In  , Thomas Jech says     Classes      Although we work in ZFC which, unlike alternative axiomatic   set theories, has only one type of object, namely sets, we introduce   the informal notion of a class. We do this for practical reasons: It   is easier to manipulate classes than formulas.   Sadly I don't get the difference between them. My question actually is:   Set is defined by the axioms of ZFC. Which axioms of set are required on class, which are not?  Axioms of Zermelo-Fraenkel from Jech's book:     1.1. Axiom of Extensionality. If X and Y have the same elements, then X=Y.      1.2. Axiom of Pairing. For any a and b there exists a set {a,b} that contains exactly a and b.      1.3. Axiom Schema of Separation. If P is a property (with parameter p), then for any X and p there exists a set $Y = \{u ∈ X : P(u,p)\}$ that contains all those u ∈ X that have property P .       1.4. Axiom of Union. For any X there exists a set $Y = \cup X$, the union of all elements of X.       1.5. Axiom of Power Set. For any X there exists a set Y = P(X), the set of all subsets of X.      1.6. Axiom of Infinity. There exists an infinite set.      1.7. Axiom Schema of Replacement. If a class F is a function, then for any X there exists a set Y = F(X) = {F(x) : x ∈ X}.      1.8. Axiom of Regularity. Every nonempty set has an ∈-minimal element. 1.9. Axiom of Choice. Every family of nonempty sets has a choice function.      The theory with axioms 1.1–1.8 is the Zermelo-Fraenkel axiomatic set   theory ZF; ZFC denotes the theory ZF with the Axiom of Choice.   Definition of class, also from Jech's book:     If $φ(x,p_1,...,p_n)$ is a formula, we call $C = \{x : φ(x,p1,...,pn)\}$ a class.    Members of the class C are all those sets x that satisfy $φ(x,p_1, . . . , p_n)$:      x ∈ C if and only if $φ(x,p_1,...,p_n)$.   Note here the definition is by formula, not by axioms, this is where I'm lost.  There are similar posts on MSE, but I'm still not sure  In  , Sylvain's answer says      The powerset axiom is the main exception, the one axiom that cannot be   properly justified.   OK, so Axiom of Powerset is out. How about others?  HellCat's answer says      the axioms of Set Theory apply only to sets and not to classes.   Also in  , Asaf says:     Sets are elements of the model of set theory, and they have to satisfy   the axioms, e.g. the axiom of power set. Classes are collections of   elements from a model of set theory, but they don't have to correspond   to any element in the model, and they don't have to obey the axioms.   Does this mean all the axioms above are not applicable to classes? Then what shall the actions on classes be restrict to?
I know what a set is. I have no idea what a class is.   As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class").  This obviously makes no sense whatsoever, since sets are of unlimited size. It's not like you get up to 25 elements and go "oh, hey, sets are limited to a maximum of 20 elements, so we'll have to call this 25-element thing something else".  Does anybody know what the actual difference between a set and a class is?  Several people have pointed out that you can write paradoxes which involve sets. I get that. But where do classes come into the picture?
What are badges? How do users get badges? How can users get tag badges? Can badges be lost/revoked/taken away after they are awarded? If so, how and when? How can I suggest a new badge?   For more information, see &quot;&quot; and the  in the . Probability that median falls within a range? If I draw 5 samples from a distribution that is cont., what is the probability that the median of the population is between the xmin and xmax of the draw samples?  What is the formula to apply here? Working with systems with Perfect Multicollinearity I am working with a time-series dataset that is based on demand-supply dynamics with several variables.  THe sample data for one time period is: &gt; data # A tibble: 19 x 6    Pred_error_A Pred_error_B  Pred_error_C  Pred_error_D  Pred_error_E  Pred_error_Target                  &lt;dbl&gt;         &lt;dbl&gt;          &lt;dbl&gt;      &lt;dbl&gt;      &lt;dbl&gt;           &lt;dbl&gt;  1            -0.45           1.72          -0.170      1.85       -0.01          -0.7    2            -0.45           1.72          -0.170      2.35       -0.01          -1.2    3            -0.45           1.92          -0.170      2.35       -0.01          -1      4            -0.0500         0.37          -0.12       1.65       -0.01          -1.5    5             0             -0.840         -0.12       0.950      -0.01          -1.7    6             0             -0.200         -0.12       1.35       -0.01          -1.5    7             0             -0.460         -0.12       1.35       -0.01          -1.8    8             0             -0.520         -0.12       1.05       -0.01          -1.5    9             0             -0.340         -0.12       1.05       -0.01          -1.4   10             0             -0.340         -0.12       1.35       -0.01          -1.7   11             0             -0.110         -0.12       1.05       -0.01          -1.2   12             0             -0.110         -0.12       0.850      -0.01          -1     13             0              0             -0.12       0.350      -0.01          -0.4   14             0              0             -0.12      -0.15       -0.01           0.100 15             0              0             -0.12      -0.350      -0.01           0.300 16             0              0             -0.02       0          -0.01          -0.1   17             0              0              0          0           0              0     18             0              0              0          0           0              0     19             0              0              0          0           0              0   I was working on a regression for each time period with data and wanted to find the relative importance of regressors in explaining the target variable ( Pred_error_A through E and dependent variable is Pred_error_Target ) using relaimpo package in R. However, I soon realized I can't use regression setting because of near to perfect collinearity and hence no relative importance values. Is there a way to go around this? Or, can I use any other method (instead of regression)? Any suggestions will be useful. The predictors are near to perfect and each one is important. I can't exclude any single one and I need the relative importance of all 5 regressors. I am also trying XGBoost as an alternative. However, I am not sure if it takes care of multicollinearity. I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? Converting from shapefile or raster formats to LAS? Are layer files compatible between ArcGIS 10 and 9.3? If I save layer files in ArcGIS 10, can they be opened and modified in ArcGIS 9.3? What happens if my order exceeds the bid or ask sizes? What will happen if my order exceeds the bid or ask size? For example, if the bid or ask is 100 shares, but I want to buy 100,000 shares, is it possible to do so? Need help disabling imessage? I recently changed from iphone to android and i have been having trouble recieving messages from my iphone friends since my iphones screen is broken i cant do anything on it. Any way to fix this problem through icloud or something? Did the Big Bang happen at a point? TV documentaries invariably show the Big Bang as an exploding ball of fire expanding outwards. Did the Big Bang really explode outwards from a point like this? If not, what did happen? How to make \left, \right pairs of delimiter work over multiple lines? If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$. Can we get x comment votes per y minutes, rather than one vote per 5 seconds? Below is an example from I.N. Herstein:       Let  $R=\Bigg\{\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of complex numbers. Define $\psi :R\to \mathbb C$ by $\psi\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)=a+bi.$       It is asked to prove that $\psi$ is a ring isomorphism of $R$ onto $\mathbb C$ .    can anyone help me with some hint how to prove the ring isomorphism. What is a “protected” or "highly active" question? Questions can be protected (i.e. marked "highly active").   What does it mean for a question to be protected? Why are some questions protected? Who can protect and unprotect questions? Who can answer a protected question? How are protected questions displayed? When should I protect or unprotect a question? When does the Community user protect a question? Are there any downsides to passing structs by value in C, rather than passing a pointer? Are there any downsides to passing structs by value in C, rather than passing a pointer?  If the struct is large, there is obviously the performancd aspect of copying lots of data, but for a smaller struct, it should basically be the same as passing several values to a function.  It is maybe even more interesting when used as return values. C only has single return values from functions, but you often need several. So a simple solution is to put them in a struct and return that.  Are there any reasons for or against this?  Since it might not be obvious to everyone what I'm talking about here, I'll give  a simple example.  If you're programming in C, you'll sooner or later start writing functions that look like this:  void examine_data(const char *ptr, size_t len) {     ... }  char *p = ...; size_t l = ...; examine_data(p, l);   This isn't a problem. The only issue is that you have to agree with your coworker in which the order the parameters should be so you use the same convention in all functions.  But what happens when you want to return the same kind of information? You typically get something like this:  char *get_data(size_t *len); {     ...     *len = ...datalen...;     return ...data...; } size_t len; char *p = get_data(&amp;len);   This works fine, but is much more problematic. A return value is a return value, except that in this implementation it isn't. There is no way to tell from the above that the function get_data isn't allowed to look at what len points to. And there is nothing that makes the compiler check that a value is actually returned through that pointer. So next month, when someone else modifies the code without understanding it properly (because he didn't read the documentation?) it gets broken without anyone noticing, or it starts crashing randomly.  So, the solution I propose is the simple struct  struct blob { char *ptr; size_t len; }   The examples can be rewritten like this:  void examine_data(const struct blob data) {     ... use data.tr and data.len ... }  struct blob = { .ptr = ..., .len = ... }; examine_data(blob);  struct blob get_data(void); {     ...     return (struct blob){ .ptr = ...data..., .len = ...len... }; } struct blob data = get_data();   For some reason, I think that most people would instinctively make examine_data take a pointer to a struct blob, but I don't see why. It still gets a pointer and an integer, it's just much  clearer that they go together. And in the get_data case it is impossible to mess up in the way I described before, since there is no input value for the length, and there must be a returned length. Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and . Relative Pronouns are ALWAYS added right after antecedent? Is is a rule that relative clauses should be added right after the antecedent(the noun relative pronouns refer to)? What if it is confusing?   For example:  "I went for a spin with a handsome boy last night."  "The handsome boy went out with my friend."  "I went for a spin with a handsome boy who went out with my friend last night ."  "I went for a spin with a handsome boy last night who went out with my friend." What is the relation between linear purity and von Neumann entropy of a state? Given a density matrix $\rho$, its linear purity is $\mathrm{Tr} \rho^2$, its von Neumann entropy is $-\mathrm{Tr} \rho \log \rho$. Knowing one how do I calculate the other?  Edit:   My thought is that the linear purity and the von Neumann entropy are a parallel set of benchmarks for mixedness. Their inter-conversion should be like unit conversion.  See it this way, as a state goes from a pure state to a maximally mixed state, $\mathrm{Tr} \rho^2$ goes from $1$ to $0$, and $-\mathrm{Tr} \rho \log \rho$ goes from $0$ to $\log d$, like two rulers. A point on one should correspond to a point on the other. C++ iostream Corruption using stringstream Is there a way to enable/disable a wireless network interface via the command line (so I can put it in a bat file)?  When I'm in the office I use a wired connection, but Windows still occasionally bugs me about my wireless connection not being connected (even though I turned off the option for notifying me when there's no connectivity).  I'm guessing the only way to stop it from bugging me is to disable the interface, but I'd rather not have to go into the network settings every time I need to do so.  I'd like to set up a bat file or a shortcut that I can use to enable/disable the wireless (preferably a single one that toggles the current state), and then just set up a keyboard shortcut for that.  I just have no idea how to do so from the command line.