Gödel's completeness theorem: Given a set of axioms, if we cannot derive a contradiction, then  the system of axioms must be consistent.  Gödel's incompleteness theorem:'Given any consistent, computable set of axioms, there's a true statement about the integers that can never be proved from those axioms'.  Since the incompleteness theorem assumes that the given set of axioms is consistent, then, by using the completeness theorem shouldn't all statements derived from this set of axioms be true thereby rendering the claim that 'there exists a true statement that cannot be proved' false?
What is the difference between Gödel's completeness and incompleteness theorems?
I was working on a problem that resulted in the calculation: $20 \equiv 10x \pmod{11}$.  I got the answer $x \equiv 2 \pmod{11}$ with the thought process: Since $10$ is a factor of $20$ I can rewrite $20 \pmod{11}$ as $10x \pmod{11}$ with $x=2$. But isn't this basically the same as using division, which we aren't supposed to do in modular spaces?  If I solve by multiplying $20$ by the multiplicative inverse of $10 \pmod{11}$ which is $10$, I get $200 \pmod{11}$ which simplifies to $2 \pmod{11}$ anyway. Did these numbers just happen to be the same or is my original method valid? how to sort file by DD-MM-YYYY dates in a file containing :  ... 18-11-2018:othercharacters 10-11-2018:othercharacters 03-10-2018:othercharacters 30-10-2018:othercharacters 27-09-2018:othercharacters 03-12-2018:othercharacters ...   the command :  sort -t- -k2 -k1    does not sort by date, what am I missing ? HTML text input allow only numeric input Is there a quick way to set an HTML text input (&lt;input type=text /&gt;) to only allow numeric keystrokes (plus '.')? Does borrowing from my 401(k) make sense in my specific circumstance? I know the standard answer and understand that I will be losing out on that compound interest, but I am young and only need to borrow a few thousand. here are some more details:   Need to borrow $6500 to pay off a car and two student loans ($300/month in payments combined) I am 22 and work full time, making about $70K annually I contribute 8% of my income with a 4% company match I increase that by 2% every year The loan would be over 48 months ($140/month) I live very frugally and have almost no discretionary spending that can be cut (don't eat out all the time, no gym membership/ Netflix/ Apple music etc.) I'm cash-poor: 90% of my salary is going to mortgage, groceries, students loans, car payment, newborn baby medical expenses, and commute. These are not my only debts, just the ones I am seeking to pay off in the short term. I work two jobs, one full and one part time. I started a business, and my wife has two jobs. Working more is not a good answer to the original question.   It seems to me that I would be reducing middle to high interest debt while also saving myself $150 per month.   Loans:     Student loan 1: $1200 @ 5%   Student loan 2: $1100 @ 4.25%   Car Loan: $4000 @ 6% Fallout Shelter :: Does Mr. Handy Pick Up Loot? I know that Mr. Handy will pick up caps when on walkabout; will he also pick up weapons and/or outfits? I see the following error while trying run the command shown below. I read somewhere that my /boot partition is low on disk space. How can I increase the size of the /boot partition so I can install more software? I have a 500GB hard disk, so there is enough space to play with.  sudo apt-get install libdvdread4  gzip: stdout: No space left on device                                                                                                                                                                                  E: mkinitramfs failure cpio 141 gzip 1                                                                                                                                                                                 update-initramfs: failed for /boot/initrd.img-3.2.0-33-generic with 1.                                                                                                                                                 run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1                                                                                                                                            Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-3.2.0-33-generic.postinst line 1010.                                                                                                        dpkg: error processing linux-image-3.2.0-33-generic (--configure):                                                                                                                                                      subprocess installed post-installation script returned error exit status 2                                                                                                                                            dpkg: dependency problems prevent configuration of linux-image-server:                                                                                                                                                 linux-image-server depends on linux-image-3.2.0-33-generic; however:                                                                                                                                                    Package linux-image-3.2.0-33-generic is not configured yet.    dpkg: error processing linux-image-server (--configure):     dependency problems - leaving unconfigured    dpkg: dependency problems prevent configuration of linux-server:     linux-server depends on linux-image-server (= 3.2.0.33.36); however:      Package linux-image-server is not configured yet.    dpkg: error processing linux-server (--configure):     dependency problems - leaving unconfigured    No apport report written because the error message indicates its a followup error from a   previous failure.                                                                                                           No apport report written because the error message indicates its a followup error from a previous failure.  Errors were encountered while processing:  linux-image-3.2.0-33-generic  linux-image-server  linux-server N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension   Listed below is the output of du  Filesystem              1K-blocks      Used Available Use% Mounted on /dev/mapper/ubuntu-root 712660664 104095912 572363692  16% / udev                      3964792         4   3964788   1% /dev tmpfs                     1591012      1064   1589948   1% /run none                         5120         0      5120   0% /run/lock none                      3977528       684   3976844   1% /run/shm /dev/sda1                  233191    219821       929 100% /boot Why didn't Sauron find Bilbo when he put on the ring like Frodo? I'm arriving to the US by air, but departing via land. How do I end form I-94 now that it is only electronic? Brute-force attacking One Time Pads feasible? I have a ciphertext encoded with One Time Pad. I do not know the key. How can I find the possible keys which when decoded, will result in a meaningful plain text? In unix/linux, any number of consecutive forwardslashes in a path is generally equivalent to a single forwardslash. eg.  $ cd /home/shum $ pwd /home/shum $ cd /home//shum $ pwd /home/shum $ cd /home///shum $ pwd /home/shum   Yet for some reason two forwardslashes at the beginning of an absolute path is treated specially. eg.  $ cd ////home $ pwd /home $ cd /// $ pwd / $ cd // $ pwd // $ cd home//shum $ pwd //home/shum   Any other number of consecutive forwardslashes anywhere else in a patch gets truncated, but two at the beginning will remain, even if you then navigate around the filesystem relative to it.  Why is this? Is there any difference between /... and //... ? Can a plan "age" reach zero? com.mysql.jdbc.exceptions.jdbc4.CommunicationsException: Communications link failure If $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$   then prove that $f(x)=0$ for all $x \in [a,b]$.  I tried with Riemann's definite integral definition but couldn't proceed Changing trains in Lille - what's the transfer with luggage between Lille Europe and Lille Flanders stations like? Lille has two main train stations, Lille Europe and Lille Flanders. (Gare de Marchandises Saint-Saveur is no longer a station, and instead hosts art exhibitions and the like). Lille Europe is the new station on the high speed lines between London / Paris / Brussels, while Lille Flanders is the original station now used for a mixture of local and high speed trains.  For some rail journeys, changing trains in Lille is the best option (avoiding having to cross Paris to change trains), and for others it's the only option. Sometimes you get lucky and both trains use the same station in Lille, but not all.  For the cases when you find yourself needing to change between Lille Flanders and Lille Europe stations, what's the connection like? Does it take long? Is it outside / inside? Is it a faff with luggage, or is there a step-free route for most of it? Are you best to walk, or take public transport? And a lower concern, is there anything interesting to see of the town during the transfer? What are metaclasses in Python? In Python, what are metaclasses and what do we use them for? This may be a very basic question.      Let $R$ be a commutative ring with unity (may be assumed to be an integral domain, if necessary) and $I$ be an ideal of $R$. Assume that $R/I$ is isomorphic to $R$. Does this mean $I=(0)$ ? Keyboard macro to change default audio device Windows 7, Logitech G110 keyboard  2 audio devices (headphones and speakers)  I am hoping to find a quick way to macro a key to switch to headphones, and a key to switch to speakers.  Anyone have a good program or anything of that nature?  EDIT:  Using AHK to try and get this task working as hoped is sadly not doing the trick.  I have included a screenshot of my audio set, as well as a screenshot of my script as it sits now.    I have also tried to uncheck "show disabled devices" and modified the script to be Down 1, and Up 1 respectively.  It will work for one device, but it refuses to switch to the secondary. Sum of digits of repunits While working in a Java app, I recently needed to assemble a comma-delimited list of values to pass to another web service without knowing how many elements there would be in advance. The best I could come up with off the top of my head was something like this:  public String appendWithDelimiter( String original, String addition, String delimiter ) {     if ( original.equals( "" ) ) {         return addition;     } else {         return original + delimiter + addition;     } }  String parameterString = ""; if ( condition ) parameterString = appendWithDelimiter( parameterString, "elementName", "," ); if ( anotherCondition ) parameterString = appendWithDelimiter( parameterString, "anotherElementName", "," );   I realize this isn't particularly efficient, since there are strings being created all over the place, but I was going for clarity more than optimization.  In Ruby, I can do something like this instead, which feels much more elegant:  parameterArray = []; parameterArray &lt;&lt; "elementName" if condition; parameterArray &lt;&lt; "anotherElementName" if anotherCondition; parameterString = parameterArray.join(",");   But since Java lacks a join command, I couldn't figure out anything equivalent.  So, what's the best way to do this in Java? What is the meaning of this senctence:Excellent answer. Delusions of grandeur. Folks have been institutionalised for less Can an atheist at least agree that existence is a mystery that is bigger than all of us?                                                       You’ve got it backwards. It’s religious people who try to fill the mysteries of the universe with beliefs based on nothing emprical. Where atheists say “I don’t know” religious people say “It’s God.” Atheists are comfortable with not knowing everything. They know humans are infinitesimal specks in the universe. Religious people think the entire universe was created for and about them. Talk about grandiosity. Comment:Delusions of grandeur. Folks have been institutionalised for less lol