Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ This is a response to the following exchange at    [Bill constructs a $\aleph_0$ linearly independent set in $\Bbb R$]  Can you extend the idea of this proof to get the right dimension? Currently, you only have a countably infinite independent set, but the dimension is size continuum. –   I believe the answer to this question is no, for the following reason. Suppose we have a spanning set for $\Bbb R$ over $\Bbb Q$, of cardinality $\frak n$. Then by definition we have the cardinality equation ${\frak c}\le^*[{\frak n}]^{&lt;\omega}\cdot \aleph_0$ (where $[{\frak n}]^{&lt;\omega}$ is the cardinality of finite subsets, and $\frak m\le^*n$ means there is a surjection from $\frak n$ to $\frak m$). If $\frak n$ is finite, then we get ${\frak c}\le^*\aleph_0$ which is false, so $\frak n$ is infinite (but not necessarily Dedekind-infinite). If we assume the axiom of choice, then the equation reduces to $\frak c\le n$, hence every Hamel basis is size continuum. But without AC I don't know any way to simplify the original equation to get anything stronger than &quot;$\frak n$ is infinite&quot;. This is at best heuristic evidence for the original question, since it is considering a different problem (spanning sets instead of linearly independent sets). How could one show that there are no ZF-constructible $\Bbb Q$-linearly independent subsets of $\Bbb R$ of cardinality continuum?
Is there any uncountably infinite set that does not generate the reals? Does there exist an uncountably infinite set $X \subseteq \mathbb R$ such that $\mathbb R \neq \left&lt;X\right&gt;$? I can't think of any, but I'm also having trouble trying to prove that no such subset exists.  For example: $\mathbb R$ is uncountable and obviously $\mathbb R = \left&lt;\mathbb R\right&gt;$. The Cantor set $C$ is uncountable, and we know that $C - C = [0, 1]$, so then since $\mathbb R = \left&lt;[0, 1]\right&gt;$ we know that $C$ also generates $\mathbb R$. Also the set of irrationals $\mathbb R \setminus \mathbb Q$ is uncountable, but we can generate all the rational numbers by fixing one irrational number $\alpha$ and then saying the any rational number $x$ shall be $(\alpha + x) - \alpha$, since both $\alpha + x$ and $\alpha$ are irrational.  So the examples that quickly come to mind all generate the reals. Is there a simple counterexample?
List changes unexpectedly after assignment. Why is this and how to prevent it? Can a ring of positive characteristic have infinite number of elements? For curiosity: can a ring of positive characteristic ever have infinite number of distinct elements? (For example, in $\mathbb{Z}/7\mathbb{Z}$, there are really only seven elements.) We know that any field/ring of characterisitc zero must have infinite elements, but I am not sure what happens above. Number of permutations of $n$ elements where no number $i$ is in position $i$ I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5.  I know that the number of total permutations is $n!$. Is there a formula for how many are acceptable given the case that no position holds its own number? A person that professes discrimination based on race is a racist.  What is the word for defining a person that discriminates based on culture (pretty sure it is not culturist) or social habits?  E.g., Eric Cartman doesn't like hippies, he discriminates against them. But a hippie is not a race, an ethnic, a religion or a nationality. It is a life style not related to any of these. I stupidly ran sudo chown -R carlos:carlos / and carlos is my user. This changed all files to be Carlos and I could not boot up again (because the root files were now owned by #1000) and I cant get sudo acess in live cd (because the owner of /usr was changed to my user). Can anyone help me? FYI I have mounted my hard drive. I have a JavaScript widget which provides standard extension points. One of them is the beforecreate function. It should return false to prevent an item from being created.   I've added an Ajax call into this function using jQuery:  beforecreate: function (node, targetNode, type, to) {   jQuery.get('http://example.com/catalog/create/' + targetNode.id + '?name=' + encode(to.inp[0].value),    function (result) {     if (result.isOk == false)          alert(result.message);   }); }   But I want to prevent my widget from creating the item, so I should return false in the mother-function, not in the callback. Is there a way to perform a synchronous AJAX request using jQuery or any other in-browser API? How to find application's path from command line? For example, I have git installed on my system. But I don't remember where I installed it, so which command is fit to find this out? Does WaitForMultipleObjects() reset all auto-reset events? Which Standard Visitor visa should I apply for? I am using XeTeX and have the following short document:  \documentclass[a4paper,12pt]{article} \usepackage{amsmath,amsthm} \usepackage[MnSymbol]{mathspec} \usepackage[no-sscript]{xltxtra} \defaultfontfeatures{Mapping=tex-text} \setallmainfonts{Minion Pro}  \begin{document}  $\phi, \varphi, \delta \ldots A \vee B$  \end{document}    Now no matter what font I am using the punctuation marks in math mode (commas, dots, \ldots etc.) come out in Computer Modern. Is there a way to change this? How to keep application running after closing the terminal? PWM complementary square output waveforms PWM channels in certain microcontrollers such as the SAM3X8E have two complementary square output waveforms, PWMH and PWML, per channel. What is the purpose of having two waves that are the inverses of each other? what could cause a script to fail to find python when it has `#!/usr/bin/env python` in the first line? Trying to get casperjs running on Ubuntu 12.04. After installing it when I run I get:   09:20 $ ll /usr/local/bin/casperjs lrwxrwxrwx 1 root root 26 Nov  6 16:49 /usr/local/bin/casperjs -&gt; /opt/casperjs/bin/casperjs  09:20 $ /usr/bin/env python --version Python 2.7.3  09:20 $ cat /opt/casperjs/bin/casperjs | head -4  #!/usr/bin/env python  import os import sys  09:20 $ casperjs : No such file or directory  09: 22 $ python Python 2.7.3 (default, Sep 26 2013, 20:03:06)  [GCC 4.6.3] on linux2   So Python is present and runnable, casperjs is pointing to the right place and it is a python script. But when I run it I get "No such file".   I can fix it by changing the first line of the casperjs python file from:    #!/usr/bin/env python   to:   #!/usr/bin/python   Result:   $ casperjs --version 1.1.0-DEV   I managed to fix it, but I'm wondering why it didn't work with #!/usr/bin/env python, since that seems to be a normal interpreter line. Do I have something configured wrong?   Here are the steps to get casperjs:   $ git clone git://github.com/n1k0/casperjs.git $ cd casperjs $ ln -sf `pwd`/bin/casperjs /usr/local/bin/casperjs $ casperjs : No such file or directory Incorrectly answered 'No' to "Any previous visa refusals..." on a UK Standard Visitor visa application. How can I fix my mistake? Support showing image in original size in overlay The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ Clearly this is an edge case and I probably shouldn't have even tried awarding the bounty to a deleted answer, but why is the bounty award button even available?  It's not a bug since someone added in the notification seen in the final image.  But that sure seems like a roundabout way to let the user know that they can't award a bounty to a deleted answer.  The second step is particularly strange since the system really ought to know that the warning is pointless: no bounty can be awarded.  My suggestion is to rip-off the  and let bounties be awarded without the warning.  Then remove the single click button from deleted answers. Iteration over std::vector: unsigned vs signed index variable What is the correct way of iterating over a vector in C++?  Consider these two code fragments, this one works fine:  for (unsigned i=0; i &lt; polygon.size(); i++) {     sum += polygon[i]; }   and this one:  for (int i=0; i &lt; polygon.size(); i++) {     sum += polygon[i]; }   which generates warning: comparison between signed and unsigned integer expressions.  I'm new in the world of C++, so the unsigned variable looks a bit frightening to me and I know unsigned variables can be dangerous if not used correctly, so - is this correct? Why is /tmp a symlink to /private/tmp on Mac OS X? In other words, why isn't /tmp just a regular directory, like on Linux or BSD? I understand how it works and I don't mind it, I'm just interested in the (historical?) reasoning behind it. When —the bartender at the Cantina at Mos Eisley spaceport in the first Star Wars—says:     “We don’t serve their kind here.”   In reference to C-3PO and R2-D2 entering the bar, it’s certainly a bit prejudicial, but are these commonly held views in this universe? We don’t see it happen anywhere else in the films but is it implied elsewhere?