The $\aleph$ hierarchy, I think, is a sequence of infinite cardinals. $\aleph_0$ is $|\mathbb{N}|$, possibly by definition. $\aleph_1$ is the next highest cardinal.  A lot has been written about whether $\aleph_1 = |\mathbb{R}|$ or, equivalently $\aleph_1 = \beth_1$ and how the continuum hypothesis is independent of ZFC.  But there definitely isn't a cardinal $\kappa$ such that $\aleph_0 &lt; \kappa &lt; \aleph_1$.  Why do we know that there's a successor cardinal to $\aleph_0$ / least cardinal greater than $\aleph_0$?    What would break if we insist, by fiat, on the existence of a hierarchy of cardinals indexed by the non-negative reals or non-negative rationals (in the way that $\aleph_n$ is indexed by $n \in \mathbb{N_0}$)? If cardinals aren't primitive things and the existence of a successor falls right out of how they're defined, this question might not make sense.
From what I understood, cardinal numbers are defined as:  $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$  The continuum hypothesis states that the cardinality of $\mathbb{R}$ is  $\aleph_1$. I more or less understand this cannot be proven (does not follow from ZFC).  However, can it be proven dat there in fact is a 'least cardinal number' greater than $\aleph_0$?  Is it possible that there are sets with cardinality $\aleph_{1/2}$, $\aleph_{1/3}$, et cetera, so to speak?  Or expressed differently: is it possible that for every set A with cardinality CA > $\aleph_0$, there exists another set B with cardinality CB where CA > CB > $\aleph_0$?
Anyone seen this one yet? Vote too old to be changed, unless post is edited My vote was only a few hours old. Then I realized the answer was wrong. Updated my question and went to remove the vote. Even after updating question I still have the issue. Small issue, but frustrating.  See also: Motivation for Ramanujan's mysterious $\pi$ formula The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$   Does anyone know how it works, or what the motivation for it is? This should be a very easy question but it might just be that I'm confusing myself. So we have the definition of a function $f$ on $S$ being continuous at $x_0$:     For any $\epsilon$>0, there exists $\delta&gt;0$ such that: whenever $|x-x_0|&lt;\delta$, we have $|f(x)-f(x_0)|&lt;\epsilon$   And I assume the negation is     There exists $\epsilon$>0 such that for all $\delta&gt;0$, $|x-x_0|&lt;\delta$ yet $|f(x)-f(x_0)|\ge \epsilon$.   Now I want to show that the function $f(x)=\sin(\frac{1}{x})$ together with $f(0)=0$ cannot be made into a continuous function at $x=0$. So I need to show that there exists $\epsilon&gt;0$ such that for all $\delta&gt;0$, $|x|&lt;\delta$ yet $|f(x)|\ge\epsilon$.  Let $\epsilon = \frac{1}{2}$. Then no matter what $\delta$ we choose, let $|x|&lt;\frac{1}{2}$. It is certainly possible that $|f(x)|\ge \frac{1}{2}$, because, well, $\frac{1}{x}$ can really take on arbitrarily large value as $x$ is small.  Now, what confuses me is that, as $x$ gets small, $f(x)$ can certainly be greater than $\frac{1}{2}$ for infinitely many times, but it will be less than that infinitely many times, too. But I suppose it doesn't really matter. So I think there's something wrong with my negation but I couldn't figure out where.  Update: The correct version can be found . Watch for Lemma 4.6 Is there any performance reason to declare method parameters final in Java?  As in:  public void foo(int bar) { ... }   Versus:  public void foo(final int bar) { ... }   Assuming that bar is only read and never modified in foo(). I install virtualBox on Ubuntu 16.04, but after install and create new virtual Host shows this error:     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or there is a permission problem with /dev/vboxdrv. Please install virtualbox-dkms package and load the kernel module by executing  modprobe vboxdrv       as root. If it is available in your distribution, you should install the     DKMS package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary.  where: suplibOsInit what: 3 VERR_VM_DRIVER_NOT_INSTALLED (-1908) - The support driver is not installed. On linux, open returned ENOENT.     How to address this modprobe vboxdrv error? I'm trying to debug a Python script by placing a print("foo=" + foo) statement in my code.  But where does the output appear?     It doesn't appear on the Python Console, and it doesn't appear on the area that appears when you drag the lower edge of the infobar downwards (what is the name for this area? My script errors show up in red here.) Why does the standard error of the intercept increase the further $\bar x$ is from 0? The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^2}\right]$$ where $\bar{x}$ is the mean of the $x_i$'s.  From what I understand, the SE quantifies your uncertainty- for instance, in 95% of the samples, the interval $[\hat{\beta}_0-2SE,\hat{\beta}_0+2SE]$ will contain the true $\beta_0$. I fail to understand how the SE, a measure of uncertainty, increases with $\bar{x}$. If I simply shift my data, so that $\bar{x}=0$, my uncertainty goes down? That seems unreasonable.  An analogous interpretation is - in the uncentered version of my data, $\hat{\beta}_0$ corresponds to my prediction at $x=0$, while in the centered data, $\hat{\beta}_0$ corresponds to my prediction at $x=\bar{x}$. So does this then mean that my uncertainty about my prediction at $x=0$ is greater than my uncertainty about my prediction at $x=\bar{x}$? That seems unreasonable too, the error $\epsilon$ has the same variance for all values of $x$, so my uncertainty in my predicted values should be the same for all $x$.  There are gaps in my understanding I'm sure. Could somebody help me understand what's going on? Pass C# ASP.NET array to Javascript array Here's my problem  I'm trying to do a cleanup of my hard drive, and I'm constantly seeing this scenario:  (1) Aggregate folder size: I check the folder size viewing properties right clicking the folder  (2) Individual folder/file sizes: I check the individual file/folder sizes belonging to that folder viewing properties on the individual file/folder sizes, and then adding them up (with hidden and system files shown).   These two figures are always different, sometimes by a large margin.   For instance, my C drive now, by method (1), shows a consumed space of 42.4 GB.   However, by method (2), by inspecting the respective files/folders, the total consumed space adds up only to 33.1 GB. This is a 28% discrepancy (over the smaller file size) and I have no idea where the rest of the space is going.   My hard drive is constantly in danger of being full (it's bootcamp on a mac with 50 GB of partitioned space), and there's not much I've got installed in terms of software, nor do I have large media files like movies, downloads, temp files, application temp files, torrents, etc.   Most of it's just consumed by the files in C:/Windows, but last I checked, I don't know enough to know which files to delete, and I've heard that you really can't do anything about the space boggling winsxs and system32.   Can you guys recommend a starting point to free up disk space?  Thanks!!  Baggio. Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? 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Chat is our place for:        Real-time collaboration   Meeting fellow members of the community   Less structured, casual (but still roughly on-topic) conversation      Our chat system exists to serve the main site; users of each site can gather and commune with each other.  When in chat, we expect all users to abide by our  policy, which means that we expect community members to treat each other with respect.    While it is a place where the community can go to interact on a more casual level, chat still needs to be moderated by the community and there are tools in place to help with the moderation. This raises a few questions, that are addressed here: I'm not sure how password hashing works (will be implementing it later), but need to create database schema now.  I'm thinking of limiting passwords to 4-20 characters, but as I understand after encrypting hash string will be of different length.  So, how to store these passwords in the database? 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