I searched all over the internet but didn't find a formal proof for this , so here is my attempt:  $\exists x[P(x)\implies \forall yP(y)]$  Let $x=x_0$. Thus $P(x_0)$ is given.  Let $y$ be arbitrary. So we have to prove $P(y)$.  $y$ being arbitarary means that either $y=x_0$ or $y\ne x_0$.  So now we prove $P(x_0) \lor P(z)$ where $z$ is arbitrary and $z\ne x_0$.  $P(x_0)$ is given. So the paradox is true. $\square$  Is this correct ? If not what is wrong ?
Why is this true?  $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
Tools to get a pictorial function call graph of code Is it possible to have a rate of change of acceleration? I know this may seem a weird question, but it always bothers me. My physics book (Resnick,Halliday,Walker), and also various sites never say anything beyond acceleration.   But when a moving body is being acted by a variable force , its acceleration will definitely change: it will either increase or decrease. Then there will be rate of change of acceleration with respect to time. So, why don't books mention this? What is the cause for not measuring $\frac{d\vec{a}}{dt}$ ? If it exists, what is the use of it? Single quote within double quotes and the Bash reference manual In section 3.1.2.3  titled Double Quotes, the Bash manual says:      Enclosing characters in double quotes (‘"’) preserves the literal   value of all characters within the quotes, with the exception of ‘$’,   ‘`’, ‘\’, and, when history expansion is enabled, ‘!’.   At the moment I am concerned with the single quote(').  It's special meaning, described in the preceding section, section 3.1.2.2 is:     Enclosing characters in single quotes (') preserves the literal   value of each character within the quotes. A single quote may not   occur between single quotes, even when preceded by a backslash.   Combining the two expositions,    echo "'$a'"   where variable a is not defined (hence $a = null string), should print $a on the screen, as '', having it's special meaning inside, would shield $ from the special interpretation. Instead, it prints ''. Why so? Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module? Let     $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$   be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that $ \mathbb{Z}^{'}_{\mathbb{C}} $ is not finitely generated as a $ \mathbb{Z} $-module.  My attempt: Suppose $ \mathbb{Z}^{'}_{\mathbb{C}}=\mathbb{Z}\alpha_{1}+...+\mathbb{Z}\alpha_{n} $ for some $ \{ \alpha_{i}\}_{i=\overline{1,n}} \subset \mathbb{Z}^{'}_{\mathbb{C}} $. I know that given $ \alpha , \beta \in \mathbb{Z}^{'}_{\mathbb{C}} $ with minimal polynomials of degrees $ m $ and $ n $ respectively, then we can construct a monic polynomial with integral coefficients $ P(z)=\prod(z-(\alpha_{i}+\beta_{j})) $ where $ \alpha_{i},\beta_{j} $ are the conjugates of $\alpha $ , $\beta $ and therefore the minimal polynomial of $ \alpha + \beta $ has degree $ \leq mn $.   I was thinking that then the degrees of the minimal polynomials of the elements of $ \mathbb{Z}\alpha_{1}+...\mathbb{Z}\alpha_{n} $ could be somehow bounded by a certain constant, whereas $ \mathbb{Z}^{'}_{\mathbb{C}}$ contains elements whose degrees of the minimal polynomials are arbitrarily large, take $ 2^{\frac{1}{n}} $ for example,$ n\geq 2$. I think this works when we are asked to prove that the set of all algebraic numbers $ \overline{\mathbb{Q}} $ is not a finitely dimensional $\mathbb{Q}$-vector space, but I don't know how to proceed in my case.  I would really appreciate any ideas, hints or solutions. Thank you very much! I'm looking for the title of a short story by Ursula Le Guin about a scientist (?) who is given a drug that slows the timescale of his perception enormously, so that he can perceive that an alien (?) forest is actually behaving as a sentient being, reacting to current events, but at an enormously slow (from a normal human point of view) timescale. It might be in , but I don't remember the title (and it doesn't jump out at me from the list of titles) and don't have access to that collection right now. It is not &quot;The Word for World is Forest&quot;, nor &quot;Direction of the Road&quot; (from ), nor &quot;Vaster than Empires and More Slow&quot;, the latter two of which look at similar plant perception/time-scale issues. Update: it turns out that the description was right, but I had most of the details wrong: not Le Guin, not alien, not a scientist ... How do I set up SSH so I don't have to type my password? How do I set up SSH so I don't have to type my password when connecting to a host? Is the sign of principal components meaningless? Does the sign of the vectors returned by PCA have any significance after I center my variables? What does {0} mean when found in a string in C#? In a dictionary like this:  Dictionary&lt;string, string&gt; openWith = new Dictionary&lt;string, string&gt;();  openWith.Add("txt", "notepad.exe"); openWith.Add("bmp", "paint.exe"); openWith.Add("dib", "paint.exe"); openWith.Add("rtf", "wordpad.exe");  Console.WriteLine("For key = \"rtf\", value = {0}.", openWith["rtf"]);   The output is:     For Key = "rtf" value = wordpad.exe   What does the {0} mean? Who in the DC Universe can stop Superman if he has a breakdown? Is there anything or anyone in the DC universe that can defeat Superman just in case he goes berserk on everybody?  Maybe Batman stores kryptonite in his Cave or something. I don't know if Batman actually has kyptonite, the example was just a theoretical example.  Who can stop Superman if needed, and how? “between” vs. “among” Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? What are Harry, Ginny, Ron and Hermione's professions? After the end of the Second Wizarding War Harry Potter later married Ginny Weasley, and the couple had three children of their own: two sons James Sirius, Albus Severus, and one daughter Lily Luna. What are Harry and Ginny's professions? I heard that Harry became an Auror, but what about Ginny, Ron, and Hermione? Can you negate the Frightened condition by not looking at the source of your fear? As the title says, if you cause a creature to be Frightened of you, as in the Frightened condition, but they can somehow manage to not see you, do they stop rolling with disadvantage to attacks/ability checks?  Below are scenarios to illustrate my question.  Scenarios:   If a creature is frightened of you, and they turn their back against you to hit someone else, would they no longer have disadvantage to attack rolls/ability checks? If a creature is frightened of you and they have Blindsight, can they close their eyes to stop seeing you, and hence stop rolling with disadvantage? If a creature has 60 ft of movement and is frightened of you, and there is a corner they can reach in 15 ft that will block you from sight, can they run to that corner to stop being frightened, then run back to attack without disadvantage? Assume they would not provoke OA to do this, although the OA is immaterial to the question. Changing pemalink structure to /%post_id%/%postname%/ I'm currently using permalink structure: /%year%/%monthnum%/%day%/%postname%/  I want to change that to: /%post_id%/%postname%/  I expect it to be:   faster (database is sorted by post_id?) shorter URLs (and clearly visible short verstions, just leaving the %postname%/ out) more SEO friendly - since my content is "evergreen" so publishing date is irrelevant (and written at each post, along with last update date) - so this would give shorter URLs, with no downsides compared to using year/month/day structure.   THE QUESTION: Does WordPress handle all the 301 redirects automatically? I can fix internal links to the new structure (though I'd expect that to be automatic as well), but worry about outsite links, pointing to my posts/articles.  If not, what's the best way to implement this change? Any links or full answers here are welcome.  I have about 120+ posts already, will this change affect the speed since it's not been done from the start? Change post permalink to external URL from custom field How to remove snap completely without losing the Chromium browser? What is the remainder when $24^{1202}$ is divided by $1446$? I tried remainder theorem but that does not simplify it.   I tried factorizing $1446$ as $2\cdot3\cdot241$ and got remainders when numerator is divided by $2,3$ and $241$ individually but then I did not know what to do next? Stuck! Why does a lightsaber turn off when a Jedi dies or drops it? I rewatched the three newer movies and noticed that whenever a Jedi was killed, or let go of their lightsaber, it immediately turned off.  What was curious was that the lightsaber turns off right away, when a Jedi is killed, even before it leaves their hand (if it does fall out).  I could understand if the button needs to be held for it to stay on, but when a Jedi is shot and is still holding the lightsaber why would it turn off? Mapping of zero vector under a linear transformation Let $T$ be a linear transformation from a vector space $V$ to $W$ that are defined over a field $\mathbb F$.   Now, $T(\mathbf0)=\mathbf0$.   Which implies the zero vector in $V$ can't move or be transformed to a new vector in $W$. Why is it so?   Does it also have a geometrical significance? What's that ship in the opening credits of 'Star Trek: Enterprise'?