It is written on Wikipedia:  In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers.  Suppose that CH is not true, that is that there is such a set, call it $C_1$, so, $|\mathbb Z|&lt;|C_1|&lt;|\mathbb R|$ But, does some principle &quot;forbids&quot; us to suppose that there are also sets $C_2$ and $C_3$ so that we have $|\mathbb Z|&lt;|C_2|&lt;|C_1|&lt;|C_3|&lt;|\mathbb R|$, and $4$ more sets with cardinalities between those $5$, and so on and so on...  That is, if CH is not true, can there be an infinitely countable number of sets $S_i$ between $\mathbb Z$ and $\mathbb R$ so that we have $|\mathbb Z|&lt;| S_1|&lt;...&lt;|S_{\infty}|&lt;|\mathbb R|$. And, if there cannot be an infinitely countable number of sets between $\mathbb Z$ and $\mathbb R$, all with different cardinalities, can there be a finite number of them (a finite number greater than $1$)?
Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably many? Uncountably many? What are the possibilities to the quantity of counterexamples?
Use HERE file and redirect output of command I have the following code in a batch script:  mpirun -np 6 ./laplace &lt;&lt;END 100 100 100 0.01 100 3 2 1 END | tail -n 1 &gt; output   But it isn't working. What I want it to do is to use the HERE file as input to the mpirun command, and then pipe the output to the tail command. However, I think the HERE file and tail output things are getting confused.  How should I write this so that it does what I want? Principle of Reflection on atomic level This well-observed phenomenon has, besides several others, always been a fascination to me. We are well aware of several theories, experiments, and practical applications of this well-known phenomenon, but is it established what goes on at the grassroots of reflection?  My high school teacher once told me that whenever light is incident on any reflecting surface, its electrons absorb the energy of the photon and release back the same energy. If this is actually true, then I have plenty of questions to continue, but if it is not, then what is reflection, and how and why does it take place? Will I be able to use my NTFS data partition in Ubuntu? My machine is currently running Windows 7. I have 2 partitions: OS (C:) and DATA (D:). I intend to do a native Ubuntu 14.04 install. Assuming I format my Windows OS partition, will I have full access to my D partition in Ubuntu ? Limits, find a limit that exist in absolute value but not outside the absolute value. Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist.   Not really sure of an example for this? Problem with option border of class standalone I tested a MWE given in question . Following the documentation (page 9) I used the class option border={left right bottom top} with left=right=top=0pt and bottom=20pt (20pt only to show the effect here).  With the version 2012/09/15 v1.1b of standalone and the following MWE I get no 20 pt margin at the bottom of my picture.  What is the problem?  MWE:  \documentclass[% % border=1pt   border={0pt 0pt 20pt 0pt} %left right bottom top ]{standalone}% http://ctan.org/pkg/standalone \begin{document} $2^5 = x_5 \times y^8$ \end{document} I read this book a few years ago which was part of a series, but I only found the first book and now I've forgotten the name.  The first book featured 3 children/teens from different places that end up on a quest to search for magic gems/stones together and the story takes place in a medieval kind fantasy world. One is a girl, while the other 2 are boys. One of the boys actually finds a stone/gem while he is on a school field trip (I think the gem/stone is red in colour), and when he picks it up, he gets transported to the fantasy land but ends up in some sort of military training ground/camp somewhere in a desert type area.  He isn't familiar with the fantasy world. The other 2 characters each start out on their own on a quest to collect some magical gems/stones which once collected, can be used to fight an evil ruler (I guess).  But halfway through the book, the girl and boy meet up and continue the quest together while the third boy(transported one) doesn't know that the stone he picked up is actually one of the magical stones others are looking for.  A few other details I remember:   The stone the boy finds is red in colour I think  The cover of the book had a border depicting the stones  It's part of a series or trilogy    Some books with similar storylines, but not the one I'm looking for:     Deltora Quest The Seven Towers Sword of Heaven series Westeria series Diadem series Proof that in a field $x=0$ is equivalent to $x=-x$ Why are imported PDFs blurry in InDesign? How to make overlay still work inside lstlisting environment? Problem  I want to explain C# programming language step by step using overlay in beamer.cls. But it does NOT works as shown in the following figure.  How to solve this problem?    Code Snippet  \documentclass[dvipsnames,cmyk]{beamer}  \usepackage{listings}  \begin{document}  \defverbatim[colored]\Lst{% \begin{lstlisting}[language={[Sharp]C}] using System; public delegate void Foo(object o); \uncover&lt;1&gt;{public class Foo} \uncover&lt;2&gt;{\{} \uncover&lt;3&gt;{public static void Main()} \end{lstlisting}}  \begin{frame}{MyListing} \Lst \end{frame}  \end{document} Is it possible for a system to have negative potential energy? With respect to both gravity and electromagnetism, to the best of my understanding potential energy is added or subtracted from a system based on the distance between two objects such as charged or massive objects.  Would this then imply that when oppositely charged objects or two massive objects are brought together within a small enough radius that the drop in potential energy would be greater than the combined mass of the two objects making the net mass/energy of the system negative? I recently installed Ubuntu Server edition 13.10 (Saucy Salamander). But I have run into a bit of problem. At the end of the installation, it asks about several extra packages whether you want them or not, like OpenSSH server and virtual machine host. Here's what I'm looking at:  Now, my problem is that I can't remember if I have checked the &quot;virtual machine host&quot; checkbox. How do I check if my server has the corresponding packages? What packages come with the virtual machine host? Is there any reference where Captain America's shield is made from stolen Wakanda vibranium? How do I set read/write permissions my hard drives? Why does a helium filled balloon move forward in a car when the car is accelerating? Can I prepay a Credit Card to make a Large Purchase greater than my credit limit? I want to make a large purchase at a retail store to the sum of $10,000 but the credit limit on my card is set at $2000.  If I send $10K to my credit card will the purchase be allowed to go through?  Not sure if the credit limit is a hard limit or not?  I don't want to be left embarrassed if my card is declined in the store lol.  I called my bank but the answer they gave wasn't too convincing. It was kind of "Errr yes i think its OK!"  Does any one know first hand? If banks offer a fixed rate lower than the variable rate, is that an indication interest rates may head down? This is one of the major banks in Australia:   At present time, the variable rate is 7.80% but the fixed rate is 6.99% for one year.  And it seems most banks in Australia, if you fix the rate (at the moment), you will get a lower rate.  Is this an indication that in the eyes of the banks, that they believe the interest rates are going to down? I just ask because it seems most people in the media think the rates are going up and they've risen about 6 times in a row now.  Could it be that the banks are secretly preparing for rates to drop? So a company decides to raise money by selling shares of its stock. The stock is offered at an initial price of their determination. The stock is then traded in a market where forces of supply and demand, as well as speculation, affect its price.  When a company performs well (or makes decisions that will likely increase the company's earnings in the future), more people want to buy that stock than do people wanting to sell it. This creates demand for that stock which causes its sell/bid prices rise. When a company performs poorly, the opposite occurs; more people want to sell the stock than want to buy. This creates a surplus that causes sell/bid prices to fall.  Understanding this, people speculate over company performance to determine the future behavior of a stock, and attempt to "buy low, sell high" or "borrow high, return low" in order to make a profit.  But my question is regarding the first part of that process. What is it about company performance that causes the perceived value of its stock to rise? This is especially perplexing since I would guess that most people don't exercise their voting rights, and many stocks don't pay dividends.  It's not like a shareholder has any direct financial gains from increased company earnings (speculative stock pricing notwithstanding).  What is the intrinsic value of the stock?  It seems to me that all participants in a stock market tacitly agree that stock price will have a direct correlation with company performance. So another way of asking the question is, would an opposite agreement hold? What financially (this isn't a question about logistics, communication, or feasibility) stops all participants in a market from agreeing to value a particular company's stock inversely with respect to the company's performance?  Since I feel like I can't quite nail down my question, another form of it would be, why is stock market behavior driven by company performance? Overstaying Schengen Visa for 1 day I was in Germany last September, my visa said I had 14 days to stay in the Schengen area.  When I was leaving back for Egypt the passport control officer noticed that I was leaving on the 15th day. I didn't even realise that as I had written previously in my visa application that I was staying for 15 days and the embassy in Cairo had my complete airline ticket with specific dates.  Anyway the officer just filed a report on the incident but didn't stamp my passport with my exit date from the Schengen area.  Do any of you guys know if that would make it harder for me to be granted a visa in the future? What's the best way to animate an illustration for the web? I have a couple of illustrations done in Illustrator and I am planing to animate it for a website that I am working on, I've heard of  toolkit with flash, but is it the best way to go or is there another "better" way of doing it?  Here's an example of the kind of animations I am aiming for: How do I split a string so I can access item x? Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"?