Show that there are exactly $\aleph_1$ nonisomorphic countable well orderings. I used the continuum hypothesis to show that $2^{\aleph_0} = \aleph_1$ and then i applied $|\mathbb{R}| = 2^{\aleph_0}$ to get $|\mathbb{R}| = \aleph_1$. Then $\aleph_1$ is equal to the cardinality of all real numbers. As $\aleph_1$ is equal to $\omega_1$ and $\omega_1$ is a well ordered set, so is $\aleph_1$. Then I assumed that $\aleph_1$ is a countable well ordering with the cardinality $|\mathbb{R}|$. My teacher told me that I shall not use the continuum hypothesis here, because it was not told to use it. I am not sure if I have done anything correct, any help would be appreciated, thanks!
Let be the next equivalence relation on $B= \{R \subseteq \mathbb{N}\times\mathbb{N} \ | \ R \ is \ a \ well-order \ on \  \mathbb{N}\}$ : $$"R \equiv R' \leftrightarrow \ (\mathbb{N},R) \cong (\mathbb{N},R') \ are \ well-ordering \ sets \  isomorphic" $$  Which is the cardinality of $B/\equiv$?  Firstly, we can notice that each well-ordering set is isomorphic to an ordinal $\alpha \ $such that $\alpha &lt; \aleph_{1}$, and so it is enough to count how many ordinals are isomorphic to a subset of $\mathbb{N}$. For example, if we consider $A$ the set of the even numbers and $B$ the set of odd numbers with the usual order relation, $A \oplus B$ is isomorphic to $\omega + \omega$. Another example could be: if we consider the order of $\mathbb{N}$ that we use in Sharkowski Theorem, that order is isomorphic to $ \omega^{2} + \omega$. Working with prime numbers we could build also $\omega^{n} \ \forall n \in \mathbb{N}$, and so $\omega^{\omega}$ and maybe something bigger. However, we know that each countable ordinal is isomorphic to a subset of $\mathbb{Q}$, but I bet that the same result is not true for $\mathbb{N}$ because it is "too small".  What can I do?
I am looking for a way to get the output of a command when it is run from within a C++ program.  I have looked at using the system() function, but that will just execute a command. Here's an example of what I'm looking for:  std::string result = system("./some_command");   I need to run an arbitrary command and get its output. I've looked at , but I have not found anything that will give me what I need. I have already bought the PC version on minecraft for my account. However, when attempting to change my skin, the game simply states:"You have to buy the PC edition of Minecraft."   Which is odd because I already have.  Why is this happening? How can I fix it? Downloading all my questions with the answerer names I am writing several papers in which I use answers which I received in various StackExchange websites. Naturally I would like to acknowlege and thank the SE members that gave the useful answers. But there are many such answers, and each answer is relevant to a different paper.   So, I would like to download all the questions that I asked in the various SE websites, including the answerer names, to an Excel worksheet in which I can organize the questions by subject.  Is this possible? Pie Chart without labels in beamer using pgf-pie I would like to install  chat platform on my Ubuntu 12.04 LTS system. Please, could anyone provide me the steps to successfully install this software?. I've been trying to install this software using wine but I couldn't succeed it. Does anyone know if there is an API to get the current monitor state (on or off) in Windows (XP/Vista/2000/2003)?  All of my searches seem to indicate there is no real way of doing this.    tries to use  which according to Microsoft's docs does not work for display devices.   In Vista I can listen to  but I do not seem to get events when the monitor is turned off manually.   In XP I can hook into WM_SYSCOMMAND SC_MONITORPOWER, looking for status 2. This only works for situations where the system triggers the power off.   The WMI Win32_DesktopMonitor class does not seem to help out as well.   Edit: Here is  on comp.os.ms-windows.programmer.win32 indicating there is no reliable way of doing this.   Anyone else have any other ideas? Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. 32-bit, 64-bit CPU op-mode on Linux I'm confused. Running Fedora Linux, lscpu yields:  Architecture:            i686 CPU op-mode(s):          32-bit, 64-bit ...   But when I try to install a 64-bit program (Chrome) I get error like:     Package /....x86_64.rpm has incompatible architecture x86_64. Valid architectures are ['i686', 'i586', 'i486', i386']   I'm less interested in being able to install Chrome and more interested in why lscpu says that my CPU can run in 64-bit mode; clearly this can't mean I can run 64-bit programs. Can anyone clarify? In Excel, the window can be frozen in place in such a way that you can always see the name of the column even though you go way down below the 'fold' of the screen.  On SE, when you scroll, the whole page goes up (or down). This means the top navigation bar is scrolled off screen. I'm not sure if it would be a good idea or not to fix the top navigation to the top of the screen and let the rest of the content scroll.  What do you think about this, in terms of pros and cons?  [Update]  With the new Top navigation bar, kudos to the design team, I think this question becomes relevant again. I'm aware of the busload of downvotes, but still.  The top navigation bar doesn't occupy much space and has two notification areas:   Inbox Achievements   Having these two notifications visible at all times would be nice, IMO.  [Update]  I've just noticed that I haven't used the word "optional" in this discussion... My bad. In Excel, the Freeze function is controlled by the user. You can optionally Freeze part of the interface, and you can eliminate the Freeze at will. Bought in, bought on, bought from, bought at? How can I find unanswered questions with zero answers in some tag? Can I travel to France with Fiktionsbescheinigung (Fiction Certificate) of Germany Unexpected behavior of rule matching a pattern How can I create my own Action using Photoshop? Can anyone guide me with a simple example? Where to get 2010 Census Block data? I would like to download 2010 Census Block Data for several towns and I can't seem to find it on American Factfinder.   Is it there? And if not, does anyone know where I could find it? amsmath align environment: row spacing How can I stretch the spacing between consecutive lines in an align* environment? I have a few lines with \fracs in which both the numerator and the denominator consist of textual phrases, not math symbols, and the result looks too cluttered.  \begin{align*} X &amp;= \frac{\textit{Some text here}}{\textit{And some text here}}\\ Y &amp;= \frac{\textit{Will make everything}}{\textit{Look too close together}} \end{align*} How was Dumbledore able to defeat Grindelwald, since Grindelwald had the Elder Wand? nginx location match issue auth_basic I have this configuration :  location / {     root /etc/nginx/euchet; }  location /app/ {     proxy_pass http://172.16.184.2:8080;     proxy_redirect off;     proxy_set_header Host $host;     proxy_set_header X-Real-IP $remote_addr;     proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;     proxy_set_header REMOTE_ADDR $remote_addr; }   And I need to add another location with auth_basic enabled only for the request: /app/f?p=3000  I tried with rexexp:     location ~ \/appt\/f\?p=3000 {             auth_basic "Hello! Enter password";             auth_basic_user_file /etc/nginx/euchet.htpasswd;              proxy_pass http://172.16.184.2:8080;             proxy_redirect off;              proxy_set_header Host $host;             proxy_set_header X-Real-IP $remote_addr;             proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;             proxy_set_header REMOTE_ADDR $remote_addr;     }   Nginx use previous location /app/.  I tried another way:  location ~* /euchet/f {      if ($arg_p = "4550") {         return 403;     }      error_page 403 = @hidden; }     location @hidden {             auth_basic "Hello! Enter password";             auth_basic_user_file /etc/nginx/euchet.htpasswd;              proxy_pass http://172.16.184.2:8080;             proxy_redirect off;              proxy_set_header Host $host;             proxy_set_header X-Real-IP $remote_addr;             proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;             proxy_set_header REMOTE_ADDR $remote_addr; }   And I get 403 error on f?p=3000, and 404 errors on f?p=&lt;another_number&gt;  How can I fix this problem? Example: I type man ls, than I want to get man only.   By using !! I can get man ls but how do I get man? If $a_{1},a_{2},a_{3},a_{4}\in \mathbb{R}$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$ and $a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4}=1.$ Then Range of $$E =(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ is  Try: From $$(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ $$=2(a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4})+2(a_{1}a_{2}+a_{2}a_{3}+a_{3}a_{4}+a_{4}a_{1})$$ $$=2-2(a_{1}+a_{3})(a_{2}+a_{4})=2+2(a_{1}+a_{3})^2\geq 2$$ and equality hold when $\displaystyle a_{1}=-a_{3}$ and $a_{2}=-a_{4}$ Could some help me how to find its upper bound, Thanks