Given a countable set on which a total ordering is defined, it is not always possible to list the full set in ascending order. The set of rationals is a well-known example.    But are there conditions under which such a listing is possible? Ideally, the conditions would be both necessary and sufficient conditions, but I suppose weak sufficient conditions will, eh, suffice. For instance, can the support of a discrete random variable on a probability space always be listed in ascending order?
This is problem #6 from Section 1.2 of Ash's Real Variables With Basic Metric Space Topology.   I am asked to show that it is impossible to list the rational numbers in increasing order.   While I know it is possible to list a finite subset of the rational numbers in increasing order, I was wondering if the reason for the impossibility in this case is because there is no least element of $\mathbb{Q}$?   I mean, it's possible to impose an ordering on $\mathbb{Q}$ (correct me if I'm wrong, but it seems possible to compare any two rationals).  But the set $\mathbb{Q}$ is of course, a subset of itself, and the Well-Ordering Principle says that a set $S$ is Well-Ordered only if any subset of $S$ contains a minimal element. Since $\mathbb{Q}$ does not contain a minimal element, it does not appear that it is a Well-Ordered set.   Is this the correct idea as to why it's impossible? If not, could somebody explain to me what the correct idea is and a strategy for proving it?  Thank you ahead of time.
Ownership of content in Community Wiki posts Why wouldn't the `which` command work for `cd`? I can't find the executable for `cd` either! I tried which cd and it didn't give a path but instead returned the exit code 1 (checked with echo $?). The coreutil cd itself is working, so the executable should be there, right? I also ran a find for cd, but there was no executable file shown. How is it implemented then?  Update:  I don't know if I should ask this in another post but since I think it is good here, I'm expanding (?) the post... So the answer was actually quite simple, there's no executable for that — because it's a builtin — But I've found some builtins (bash shell in Fedora) have the executable files! So builtin -> no executable isn't right I suppose? Maybe an answer explaining what builtins actually are (builtin commands?), which actually is the matter here, rather than focusing more on cd... Some good links posted previously indicate that builtins are not programs... so what are they? How do they work? Are they just functions or threads of the shell? When customers buy things from a store, you say:  (1) Customers give ABC Store business.  If I "turn this around", can I say:  (2) ABC Store obtains or acquires business from its customers.  Do these verbs work in (2)? Can "real" be used as an adverb to describe an adjective? I work at a nursery and I have a hard time managing the toys that the kids play with. I watch them put things in their mouths, cough and sneeze on the toys, and throw everything on the ground. Even though I see all of these things, it is hard to find the time to clean all of the toys, simply because there are too many and I do not have that much time. Any input would be appreciated! In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base? Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? I know from reading  that the "primary" use of the IDisposable interface is to clean up unmanaged resources.  To me, "unmanaged" means things like database connections, sockets, window handles, etc.  But, I've seen code where the Dispose() method is implemented to free managed resources, which seems redundant to me, since the garbage collector should take care of that for you.  For example:  public class MyCollection : IDisposable {     private List&lt;String&gt; _theList = new List&lt;String&gt;();     private Dictionary&lt;String, Point&gt; _theDict = new Dictionary&lt;String, Point&gt;();      // Die, clear it up! (free unmanaged resources)     public void Dispose()     {         _theList.clear();         _theDict.clear();         _theList = null;         _theDict = null;     }   My question is, does this make the garbage collector free memory used by MyCollection any faster than it normally would?  edit:  So far people have posted some good examples of using IDisposable to clean up unmanaged resources such as database connections and bitmaps.  But suppose that _theList in the above code contained a million strings, and you wanted to free that memory now, rather than waiting for the garbage collector.  Would the above code accomplish that? Blank screen after installing nvidia restricted driver $n$-th digit in the sequence of natural numbers SD card not showing files when connected to pc When my phone MOTOROLA G3(2015) is updated to Marshmallow, its not showing files when connected to PC. While connecting, it shows options as charing only, PTP, MTP, MIDI. When I select MTP, it shows the external card, but not showing the contents, where as internal memory is detected and all files are shown. I had Windows 7 and Ubuntu installed on the same partition   Some time later, decided to uninstall Ubuntu.So I booted Ubuntu,Changed the hd related to it to NTFS,I did NOT extended my main drive   restarted-> GRUB rescue&gt;. No surprise since I had just wiped out the whole GRUB directory staff   Now I just want to boot into my windows.Move my necessary files to an external memory and reinstall the windows seven     What I have in my hands: Ubuntu live CD, A copy of win7 DVD I tried to make use of the win7 DVD(I had lost my DVD) to fix the mbr but the problem is when I choose repair, It says that this version of Windows is incompatible with yours What should I do? Prime ideal in a polynomial ring over an integrally closed domain How do I install and manage GNOME Shell extensions? Proof Check: Every Cauchy Sequence is Bounded How do I check if an array includes a value in JavaScript? What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? Variables are often adjusted (e.g. standardised) before making a model - when is this a good idea, and when is it a bad one? In what circumstances would you want to, or not want to scale or standardize a variable prior to model fitting?  And what are the advantages / disadvantages of scaling a variable? Chapterwise page range in the TOC Can you give me some advice for how to obtain the following format of the TOC?        Chapter’s heading 1-7   Chapter’s heading 9-10   Chapter’s heading 11-15 I install virtualBox on Ubuntu 16.04, but after install and create new virtual Host shows this error:     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or there is a permission problem with /dev/vboxdrv. Please install virtualbox-dkms package and load the kernel module by executing  modprobe vboxdrv       as root. If it is available in your distribution, you should install the     DKMS package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary.  where: suplibOsInit what: 3 VERR_VM_DRIVER_NOT_INSTALLED (-1908) - The support driver is not installed. On linux, open returned ENOENT.     How to address this modprobe vboxdrv error? How can I make hibernate my desktop in UEFI with Secure boot on.  I used command "sudo pm-hibernate" to make hibernate my desktop but my machine did nothing (my desktop is booted up on UEFI with Secure boot on). But, if I boot my desktop on UEFI with Secure boot off, Hibernation is normal.  Sorry for my English, thanks,