Prove that a theory $\Gamma$ is consistent if and only if there is a structure $M$ so that $M$ $\models$ $\Gamma$. $\Gamma$ is consistent, then either $\Gamma \models \varphi$ or $\Gamma \models \neg \varphi$ (but not both) for all sentences $\varphi$.  Can we assume $M$ is a structure such that $M \models \varphi$ for all $\varphi \in \Gamma$? I think this is incorrect because if so, $M \models \Gamma$ will be true trivially. We are assuming what we want to prove. If this is wrong, what else can I try?   On the other side, from $M \models \Gamma$, we have $M \models \varphi$ for every $\varphi \in \Gamma$.  Then can we say $M \not\models \neg\varphi$? And how can we finish the prove with $\neg (M \models \varphi \land \neg\varphi)$?
If $\Gamma$ is satisfiable then $\Gamma$ is consistent We say that $\Gamma$ is consistent if for no $\phi$, then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$. We say that $\Gamma$ is satisfiable if there is a structure $M$ s.t. $\models_M\gamma[s]$ for alle $\gamma\in\Gamma$.  I want to show that if $\Gamma$ is satisfiable, then $\Gamma$ is consistent.  I'm thinking we assume that $\Gamma$ is not consistent, and then show that $\Gamma$ is not satisfiable. So if $\Gamma$ is not consistent then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$ for some $\phi$. But by soundness theorem then $\Gamma\models\neg\phi$ and $\Gamma\models\phi$, meaning for structure $M$, we have $\models_M\neg\phi[s]$ (whereas $\not\models_M\phi[s]$) and $\models_M\phi[s]$. So would this mean that $\Gamma$ is not satisfiable? I'm a little confused, because wouldn't this instead be a contradiction?
Integral representation of Euler's constant Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$  where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).    This integral was mentioned in  as in  , but the solutions I've got uses corollaries from . Can you give me a simple solution (not using much advanced theorems) or at least some hints. How is well-ordering in real line possible?  I know that the axiom of choice provides possible well-ordering, but intuitively, this does not seem to make sense.  How can you compare 1.111111.... and 1.111111... when we can never know the full digits of each number? Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands? It has been said that RSA uses a modulus product of two prime numbers for security reasons. But does RSA even work correctly if we allow composite integers instead?  I think that the answer is "NO".  Define “work correctly” as:   allows encryption and decryption to be carried out as in normal RSA, yielding the original message (message padding and unpadding can remain unspecified, or absent as in texbook RSA); a security level can be given in terms of the key generation process (which may have alternate requirements to testing that $p$ and $q$ are prime). Can you fake a closed question? [‌closed] I just had a random thought; what if you append "[closed]" to the end of a question title. Is it filtered out, or in some way disallowed?  If you are seeing this, it isn't. It could potentially be used for some trickery, right? Or maybe not. I don't know, that's why you're here: discuss!  EDIT: Just a recap: Juan Manuel edited and removed "[closed]" from the end of the title and then answered (see below) - but it is technically possible.  Edit 2: Looks like it's still possible, if you play silly tricks with Unicode. (There's a  character in the new title.) Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd.   My attempt: If $a$ is even, let $a = 2^{s}k$ for some integers $k, s$ Then, $$a^{2^{n}} + 1 = 2^{2^{n}s}\cdot k^{2^n} + 1$$ and $$a^{2^{m}}  + 1 = 2^{2^{m}s}\cdot k^{2^m} + 1$$ To prove that they're relatively prime, we need to show that their gcd = 1. And I was stuck here, how could I prove that gcd of two numbers is $1$?     A hint would be sufficient. Thanks. What is the correct way to install proprietary ATI Catalyst Video Drivers (fglrx) directly from AMD? incoming mail just sits in the drop folder Trigger a php file on every post or page if a condition is met I was wondering if we can trigger a PHP file on every WordPress post or page if a certain condition is met. Let me detail what I want.   Can we write a simple PHP script that will be automatically loaded on every WP post and page. Once loaded, the script should scan the HTML and if it finds a specific DIV tag with the ID myID it should trigger another PHP file (let's say script.php). Once the target PHP file has been triggered or executed, the target PHP file (script.php) should be able to automatically fetch the URL of the post or page that triggered the file and store it in a variable.   Any ideas?? Which merchants pay best? I noticed at some point that different merchants don't all pay the same for what you're selling. Depending on the category, or even the specific item, some merchants simply pay significantly more.  Are there some general rules on which merchants pay better for which items? My website is javatechig.com and is around 6 months old. I have used All-in-One SEO WordPress plugin for SEO tricks. Definitely after using SEO plugin, my website page view is increased, but still it is not giving one good site-link in Google Search.   I have followed the following steps:   Created sitemap.xml and submitted to Google Webmaster Tools Edited all posts with meta tags Header tags are carefully taken care   I want to make a sitelink similar to the attached image: How to reverse name and first name? With biblatex and verbose-trad2 style, I would like to reverse the order of the last name and the first name for inline references.  The default behavior is:     John DOE, Title, place : publisher, year   for inline references (in my case, in a footnote with footcite command) and:     DOE, John, Title, place : publisher, year   in the bibliography at the end of the document.  I would like to have both inline and final references in the following format (similar to the bibiography style).      DOE, John, Title, place : publisher, year   P-S : There are some close questions (like ) but only about the final bibliography (and no inline references). \DeclareNameAlias{sortname}{last-first} works only for the ending references (not inline ones). Can someone show me a straightforward proof that  $\lim_{m\to\infty}(1+\frac{r}{m})^{mt}=e^{rt}$  Thanks! How can I tell if a DOM element is visible in the current viewport? Is there an efficient way to tell if a DOM element (in an HTML document) is currently visible (appears in the viewport)?  (The question refers to Firefox.) Why is the change in Hamiltonian for an active infinitesimal canonical transformation defined the way it is? I'm trying to understand infinitesimal canonical transformations and conservation theorems (section 9.6 Goldstein ed3). My specific problem is with understanding eq 9.104,         $\partial H = H(B) - K(A^{'}) $, where $\partial H$ represents the change in the hamiltonian under an infinitesimal canonical transformation (from the active view point ofc). The issue only arises for generating functions that explicitly depend on time.  Here's what I've thought of so far. I understand what $K$ is from the passive viewpoint (or so I think), its the hamiltonian that would furnish Hamilton's equations for the transformed variables $Q=q+ \delta q$ and $P = p + \delta p$. In the specific case of the generating function being a time dependent hamiltonian, $K = H + \delta t \frac{\partial H}{\partial t} = H + \delta t\dot H$, which makes sense because once you transform the variables to what they'll be in time $\delta t$, you'll also need to add on a term that comes from the Hamiltonian being at a later time $t+ \delta t$.   In the book, this is explained as a transformation depending on time changes the hamiltonian, so $H(A) \rightarrow K(A^{'}) $. Which I sorta get. But why not for $H(B)$? And what about the case when the Hamiltonian is time dependent and it is the generating function? Is $H(B)$ at a later time then?   Note: $A$ represents the original variables, $B$ is the transformed variables in the active sense, so in the same phase space as $A$, and $A^{'}$ is the same point as A but in the transformed phase space (passive). Sending the output from 'dd' to awk/sed/grep I'm trying to approximate the computer's write speed using dd:  dd if=/dev/urandom of=/dev/null bs=1K count=10000   which gives the following output  10000+0 records in 10000+0 records out 10240000 bytes (10 MB) copied, 0.746313 s, 13.7 MB/s   How can I get the '13.7 MB/s' into a bash variable? I've tried piping the output from dd to progs like awk, sed and grep to no avail.  Ultimately, I'm calling this via os.system(...) from within a python script. If anyone knows of a more direct way to get a similar result inside python I'd be interested in that also. I'm trying to predict how long a file copy will take, based on the file size.  Any help would be greatly appreciated. How to change css property using javascript My computer boots to a black screen, what options do I have to fix it? Is "funnest" a word? We seem to be stuck at an impasse on this issue.  Is funnest a word or not?  If so, does it mean "most fun"? How to prove that a number is irrational