All explanations I found on the axiom of choice seem to say more or less the following: Although the axiom of choice is &quot;intuitively obvious,&quot; it is necessary because making infinitely many choices is not &quot;built into&quot; the first order logic. I'm not so familiar with the first order logic, but which part of the following causes problem with the first order logic?  Let $X_1,X_2,X_3,\ldots$ be a sequence of nonempty sets. For each $X_i$, there exists an element $x_i\in X_i$. Let $f$ be a function on the set $\{1,2,3,\cdots\}$ defined by $f(i)=x_i$.  All statements look &quot;first order&quot; to me.
In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises.
What program can I use to decompile a class file? Will I actually get Java code, or is it just JVM assembly code?  On Java performance questions on this site I often see responses from people who have "decompiled" the Java class file to see how the compiler optimizes certain things. How to completely delete and app from iphone (including all the data and history) So this problem particularly emerged with the Tinder app. I was using Tinder with a user account (A) and one day I logged out and created a new user account (B) which I never succeeded to activate due to sms verification problems, so I decided to login again with my old account. But even when I delete the Tinder app and reinstall it again, it still shows the B account logged in to the app. This means that IOS is somewhere storing a history for this app. Now I'm looking for a way to delete this history.  Note that this question is not necessarily about a certain app, but rather about where IOS stores historical data of apps and how to delete those histories alongside with the app.  P.S: If there are solutions like changing UUID of my i-phone that wouldn't damage my phone (I'm new to iphone so I'm not sure what I'm saying. For instance this jailbreaking thing which I keep hearing about and I'm not sure what it exactly does to my phone, doesn't sound like an I idea that I like), I'm open to them. How find this limit $\lim\limits_{x\to 0^{+}}\frac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$ Volume of the torus with an integral Calculating days of week given a week number Given a week number, e.g. date -u +%W, how do you calculate the days in that week starting from Monday?  Example rfc-3339 output for week 40:  2008-10-06 2008-10-07 2008-10-08 2008-10-09 2008-10-10 2008-10-11 2008-10-12 Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.) geometric interpretation of $\vec v \cdot \operatorname{curl} \vec v = 0 $ There is a family of surfaces orthogonal to the vector field $\vec v \in \mathbb R^3$ iff $\vec v \cdot  \operatorname{curl} \vec v = 0 $. Now the necessity part is trivial, but the proof of sufficiency I have seen in physics textbooks, e.g. Kestin: Thermodynamics, is kind of mindless integration similar to the one usually offered to prove Poincare's lemma as in Flanders, or just asserted as in Born&amp;Wolf: Optics. Is there an intuitive and geometric interpretation of this condition that would make it obvious why its sufficiency must be true? How to calculate: $\sum_{n=1}^{\infty} n a^n$ I've tried to calculate this sum:  $$\sum_{n=1}^{\infty} n a^n$$  The point of this is to try to work out the "mean" term in an exponentially decaying average.  I've done the following:  $$\text{let }x = \sum_{n=1}^{\infty} n a^n$$ $$x = a + a \sum_{n=1}^{\infty} (n+1) a^n$$ $$x = a + a (\sum_{n=1}^{\infty} n a^n + \sum_{n=1}^{\infty} a^n)$$ $$x = a + a (x + \sum_{n=1}^{\infty} a^n)$$ $$x = a + ax + a\sum_{n=1}^{\infty} a^n$$ $$(1-a)x = a + a\sum_{n=1}^{\infty} a^n$$  Lets try to work out the $\sum_{n=1}^{\infty} a^n$ part:  $$let y = \sum_{n=1}^{\infty} a^n$$ $$y = a + a \sum_{n=1}^{\infty} a^n$$ $$y = a + ay$$ $$y - ay = a$$ $$y(1-a) = a$$ $$y = a/(1-a)$$  Substitute y back in:  $$(1-a)x = a + a*(a/(1-a))$$ $$(1-a)^2 x = a(1-a) + a^2$$ $$(1-a)^2 x = a - a^2 + a^2$$ $$(1-a)^2 x = a$$ $$x = a/(1-a)^2$$  Is this right, and if so is there a shorter way?  Edit:  To actually calculate the "mean" term of a exponential moving average we need to keep in mind that terms are weighted at the level of $(1-a)$. i.e. for $a=1$ there is no decay, for $a=0$ only the most recent term counts.  So the above result we need to multiply by $(1-a)$ to get the result:  Exponential moving average "mean term" = $a/(1-a)$  This gives the results, for $a=0$, the mean term is the "0th term" (none other are used) whereas for $a=0.5$ the mean term is the "1st term" (i.e. after the current term). Taking the second derivative of a parametric curve I understand that for the parametric equations  $$\begin{align*}x&amp;=f(t)\\   y&amp;=g(t)\end{align*}$$  If $F(x)$ is the function with parameter removed then $\displaystyle F&#39;(x) = \frac{\text{d}y}{\text{d}t}\big/\frac{\text{d}x}{\text{d}t}$  But the procedure for taking the second derivative is just described as " replace $y$ with dy/dx " to get   $$\frac{\text{d}^2y}{\text{d}x^2}=\frac{\text{d}}{\text{d}x}\left(\frac{\text{d}y}{\text{d}x}\right)=\frac{\left[\frac{\text{d}}{\text{d}t}\left(\frac{\text{d}y}{\text{d}t}\right)\right]}{\left(\frac{\text{d}x}{\text{d}t}\right)}$$  I don't understand the justification for this step. Not at all.   But that's all my book says on the matter then it launches in to plugging things in to this formula, and it seems to work well enough, but I don't know why.  I often find answers about question on differentials are beyond my level, I'd really like to get this, it'd mean a lot to me if someone could break it down. What does "cpu MHz" field mean in the /proc/cpuinfo of a hyper-threading cpu? The CPU is a i7-3770@3.40GHz. It has 4 cores and each core has 2 threads. Here is the dmidecode output:  # dmidecode -t 4 # dmidecode 2.9 SMBIOS 2.7 present.  Handle 0x0042, DMI type 4, 42 bytes Processor Information     Socket Designation: SOCKET 0     Type: Central Processor     Family: &lt;OUT OF SPEC&gt;     Manufacturer: Intel(R) Corporation     ID: A9 06 03 00 FF FB EB BF     Version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz     Voltage: 1.1 V     External Clock: 100 MHz     Max Speed: 3800 MHz     Current Speed: 3400 MHz     Status: Populated, Enabled     Upgrade: &lt;OUT OF SPEC&gt;     L1 Cache Handle: 0x003F     L2 Cache Handle: 0x003E     L3 Cache Handle: 0x0040     Serial Number: Not Specified     Asset Tag: Fill By OEM     Part Number: Fill By OEM     Core Count: 4     Core Enabled: 4     Thread Count: 8     Characteristics:         64-bit capable   It will be 8 logic core in a system, like what shows in /proc/cpuinfo. But can any one tell why the cpu MHz of a core is 1600MHz? I guess there is 2 threads in a core, so a hw thread freq may be about the half of the core's? How this number is calculated?  processor   : 7 vendor_id   : GenuineIntel cpu family  : 6 model       : 58 model name  : Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz stepping    : 9 cpu MHz     : 1600.000 cache size  : 8192 KB physical id : 0 siblings    : 8 core id     : 3 cpu cores   : 4 apicid      : 7 initial apicid  : 7 fpu     : yes fpu_exception   : yes cpuid level : 13 wp      : yes flags       : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp lm constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid bogomips    : 7013.49 clflush size    : 64 cache_alignment : 64 address sizes   : 36 bits physical, 48 bits virtual power management:   Also, here is the output of lshw and lscpu command. There are also 1600MHz mentioned.   lshw info:  #lshw -class processor   *-cpu                           description: CPU        product: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        vendor: Intel Corp.        physical id: 42        bus info: cpu@0        version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        slot: SOCKET 0        size: 1600MHz        capacity: 3800MHz        width: 64 bits        clock: 100MHz        capabilities: fpu fpu_exception wp vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp x86-64 constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid cpufreq   lscpu info:  #lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit CPU(s):                8 Thread(s) per core:    2 Core(s) per socket:    4 CPU socket(s):         1 NUMA node(s):          1 Vendor ID:             GenuineIntel CPU family:            6 Model:                 58 Stepping:              9 CPU MHz:               1600.000 Virtualization:        VT-x L1d cache:             32K L1i cache:             32K L2 cache:              256K L3 cache:              8192K Expected value of x in a normal distribution, GIVEN that it is below a certain value Using dconf-editor I tried enabling   org -&gt; gnome -&gt; nautilus -&gt; list-view -&gt; use-tree-view     but even after the check box is selected, plus a reboot, it still doesn't switch to the tree view on the side bar.  In previous versions of Nautilus / Ubuntu the side-pane-view setting used to be in the window-state section (which is now missing)    I'm I missing something, or is there no way to enable Tree View on the side bar in 14.04?  Is there a way to manually create the missing entry back in window-state ? Probability of $p(x_1<x_2<\max(x_3,x_4))$ Any way to sync directory structure when the files are already on both sides? I have two drives with the same files, but the directory structure is totally different.  Is there any way to 'move' all the files on the destination side so that they match the structure of the source side? With a script perhaps?    For example, drive A has:    /foo/bar/123.txt /foo/bar/234.txt /foo/bar/dir/567.txt   Whereas drive B has:  /some/other/path/123.txt /bar/doo2/wow/234.txt /bar/doo/567.txt   The files in question are huge (800GB), so I don't want to re-copy them; I just want to sync the structure by creating the necessary directories and moving the files.  I was thinking of a recursive script that would find each source file on the destination, then move it to a matching directory, creating it if necessary. But -- that's beyond my abilities!  Another elegant solution was given here: Read columns from file, then column into an exsisting csv file Essentially, I have a csv file that contain multiple columns, called cols.csv  1,a,100 2,b,200 3,c,300 4,e,400   and I have a new csv file that has one column, called col.csv  f g h i   I want to copy the items in col.csv and append them to the end of each line in cols.csv so that, cols.csv now contains those  1,a,100,f 2,b,200,g 3,c,300,h 4,e,400,i   is that possible at all? i tried join, paste, nothing worked Proving divisibility of expression using induction When are "because", "since","for" and "as" interchangeable? I am not a native speaker. On a previous question of mine, I thanked for an answer by saying:     So the phrase is not an idiom, since it is applied in its literal sense.   I consciously chose since over because, because — well, I like that usage of since, even though I know that some (native speakers) don't like it. Nonetheless, that choice got me thinking for a while, and it occurred to me that I even could have used more alternatives, namely as or for, without changing the meaning.   So, on a general note, here are the alternatives:        Statement A (is true), because statement B (is true).   Statement A (is true), since statement B (is true).   Statement A (is true), as statement B (is true).   Statement A (is true), for statement B (is true).      Are these completely interchangeable? Or interchangeable in certain situations? Or can you point out any (not-so-)subtle differences between any of them? Are there even more synonyms? What is the noun for someone who receives a referral? When a referrer (noun) gives a referral (noun) to another person, what is the term (noun) used for the recipient of the referral? "The referred" may work, but am I missing another term? "Referred" indicates past-tense, but I intend to use the term in future and present tense as well. How to Find Nearby iOS Devices (GPS locations) using same app In Nginx, how can I rewrite all http requests to https while maintaining sub-domain? I want to rewrite all http requests on my web server to be https requests, I started with the following:   server {     listen      80;      location / {       rewrite     ^(.*)   https://mysite.com$1 permanent;     } ...    One Problem is that this strips away any subdomain information (e.g., node1.mysite.com/folder), how could I rewrite the above to reroute everything to https and maintain the sub-domain?