Does there exist a set $A$ such that $A \in A$?  The empty set doesn't work. Also, it seems clear that if $A$ exists, then it must be infinite.  If no such set exists, then is there a simple proof of this?
I am trying to understand   How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself.
University rank/stature - How much does it affect one's career post-Ph.D? What I'm trying to understand is, to what degree does the status/rank of the University (where one completes his/her Ph.D) matter while shaping his/her career after graduation? I would like to know the weight given to one's school in both the following cases:   While applying for post-docs/faculty positions in academia While applying to industrial research labs   For instance, I've read on some forums (I can't locate the link now) that while considering prospective applications for tenure-track faculty positions, very few Universities accept a candidate who has completed his/her Ph.D from a lower ranked school having a lesser "brand" value, irrespective of the fact whether he/she has published equally original work as his/her counterpart from an Ivy league college. How much truth is in this statement? It would be really great if someone already in academia, either as a newly-accepted faculty or someone on the Faculty Hiring committee could share their experiences/statistics on this regard. I'm simply interested to know the answer, without commenting at all on whether such a practice is justifiable.  Similarly, what about recruitment to internationally acclaimed research labs (like  or ) - what importance do they place on the pedigree of a candidate's college, before taking into consideration what they published ?  I'm personally interested in answers related to the field of Computer Science (theory), but the question is applicable to any prospective grad student in any discipline in my opinion. Feel free to share your personal experiences post-Ph.D in detail, as that would give me (and future viewers of this question) about what its like to carve a career once you are out of school! How to apply for a Schengen visa when the travel dates are uncertain My husband has a Chinese passport and I am a U.S. citizen; I don't need a visa to travel to Schengen countries. I am helping him to apply for a visa.  Our plans are to go to Paris, Italy and Spain, but we haven't nailed down definite dates; the approximate dates are August 18 to September 11. We made an appointment with Spain consulate to apply for the visa. We need to show air flight reservations, I could book it online without any problems.    Since we don't have definite dates, what dates should he put down on the application?  I know that, once they approve you, they give you 90 days, so does it matter what dates I put down on the visa application?   Our appointment at the Spanish Consulate is in a few days. I've seen many people who say:     This is a HTML page.   Yet I've also seen many people who say:     This is an HTML page.   Are both usages equally correct?  Or, which is the grammatically correct one?       Possible Duplicates: Three inches of snow is/are expected? I'm thinking that the singular verbs "is" and "has" are correct below because we're referring to a singular lump of precipitation. Although the subject "inches" is plural, I think the singular verbs are correct below. We wouldn't say, "Fifty dollars are a lot of money." I think this is basically the same concept.  Good for the singular verbs below?  One to 3 inches of snow "is" expected. (Not: ... are expected.)  Ten inches of snow "has" fallen. (Not: ... have fallen.) Elegantly determine if more than one boolean is "true" Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ Injective functions also surjective? Is it true that for each set $M$ a given injective function $f: M \rightarrow M$ is surjective, too?  Can someone explain why it is true or not and give an example? Create a local texmf tree in MiKTeX How to perform hard / cold reboot of iPhone or iPad? I understand that when I press and release the power button on the top of my iPhone or iPad, it stays on but turns off the screen.  Whereas if I press and hold the power button, a "slide to turn off" control appears on the screen.  Sliding this, as I was told, does turn the unit off, but it does so by suspending the RAM to disk (flash memory, actually.) So, when turning on the iPhone or iPad, it isn't so much rebooting the OS as it is resuming it from flash storage.  How to force the OS to completely reboot on these devices? Best way to parse RSS/Atom feeds with PHP I'm currently using  but it sometimes falls over when the RSS or Atom feed isn't well formed. Are there any other options for parsing RSS and Atom feeds with PHP? How do I add a DNS server via resolv.conf? Is /etc/resolv.conf useless in  (Precise Pangolin)?  I see that the DNS server information is stored in  now. The nmcli command line tool can list that for you.  If I want to add one more  server, will adding it to /etc/resolv.conf by using the resolvconf package help? Let $R$ be a ring such that $x^4=x$ for every $x\in R$. Is this ring commutative? I recentely tried to log-in / sign-up for  using my SE account (which works on ,  and sign-up here) but it showed me an OpenID related error. (See below)  I could reproduce this error across multiple days and browser refreshes and using my other OpenID provider.  Error message (shown in red, obfuscated my e-mail address):  Unable to log in with your OpenID provider:  The openid.return_to parameter (http://area51.stackexchange.com/users/authenticate/?s=bd394805-cf55-4e8e-ad73-21485a09a989&amp;dnoa.userSuppliedIdentifier=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1) does not match the actual URL (https://area51.stackexchange.com/users/authenticate/?s=bd394805-cf55-4e8e-ad73-21485a09a989&amp;dnoa.userSuppliedIdentifier=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.claimed_id=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.identity=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.sig=4emyzuA31ChHU0LabPGJXldbN8GeKN%2FFBJXJy69yD10%3D&amp;openid.signed=claimed_id%2Cidentity%2Cassoc_handle%2Cop_endpoint%2Creturn_to%2Cresponse_nonce%2Cns.alias3%2Calias3.mode%2Calias3.type.alias1%2Calias3.value.alias1%2Calias3.type.alias2%2Calias3.value.alias2&amp;openid.assoc_handle=hrQL%21IAAAAGzBtFeNGBLU2o04XSWr7ISRAbO-B2tZCK3L61IzAsuQQQAAAAEdoTN6rf9fqaP1oekJGXGXJeB8fNSVuByptp5yx1VpQbAzQwF6HJzh6a0DwedydNJ-wwpJQoN5dOLEFmndZt-L&amp;openid.op_endpoint=https%3A%2F%2Fopenid.stackexchange.com%2Fopenid%2Fprovider&amp;openid.return_to=http%3A%2F%2Farea51.stackexchange.com%2Fusers%2Fauthenticate%2F%3Fs%3Dbd394805-cf55-4e8e-ad73-21485a09a989%26dnoa.userSuppliedIdentifier%3Dhttps%253A%252F%252Fopenid.stackexchange.com%252Fuser%252F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.response_nonce=2015-06-12T12%3A30%3A42Z7tADDeYZ&amp;openid.mode=id_res&amp;openid.ns=http%3A%2F%2Fspecs.openid.net%2Fauth%2F2.0&amp;openid.ns.alias3=http%3A%2F%2Fopenid.net%2Fsrv%2Fax%2F1.0&amp;openid.alias3.mode=fetch_response&amp;openid.alias3.type.alias1=http%3A%2F%2Faxschema.org%2Fcontact%2Femail&amp;openid.alias3.value.alias1=mycorrectaddress%40web.de&amp;openid.alias3.type.alias2=http%3A%2F%2Faxschema.org%2FnamePerson&amp;openid.alias3.value.alias2=SOJPM) the request was made with.    As noted in the comments I'm not the only one, so I'd guess it may be not something I can fix. So will this be fixed?  Repro steps (using SE OpenID):   Get an Stackexchange OpenID account (like I have) (optional?) Add other OpenID providers and check they're working (StartSSL in my case) Please note: "You" don't have an "account" for the Area 51 subpage yet and still need to do the initial sign-in. Go to the Area 51 SE page Hit "Log-in" in the top-bar Hit the "log-in using stackexchange" button Enter your e-mail and password and submit Get the above error   Repro steps (using other OpenID)   Perform as above (until step 4), but this time step 2 is mandatory Hit "more OpenID options" Provide your OpenID link XXX.provider.com (works on crypto, server fault, ...) Authenticate against your provider Receive a variation of the above error (can be included if needed) How can I mitigate edge halo when darkening skies in post-processing? What is an observer in quantum mechanics? My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.  What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally? I'm attempting to install Ubuntu 14.04.1 (the only version I have handy) onto my old Compaq Presario R3000 (this Ubuntu was recommended for this computer). The installation has stalled at about 60% and hasn't moved in an hour.  The message at the bottom says: Configuring bcmwl-kernal-source (i386).  I have no experience with Ubuntu (or with Linux, for that matter), coming from a strictly-Windows background). Can anyone offer some advice? QGIS query syntax differs between attribute table and feature filter Using QGIS 2.16.2, I can successfully select records from a shapefile attribute table using the following expression:  left("start_time", 10) = '2015-08-01'   Please note that start_time is a string field.  However, when I copy and paste the same expression into the shapefile's Layer Properties>General>feature filter>Query Builder I get the following error:    I would expect that the same expression would work everywhere in QGIS.  What am I doing wrong? What is the difference between DX format and FX format lenses, and which to choose for what purpose?  What are the pros and cons of both? Will database replication increase the size of the replicated database? First thank you for your time. We have a SQL Server database that was assigned to be replicated. the original size of the database was 5 GB for the datafile, not the log file. Now after the replication the data file is 4 times bigger around 19 GB. Any pointers to see what could happened will be appreciated. I know sql as developer and some little management but nothing about replication. The performance of the database has been damaged very bad so I am trying to see what need to be done to improve it. I do apologize for my English as well. I am wondering whether there is any difference between a gerund acting as subject and an infinitive acting as a subject.