This is more of a question about philosophy of mathematics. There is an apparent circularity in set theory and logic, in the means that we use set theory to define model theory, and we use results from model theory for proofs in set theory. And, analogously, we use set theory to prove facts from logic (unique readability, for example) and we define set theory using logic. I feel that we can reduce all this notions to a single syntactical idea of logical consequence, and from there we can define things in a non-circular way. However, I can't seem to make this notions all fit together in a logical, foundational manner. Does anyone know of an article or book that addresses this issue?
I wonder about the foundations of set theory and my question can be stated in some related forms:   If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not allowed to contain the notion of sets? The axioms of Zermelo–Fraenkel set theory seem to already expect the notion of a set to be defined. Is there are pre-definition of what we are dealing with? And where? In set theory, if a function is defined as a set using tuples, why or how does first order logic and the axioms of Zermelo–Fraenkel set theory contain parameter dependend properties $\psi(u_1,u_2,q,...)$, which basically are functions?
Calculate mean of ordinal variable I've read in a number of places that calculating the mean of an ordinal variable is inappropriate. I'm trying to get an intuition for why it might be inappropriate. I think it is because, in general, an ordinal variable is not normally distributed and so calculating the mean will give an inaccurate representation. Could somebody give more detailed reasoning for why calculating the mean of an ordinal variable might be inappropriate? I need to generate random numbers within a specified interval, [max;min].  Also, the random numbers should be uniformly distributed over the interval, not located to a particular point.  Currenly I am generating as:  for(int i=0; i&lt;6; i++) {     DWORD random = rand()%(max-min+1) + min; }   From my tests, random numbers are generated around one point only.  Example min = 3604607; max = 7654607;   Random numbers generated:  3631594 3609293 3630000 3628441 3636376 3621404   From answers below: OK, RAND_MAX is 32767. I am on C++ Windows platform. Is there any other method to generate random numbers with a uniform distribution? How to mount EXT4 disk on 10.15? Since I have osxfuse installed it would be nice if I could use it. There are a  (heavily) outdated  here, saying that it is not . Is this still the case? How to remove duplicate lines inside a text file? A huge (up to 2 GiB) text file of mine contains about 100 exact duplicates of every line in it (useless in my case, as the file is a CSV-like data table).  What I need is to remove all the repetitions while (preferably, but this can be sacrificed for a significant performance boost) maintaining the original sequence order. In the result each line is to be unique. If there were 100 equal lines (usually the duplicates are spread across the file and won't be neighbours) there is to be only one of the kind left.  I have written a program in Scala (consider it Java if you don't know about Scala) to implement this. But maybe there are faster C-written native tools able to do this faster?  UPDATE: the awk '!seen[$0]++' filename solution seemed working just fine for me as long as the files were near 2 GiB or smaller but now as I am to clean-up a 8 GiB file it doesn't work any more. It seems taking infinity on a Mac with 4 GiB RAM and a 64-bit Windows 7 PC with 4 GiB RAM and 6 GiB swap just runs out of memory. And I don't feel enthusiastic about trying it on Linux with 4 GiB RAM given this experience. Got ransomware and Shadow explorer doesn't work Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste? I have an object x. I'd like to copy it as object y, such that changes to y do not modify x. I realized that copying objects derived from built-in JavaScript objects will result in extra, unwanted properties. This isn't a problem, since I'm copying one of my own literal-constructed objects.  How do I correctly clone a JavaScript object? I'm looking for a more politically correct substitute for the expression "poor man's", meaning an inferior improvised or makeshift substitute. Usage examples:   "Guncotton is the poor man's TNT" "Poor man's oyster" (mussels, or coughed up phlegm)    My hesitance is mainly because it is not as gender neutral as perhaps it once was. The closest I can think of is "pauper's", but I don't know many examples of usage. Several badge descriptions contain raw HTML I was reading through some badge descriptions and noticed that Synonymizer's description needs to be updated. As it stands, there is raw HTML (an href) in the description:    EDIT: after reading comments by @ColeJohnson, I have come to the understanding that this post comes off as whiny. I want to make it very clear that I wasn't ever trying to complain. My first reaction was to fix the perceived error myself. When I realized I was unable to do so, I figured the best course of action would be to find someone who would be able to make the necessary edits. Hence the post here. Command for taking a screenshot to a new file Show how if Alice uses the same value of $k$ to sign two different messages $m_1$ and $m_2$, using the ElGamal signature scheme, Eve can recover the value of $a$ from the corresponding signatures $(m_1, r_1, s_1)$ and $(m_2, r_2, s_2)$. (Note: you are allowed to assume that if $\gcd(a, n) = d$ then there are $d$ solutions to the congruence $ax \equiv b \pmod n$.)     ElGamal Signature Scheme:      Key Gen:      Compute $y = g^x \;\bmod p$.      The public key is $(p, g, y)$. The secret key is $(p, g, x)$      Signature Gen:      Choose a random $k$ such that $0 &lt; k &lt; p − 1$ and $\gcd(k, p − 1) = 1$      Compute $r = g^k \;\bmod p$ and $s = k^{-1}(m – xr) \;\bmod{p-1}$   Thoughts: So far, I can tell that $r_1$ and $r_2$ are equal and $s_1$ and $s_2$ are closely related since the only variation is $m$.  We can relate the two equations for $s$, by solving them for $-xr$:  $$s_1k - m_1 \equiv s_2k - m_2 \pmod p$$  $$(s_1 - s_2)k \equiv m_1 - m_2 \pmod p$$  Let $a = s_1 - s_2$, then we know from the question that for $\gcd(a, n) = d$ there are $d$ solutions for $k$.  Now the forger can compute $g^k$ for each solution to k, until $r$ is found.  Then compute $xr \equiv m_1 - ks_1 \pmod{p-1}$  There are $d&#39; = \gcd(r, p-1)$ solutions for $x$.  The forger can compute $g^x$ for all the $x$s found until she finds $y$.  Once she has $y$ she knows the proper $x$ which can be used to find $m_1$ and $m_2$.  How does this look? When will my device get the Android 4.4 update (KitKat)? "(" or "\left(" parentheses? What is the difference between ( and \left( in LaTeX? Sometimes, when the content is small, it does not seem to matter which pair I use.  What is the best practice when it comes to which parentheses use? Limit of a stochastic integral Are Windows file timestamps timezone aware? I know this may be a tough or confusing question, but I have numerous existing materials I created and reuse for a project, which were all prepared for Blender Internal.  Now, I want to start the transition to Cycles, but I've hit a roadblock. How can I transition my materials without completely starting over? Or should I just forget it and start with fresh materials? 2015 13" MacBook Pro Trackpad and Keyboard stop working Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$ Column vs Field: have I been using these terms incorrectly? I feel kind of embarrassed here, I've always used the terms "column" and "field" completely interchangeably, which recently caused some confusion in a technical discussion.   I was told, though, that this wasn't correct, that it should be (translating each term into spreadsheet terminology, ignoring data types and all the other stuff that make databases useful):   Database Column: like a spreadsheet column  Database Record: like a spreadsheet row Database Field: like a spreadsheet "cell" (a specific column of a specific row)   Is this right? I could have sworn that column and field are used more interchangeably than that. I certainly have been.  So we don't add fields to a table, we add columns to a table, and fields are only relevant when talking about data within a record?  Other thoughts on column vs field?  Edit: to clarify, the current context is MS SQL Server. My background before SQL server was MS Access, which might influence my use of these terms. Our son is about to turn 3 months but he is yet to sleep on his own (as in we put him to sleep, put him down on the crib then he sleeps for few hours). He only sleeps when he is carried during the day, the moment we put him down he wakes up crying. At night, he sleeps with us on the bed where he is right next to his mommy. We blame ourselves for letting this happen but now my wife can't do anything during the day but carry him.  What can we do to break him out of this habit? Any pointers would be great appreciated.