A friend of mine asked me why $\aleph_0$ is the smallest infinity and if we must use axiom of choice to say that it is the smallest infinity.   Do you know if Axiom of Choice is neccesary?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| = \aleph_i$, then (assuming continuum hypothesis) the cardinality of $2^{\Omega}$ is $|2^{\Omega}| = \aleph_{i+1} &gt; \aleph_i$ and we have $\aleph_{i+1} = 2^{\aleph_i}$.  I am fine with these argument.  What I don't understand is why should there be a smallest $\infty$? Is there a proof (or) an axiom that there exists a smallest $\infty$ and that this is what we address as "countably infinite"? or to rephrase the question "why can't I find a set whose power set gives us $\mathbb{N}$"?  The reason why I am asking this question is in "some sense" $\aleph_i = \log_2(\aleph_{i+1})$. I do not completely understand why this process should stop when $\aleph_i = \aleph_0$.  (Though coming to think about it I can feel that if I take an infinite set with $\log_2 \aleph_0$ elements I can still put it in one-to-one correspondence with the Natural number set. So Is $\log_2 \aleph_0 = \aleph_0$ (or) am I just confused? If $n \rightarrow \infty$, then $2^n \rightarrow \infty$ faster while $\log (n) \rightarrow \infty$ slower and $\log (\log (n)) \rightarrow \infty$ even slower and $\log(\log(\log (n))) \rightarrow \infty$ even "more" slower and so on).  Is there a clean (and relatively elementary) way to explain this to me?  (I am totally fine if you direct me to some paper (or) webpage. I tried googling but could not find an answer to my question)
So a couple of weeks ago my network manager icon stopped appearing; I still get automatically connected to my saved networks and the internet works fine, etc. If I go into System Settings and look at the Network settings I can still see all the networks, only the icon that's normal in the top bar is missing.  I have no idea what caused this, I don't remember installing anything. I think it just happened after a regular update.  I tried these commands to make sure I was properly up-to-date, (which I found on another forum post):  sudo apt-get -f install sudo apt-get update sudo apt-get upgrade   That didn't do it so I found this post:  and I followed the instructions in those answers:  sudo apt-get purge network-manager-gnome &amp;&amp; \ sudo apt-get install network-manager-gnome   And, quoted from an answer there:     Add nm-applet to the Startup Applications (which may be found by   entering "Startup Applications" in the Dash).   I restarted and then saw no Unity interface at all! So I tried those 3 commands above to update again and thankfully that fixed Unity but I still don't have my network manager.  If I try the command nm-applet in a terminal it says:  nm-applet-Message: using fallback from indicator to GtkStatusIcon   So.... I'm pretty much stumped. Any help would be appreciated. Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$ Assuming that $\gcd(a,b) = 1$, prove that $\gcd(a+b,a^2+b^2) = 1$ or $2$.   I tried this problem and ended up with $$d\mid 2a^2,\quad d\mid 2b^2$$ where $d = \gcd(a+b,a^2+b^2)$, but then I am stuck; by these two conclusions how can I conclude $d=1$ or $2$?  And also is there any other way of proving this result? How do I get the number of days between two dates in JavaScript? How do I get the number of days between two dates in JavaScript? For example, given two dates in input boxes:  &lt;input id="first" value="1/1/2000"/&gt; &lt;input id="second" value="1/1/2001"/&gt;  &lt;script&gt;   alert(datediff("day", first, second)); // what goes here? &lt;/script&gt; Biliniear form to inner product Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V.  If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each $v,u\in V$, $f(u,v) = (Tu, v)$ ?  Would that sill be the case if $F=C$? Automotive electronic circuit design, Voltage divider to uC ADC pin Automotive Circuit requirement:   If input pin is connected to Battery(T87 line), the output of the circuit shall provide a output voltage of 5V to ADC pin of microcontroller, if the input pin is connected to GND, the output of the circuit shall provide a output voltage of 0V to ADC pin of microcontroller and if the input pin is left floating, the output of the circuit shall provide a output voltage of 2.5V to ADC pin of microcontroller.   The battery voltage in load dump condition could be up to 60V, but in normal operation without any fault condition, it would be up to 32V.  Would a simple voltage divider (with a pull-up to 5V) with a blocking diode be enough? Any other ideas? What parameters of the ADC of the microcontroller should be considered? Injection current into the uC- ADC pin would be critical, right? How do I find out the voltage thresholds? Also, what are the parameters of the Schottky diode that I should look to select? The diode would decouple the input pin from the centre tap of the voltage divider, because in case the input pin is connected to battery, the injection current into the uC ADC pin would be too high.  I have used a switching diode, since it was already available in the Bill of Materials. I have a question about editing practices:  Does it look good to edit a question you have just answered?  For example: say I had just answered a question, then saw that the question had some minor formatting issues. Should I edit it? Does it turn up voters off when they see that I am answering a question I changed, possibly to better suit my answer? How to compute the marginal likelihood given two multivariate Normal distribution? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot! Word caption style -- different style for label and number? This should be a common problem for anyone involved in technical writing.  When you insert an image (or table etc.) caption in Word, you get  dialog. You enter your caption,  "A happy bicycle" (in yellow).  This might be fine for a happy bicycle, but I want to have captions that have one style for the "enumeration" (label and number) and one style for the description (caption text, as called by Word), as such:    I have tried searching for this but I haven't found anything yet... generic function with a "has property X" constraint? How to model non-negative zero-inflated continuous data? I'm currently trying to apply a linear model (family = gaussian) to an indicator of biodiversity that cannot take values lower than zero, is zero-inflated and is continuous. Values range from 0 to a little over 0.25. As a consequence, there is quite an obvious pattern in the residuals of the model that I haven't managed to get rid of:   Does anyone have any ideas on how to solve this? factorization of a^n+1? $(a^{2^n}+1,a^{2^m}+1)=1 or 2$ Prove that if $m\not =n,a$ are positive integers then $(a^{2^n}+1,a^{2^m}+1)$ is $1$ if $a$ is even and $2$ if $a$ is odd.    I solve the problem in the following way: I assume that $m&gt;n$ then  $$a^{2^m}+1=a^{2^m}-1+2$$ With this expression if I take a divisor prime $p$ of $a^{2^n}+1$ then if $p|a^{2^m}+1$ this prime must divide 2 and the result follows.   But, how can I prove this statement without use of primes? This exercise appears before the concept of prime is introduced then I prefer do not use it.    Any idea?   Thanks! What made us think that Earth moves around the Sun? So far, I know the matrices $$S=\left( \begin{matrix} 0&amp;-1\\ 1&amp;0 \end{matrix} \right), \quad T=\left( \begin{matrix} 1&amp;1\\ 0&amp;1 \end{matrix} \right) $$ generate SL$(2,\mathbb{Z})$. $S^2=(ST)^3=-I$. I also know the matrices $$X=\left( \begin{matrix} 1&amp;1\\ 1&amp;2 \end{matrix} \right), \quad Y=\left( \begin{matrix} 2&amp;1\\ 1&amp;1 \end{matrix} \right) $$ with their inverses $X^{-1},Y^{-1}$ generate SL$(2,\mathbb{Z})'$, the commutator subgroup. I'm interested in what the quotient $SL(2,\mathbb{Z})/SL(2,\mathbb{Z})'$ is. In general, how would I go about computing the quotient of two matrix groups? I'm trying to use GAP but I'm having a lot of trouble! I'm trying to safely update the home directory as specified in /etc/passwd, but the standard Linux utils - usermod and vipw - for doing so aren't provided by Cygwin.  Could anyone tell me how they changed this in Cygwin? Is reusing old code for a new assignment considered self plagiarism? How to protect yourself if you consider it to be, and a group partner does not? Is the consciousness currently in oneself the same the the consciousness that used to be or in the future will be in oneself? People seem to believe that one's consciousness is a "different" consciousness than that of others, but the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future. How likely is this to be true and why?   Also, I have almost no knowledge about neuroscience, so please refrain from using neuroscience jargon if feasible. Chrome and IE 11 are fine, I'm only able to reproduce this in Firefox. Using version 27.0.1 on Windows 7. Meta as well as all the other sites are affected.        I can still see the hover effect, but the image is gone. I see this in Firebug:        The image doesn't show up in the Network tab. I am able to see the image by visiting  which goes to   I've done Ctrl-F5, I've cleared the cache and I've tried Private Browsing. Possibly related to  ? Is the double-slit experiment typically carried out in a vacuum? It is my understanding of the double-slit experiment that when a measuring device is activated, to measure which of the two slits a particle is travelling through, this measuring is responsible for a different outcome than when the device is left off. When the device is left off particles leave an interference pattern, suggesting a single particle (or its wave) interferes with itself; when the device is activated no interference pattern is left, and particles are observed to travel through a single slit.  Now, does all this imply that this experiment is typically carried out in a vacuum? If not, how is it that interaction with other particles in the environment of the experiment's set-up are not already influencing the outcome, regardless of whether the measuring device is on or left off?  This question may very well be a very naive one, due to my very limited understanding of particle interaction, vacuums and/or physics in general. If so, I would appreciate a brief, layman's, overview of where my understanding of the different subjects at play here is likely to be lacking. Short story about aliens that are parts of a spaceship