Let $(A_i \mid i \in I)$ be a family of non-empty set. Then $A_i \neq \emptyset$ for all $i \in I$, thus $\exists a_i \in A_i$ for all $i \in I$. We define $f:I \to \bigcup A_i$ by $f(i)=a_i$.   From reading other posts in MSE, I'm sure that my argument is wrong without Axiom of Choice, but I can not understand why it's wrong. The usual answer is that you can not appeal to Existential Instantiation infinite many times. However, I'm unable to understand this explanation under set theory. Maybe the problem is related to underlying logic, with which I'm not familiar.  Please explain me why you can not appeal to Existential Instantiation infinite many times?
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.
Hook into the loop via a plugin, and output something after every X post? I have a plugin where I want to output ads after X number of posts on the home page. The home page is step 1, but things like archives should be possible too once I get the code for the home page.  How do I hook in to post loops and say something like "after every loop, increment a counter, and then if the counter = my number, output an ad". I can write all the logic for this code myself, but where to hook/implement my code is confusing. Sphere rolling up a ramp In this question, I did energy conservation, with the states being the initial one and the one where it is at the top , and I got $$ .5 I w^2 + .5 mv^2 = mgh$$ However, in  the answer key they have done, $$ .5 mv^2 + .5 Iw^2 = .5 Iw^2 + mgh$$ Now why would the k.e be turned to p.e but not the rolling energy?? I am currently undergoing a full Apostol Calculus I reading, and with few chapters left I've began planning what other books to read. I was thinking about finishing both Apostol Calculus l and ll, and then read Apostol mathematical analysis. I'm not pretty sure whether these are the best books or if they will provide me with a strong math basis. I've already taken calculus i, ii, iii and linear algebra. The reason I've chosen Aapostol volume 1 and 2 is that I've heard they cover almost all the topics I've seen and will see in the next semester/s. I would appreciate if somebody could give me some advice about this. I really like to have a book that really gives me hard and challenging problems. I don't know whether to read Spivak Calculus book or not. I'm making an python script which create a lot of object and this could take a lot of time. My question is how to show to the user of my script, the "progression" of the script? For now I print an progress bar in the system console.  But you have to open the system console befor and (apparently) the system console only work with Windows.  Is there an other way to show the progression of my script with python like control the mouse percentage icon (the little icon that replace the mouse while baking) or another idea? A does the same B as does C How to draw arrows between colourboxes Countries with no visa requirements on its visitors $f$ bounded but $f'$ isn't How do I SOSL across objects, matching all records with FIND clause? Sending Text Cross Domain By Bookmarklet Downloading US Census data? Where are the outliner toggles in 2.8? The selection toggle arrows in the Outliner disappeared. Is there some preference in the settings that has to be turned on? Crash adding comment The SE iOS app crashes when adding comments to certain posts.  It was discovered to be if the "Add an Answer" button was visible, not just responding to a deleted comment.  Edited to match answer. Previous steps were not the only use case, and this error keeps getting re-reported in different ways. How to get text under mouse in any open window Disk space full but logical space available in database We have a fairly big MS SQL 2008R2 database that resides on a SSD drive. The drive itself only has ~110Gb of space, and the database files are the only files on the drive.  The database is in "Simple" recovery mode, and only has two files, .MDF and .LDF.   The disk is now nearly full: the MDF is currently 109Gb in size. However, SSMS tells me there's nearly 18Gb of "Space Available" (in the 'General' properties page), and if I go through the motions of Shrinking the file it also tells me there's 18Gb of space free. SSMS also tells me the database size is around 132Gb, which surprises me - that wouldn't fit on the drive!  From what I've read, shrink is a really bad idea. However, I'm starting to see replication errors (could not allocate space for object). We've previously tried shrinking the database, but within a few hours the file was back to it's original size.  How should we proceed - given there's apparently 18Gb of free space, should SQL automatically use that free space? Or is it as simple as: we really need more disk space? When is the present perfect tense used instead of the past tense? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. Selenium/PhantomJS raises error Orthogonal basis for infinite-dimensional vector spaces Tikz: shading region bounded by several curves I know that my effort below can be improved upon considerably.  In particular, there must be a way of shading the region in a single scope, as opposed to subdividing as I have done (thus necessitating a hidden gray boundary).  Neither am I fond of sample points, but this is where my research has led me.  Your constructive criticism is appreciated.    \documentclass{article} \usepackage{tikz} \begin{document}  \begin{tikzpicture} %graph \draw[draw=gray!50!white,fill=gray!50!white]      plot[smooth,samples=100,domain=0:1] (\x,{0}) --      plot[smooth,samples=100,domain=1:0] (\x,{1}); \draw[fill=gray!50!white] plot[smooth,samples=100,domain=1:2.71828] (\x,{ln(\x)}) --      plot[smooth,samples=100,domain=2.71828:1] (\x,{1}); \draw[domain=0:4] plot (\x,{1}); \draw[samples=100,domain=.25:4] plot (\x,{ln(\x)}) node[above left]{$y=\ln x$}; %coordinate grid \draw (-.5,0)--(4.5,0) node[right]{$x$}; \draw (0,-1.5)--(0,2.5) node[above]{$y$}; \foreach \x in {1,2,3,4}     \draw (\x,2pt)--(\x,-2pt) node[below] {$\x$}; \foreach \y/\ytext in {1,2}     \draw (2pt,\y)--(-2pt,\y) node[left] {$\y$};     %labels  \node at (.75,.5) {$\mathcal{D}$}; \end{tikzpicture}  \end{document} Using "that" and "this" interchangeably Learning and using English I'm always confused about what word to use for referring to things that have been described by me a few sentences earlier: "that" or "this".   Confusion comes from the fact that only the equivalent of "this" is always used in my native language for such referring. But I've noticed that in English for such referring "that" is used as often as "this" (or maybe even more often).  Some examples for illustrating (just tried to google something appropriate to convey my idea better):     We assess local... the demographics of the local population. What are the natural traffic drivers in the area. Things like cinemas and pubs and retail and office and all that type of things and we now put together a bit of a matrix and actually give a weighted score to each of the things we know help our business. That helps us decide in a more scientific fashion.   or:     The problem of Cervantes' origin became after that into a tough matter. Some experts believed that the Cervantes from Alcázar, in the times of the Lepanto Battle, was in the age of a child, more concerned about gathering nests and that type of things than about fighting as a soldier.   and for "this":     I believe lot of people who involved fishery industries in Mexico Gulf are suffering now but as for economic issues, U.S government and other countries will support them and give an utmost response to it. Important thing is that we learn from this mistake and make sure this type of things will never happen in the future and protect nature environment thus we have to take this technology to get energy from water very seriously. I really hope this technology will be available and used for everybody as soon as possible.   Is there a rule describing proper usage of these words in cases like this? (or should I have written "..in cases like that"?)