This a very elementary question, but I am quite new to set theory. I came across this quote on the Wikipedia page for &quot;Hereditary Set&quot;.  In formulations of set theory that are intended to be interpreted in the von Neumann universe or to express the content of Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set.  Is this quote saying that all sets in ZFC are hereditary sets? How does one then construct a set such as ℝ? Thank you
In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?    Thanks for the answers. Several answers basically said for irrational numbers that  A Dedekind cut is a pair of sets of rational numbers $\{L, R\}$. The set of real numbers is defined to be the set of all Dedekind cuts, where a Dedekind cut is a pair of sets of rational numbers $\{L, R\}$ which have no elements in common, and where all the elements of $L$ are less than any element of $R$.  Each Dedekind cut is a real number. This is where I have a problem - surely that can’t be correct. The set $L$ is a set of all rationals, and there must be a rational in the set $L$ that is greater than all other rationals in that set, even if we have no method of determining it. And similarly, there must be a rational in the set $R$ that is less than all other rationals in that set, even if we have no method of determining it. If every irrational number has a corresponding set $L$, then each irrational number has some such corresponding largest element of that set $L$, and then each irrational number has some corresponding rational number. And that would mean that the irrational numbers are countable. So, with Dedekind cuts,  the only conclusion is that there must be irrational numbers $x$ which are either greater or lesser than some irrational cut $y$ of the rationals, and between $x$ and $y$ there is no rational number.  But that is impossible, so that the Dedekind cuts cannot be the correct representation of the real numbers.  Surely the problem with Dedekind cuts is in using sets of rationals that include all rationals up to a certain rational. But there is an alternative method of representing irrationals can be defined in terms of infinite sets of rational numbers. For example, in binary notation, the non-integer part of $\pi$ is $.00100100\ 00111111\ 01101010\ 10001$. You define a set by: if the nth digit is a $1$, then  the natural number $n$ is in the set. And then we have that, for the real numbers between $0$ and $1$, that the set of real numbers is simply the set of all subsets of natural numbers. Each subset corresponds to some real number between $0$ and $1$.  And in this way, all real numbers can be considered to be some set based only on nested sets of the empty set.  But I still haven’t got a satisfactory answer for how negative numbers can be represented in terms only of sets containing the empty set. Any ideas?
Using k-fold cross-validation for time-series model selection Question: I want to be sure of something, is the use of k-fold cross-validation with time series is straightforward, or does one need to pay special attention before using it?   Background: I'm modeling a time series of 6 year (with semi-markov chain), with a data sample every 5 min. To compare several models, I'm using a 6-fold cross-validation by separating the data in 6 year, so my training sets (to calculate the parameters) have a length of 5 years, and the test sets have a length of 1 year. I'm not taking into account the time order, so my different sets are :   fold 1 : training [1 2 3 4 5], test [6] fold 2 : training [1 2 3 4 6], test [5] fold 3 : training [1 2 3 5 6], test [4] fold 4 : training [1 2 4 5 6], test [3] fold 5 : training [1 3 4 5 6], test [2] fold 6 : training [2 3 4 5 6], test [1].   I'm making the hypothesis that each year are independent from each other. How can I verify that? Is there any reference showing the applicability of k-fold cross-validation with time series. Countinuity of the identity map between different topologies How can I use pathauto to generate URLs for deeply nested vocabularies? I am generating breadcrumbs from URL arguments as described .  I have a vocabulary that looks like this:  Arts-> Performing-> Dancing-> Ballet  I want my url to look like this: arts/performing/dancing/ballet  Using the pathauto module, I can do this for taxonomy terms:  [term:parent]/[term:name]   Which, on the Ballet page, gives me the following; not what I want:  dancing/ballet    If I use parents instead of parent,  ([term:parents]/[term:name]); this is close but not quite:  arts-performing-dancing/ballet   How can I get from arts-performing-dancing to arts/performing/dancing? Could an argument with false Premises and a true Conclusion be logically valid? Subgroups of Symmetric groups isomorphic to dihedral group Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? LG Nexus 4 compatibility Airtel India Possible Duplicate:          I plan to purchase the LG Nexus 4 from the US on the release date (13th November). The phone claims to be unlocked. But will it be compatible with Airtel in India for both data and call services? How to set up reverse tethering over USB? My HTC G2 phone is rooted and running Cyanogenmod 7 I don't have a data plan. Sometimes I want to connect the phone to the Internet when there isn't Wi-Fi, to update Market apps, backup SMS messages to Gmail, sync new contacts from Gmail, etc.  Things I can't do with USB mass storage mode. I have a Windows 7 Professional computer connected to the Internet, but I'm not allowed to set up an ad-hoc Wi-Fi network.  (If I do, they will notice and hunt me down.)   Is there a way for the phone to access the Internet through the USB connection to the computer?  If so, how do I set it up? How do you distinguish between magic effects that bypass immunity and those that don't? This question was triggered by .  Basically, it goes like this: If you have a weapon with a certain magical effect, let's say the minor property unbreakable (can not be broken, requires special means) or temperate (no harm in temperatures between -20 to 120 F), it can bypass the immunity of something like a Werewolf because it is magical instead of mundane. This goes for any minor property because those properties are under the Magic Items portion of the DMG on pg.143.  How does this differ from a pair of gauntlets with the exact same effect being used as an improvised weapon (emphasis added for clarity) from dealing damage in the same manner to the same Werewolf? Or something that's clearly not a weapon like a Shield of Missile Snaring being used to bash an opponent? Deleting 'garbage' nodes after polygon dissolve I work with QGIS 2.10.1 on Windows 7 (x64). I have dissolved polygons from an attribute value and the polygons geometries being not perfectly joined together, I get some 'garbage' nodes I need to delete. I have tried to edit the layer and using nodes tool, selecting and deleting the nodes, but it seems impossible to do (nodes are not deleted at all!). Any idea on how to process this kind of clean-up?  EDIT : you can  for testing. Obtaining zero-length string from strtok() Possible to save a list template and maintain lookup relationships? My issue is that I have two lists that I would like to move with their data.  While I am able to save both as templates and then upload to the new site, none of the lookup columns in my parent table are populated with data now.  Can this be done? I am really satisfied that $(x \to a) ≠(x=a)$ and if that is not right , Then all the process of $Limits$ is dividing by zero and that is a crime.  Since $(x \to a) + h = (x=a)$ , $h ≠ 0$,So Why does $f(x \to a) = f(x = a)$ ?  NOTE:I am talking about continuous function. Stop Excel from automatically converting certain text values to dates Does anyone happen to know if there is a token I can add to my csv for a certain field so Excel doesn't try to convert it to a date?  I'm trying to write a .csv file from my application and one of the values happens to look enough like a date that Excel is automatically converting it from text to a date.  I've tried putting all of my text fields (including the one that looks like a date) within double quotes, but that has no effect. Using C# .NET 2.0, I have a composite data class that does have the [Serializable] attribute on it.  I am creating an XMLSerializer class and passing that into the constructor:  XmlSerializer serializer = new XmlSerializer(typeof(DataClass));   I am getting an exception saying:      There was an error reflecting type.   Inside the data class there is another composite object.  Does this also need to have the [Serializable] attribute, or by having it on the top object, does it recursively apply it to all objects inside? how to redirect output to multiple log files What is the path of an MTP device in Windows? I am running Windows XP and have a phone mounted through USB by MTP.  In Explorer, its address is:  My Computer\SAMSUNG-SGH-I547\Phone\DCIM   I would like to execute a command line command such as  explorer "My Computer\SAMSUNG-SGH-I547\Phone\DCIM\Camera"   so as to quickly go to the Camera folder. (Right-clicking the device or anything in it does NOT provide a "Create Shortcut" dialog.)  But "My Computer" is not a valid PATH in Windows. Unlike USB flash drives, it seems the phone mounted by MTP does not get a letter drive name.  So, what is the path of an MTP device in Windows?  If the answer is there is no command line path, is there workaround? I'm wanting to parse a string into a nullable int in C#.  ie. I want to get back either the int value of the string or null if it can't be parsed.  I was kind of hoping that this would work  int? val = stringVal as int?;   But that won't work, so the way I'm doing it now is I've written this extension method  public static int? ParseNullableInt(this string value) {     if (value == null || value.Trim() == string.Empty)     {         return null;     }     else     {         try         {             return int.Parse(value);         }         catch         {             return null;         }     } }      Is there a better way of doing this?  EDIT:  Thanks for the TryParse suggestions, I did know about that, but it worked out about the same.  I'm more interested in knowing if there is a built-in framework method that will parse directly into a nullable int? At what voltage is a NiMH battery empty? I have a battery pack of NiMH batteries. It is ten cells with 1.2V, 4000mAh each, put together in series. So rated voltage is 12V. After charging, i.e. when the charging device says that it is finished, I measure 14.3V.   When I now discharge, at what voltage should I consider my battery as "empty"? Is this the lowest voltage that I can discharge it to without damaging it? Which voltage is that? Problem with \mkbibdateapalongextra of biblatex-apa biblatex-apa gives me this:  ! Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra                                {labelyear}{labelmonth}{labelday}\iffieldu...   I'm using the default (english) language, and have done the:  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa}   thing.   Here's a short example:  \documentclass[english]{memoir} \usepackage{babel} \usepackage{roffe} \usepackage[backend=biber,date=short,maxcitenames=2,style=apa]{biblatex}  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa} \addbibresource{Papers2.bib} \addbibresource{R.bib}  \begin{document} \cite{R-base}  \printbibliography \end{document}   This one replicates the problem. What is the relation between normal extension and separable extension? What is the relation between normal extension and separable extension? Let F be the algebraic extension of K, if F is a separable extension of K,if and only if F is a normal extension of K? is this correct?