 - \frac13 +\frac15 - \frac17 + \dots=\frac{\pi}{4}$$ Now how can we prove that: $$\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + \dots$$