Every time I read about a theory in mathematics, it usually starts with axiomatizing the most fundamental concepts that are going to be treated.  Recently, I have started finding this troubling. In the foundational crisis, we tried to root all of mathematics on set theory and build it up from there. I believe this is a supremely elegant idea, but I have to ask myself why.  I understand why axioms are the brick wall against which all infinite regressions crash. We cannot, after all, ask why something is indefinitely. There must come a time when we simply say: because it is.  But why? What happens if we throw logic out of the window and attempt to start everything from scratch?   I have read about model and category theory and all types of order logics, but none of them seem to be enough because they are all rooted in something that eventually leads to a so-called "self-evident truth". What if infinite regressions are similar to infinite series: something that at first we assumed was non sensical but actually turns out to be really useful?  My question is: are there ways to build mathematics without axiomatizing? If no, is there a proof?
I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other?
Since we're all into this OpenID thing, I think it would be really useful (and quite easy) to implement a friend invitation feature that grabs a list of friends from one's webmail contact list with an option to choose who to invite to SO, SF or SU. Something in the line of Facebook's friend invitation and recommendation feature. I have 4GB RAM. When I open Firefox, IntelliJ IDEA or VS Code and some other application my memory is about used up thus my machine hangs and I can't do anything. I can't even close any applications.    Date and time are shown in the top bar so that I can view it any time without any thing typing.   If I would view memory status in this way without typing anything then I can make a decision whether to open an application or whether this application may put my machine in hanging state or not.    Is it possible in Ubuntu 18.04 LTS? What does “times” mean in the newspaper “Washington Times”?  I have always wanted to ask about this, but feared it might be so obvious that I would look like a fool. For that many times, I haven't before experienced that level of hesitation with any other question, but I think now is the right time to ask, anyway. Is it ok to answer based someone's older answer on SE? Possible Duplicate:             This question will probably be closed in a few minutes. It is a duplicate but I couldn't find the question using with meta search.  Anyway, , . In this case, is vote to close the new question as a possible dublicate not the best choice? Looks to me a little bit not ethical. What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? How can I search for one label but not another? What search query will return conversations that have the label Reading but not the label Ignore?  I've tried using label:reading -label:ignore but Gmail still returns conversations that have the label Ignore:    The query label:(reading -ignore) gives me the same results. Forward/Inverse Search Does Not work on certain tex files I have this weird issue with forward and inverse searches. I am using WinEdt + Sumatra combination. The forward and inverse search works perfectly on certain some .tex files, but fails on some others.  Currently, I am working on a .tex file and the inverse search always points to the end of a section. Where ever I double click in a particular section of the pdf, the inverse search points to the end of that section in the .tex file.  Moreover, when I try to perform forward search (Shift+F8) from the .tex file, the section in the pdf file doesn't get highlighted.  There are other .tex files where both the searches work perfectly. What could be the issue with this .tex file? Hiding winforms app from taskbar Bash built-ins with no manual entries I have installed Apache(LAMP) web server on my ubuntu machine and I want to do the above mentioned thing.I have tried creating symbolic links in the /var/www directory but it only shows files but not directories inside the directory linked.I want to redirect /var/www to my home folder so that I can access all the files and directories contained therein.Please help me. Can "zero hours contracts" be considered as not grammatical? show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$ Show that $\displaystyle\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$ for all $k,l, n\geq 0$ with $k+l \leq n.$  The $k+l\leq n$ part doesn't seem that important. The claim clearly holds for $n=0,$ and in that case $k$ and $l$ must both be zero. I think the claim can be shown by induction, but I'm not sure how to do so; I'm not sure how to use the inductive hypothesis to prove the inductive step. The sum on the LHS is not always an integer, so finding a combinatorial argument doesn't seem very feasible. For $n = 1,$ we can have $(k,l) = (0,0), (1,0), (0,1),$ and in each case the sum holds. Perhaps multiplying both sides by a number (idk ${n-l\choose k}$) might be useful? Rotate Geographic map How to have full bleed between two facing pages Indesign How to draw a return arrow from node-3 to node-1 Login required moving to Super User on iPad When moving around the sites on Stack Exchange using Safari on my iPad (not the Stack Exchange App), I am fine when moving to any site other than SuperUser.  When logged in, I can switch sites and stay logged in, but when I switch to Super User I need to log in again. If I choose not to log in and switch to another site, I am logged in there. Is there a problem with the login handling for Super User? If it helps, I log in with a Stack Exchange account, not through Facebook or Google. Update After getting fed up with the problem, I took the advice given in the answer below which was to use Chrome, and the problem still exists. The same goes for Firefox. Have I missed something?  As you can see from the screenshots below, there are no cookie settings to be found. Chrome for iOS settings page  Privacy settings page  Content settings page  Google app settings page Auto-implemented getters and setters vs. public fields I see a lot of example code for C# classes that does this:  public class Point {     public int x { get; set; }     public int y { get; set; } }   Or, in older code, the same with an explicit private backing value and without the new auto-implemented properties:  public class Point {     private int _x;     private int _y;      public int x {         get { return _x; }         set { _x = value; }     }      public int y {         get { return _y; }         set { _y = value; }     } }   My question is why. Is there any functional difference between doing the above and just making these members public fields, like below?  public class Point {     public int x;     public int y; }   To be clear, I understand the value of getters and setters when you need to do some translation of the underlying data. But in cases where you're just passing the values through, it seems needlessly verbose. Which day does "next Tuesday" refer to? At what point does next Tuesday mean      the next Tuesday that will come to pass     and no longer      the Tuesday after the Tuesday that will come to pass?   And, when does the meaning switch back? Photoelectric Effect - Dependence of current on frequency There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process?