It is easy to prove that $\omega_1$ does not have an order-preserving injection into $(\mathbb R, &lt;)$: to each $\alpha \in \omega_1$ we could assign the interval between the image of $\alpha$ and $\alpha + 1$, and this gives uncountably many disjoint open intervals in $\mathbb R$, contradicting its separability.  Intuitively it seems to be that if $\alpha &lt; \omega_1$, there should be an order-preserving injection into $\mathbb R$, but I am having trouble constructing it. If we have $\alpha \to \mathbb R$, we can compose it with a homeomorphism $\mathbb R \to (0, 1)$, and find $\alpha + 1 \to \mathbb R$ by sending $\alpha$ to $(0, 1)$ and $\{\alpha\}$ to 2. However, the limit step is more difficult: if $\lambda &lt; \omega_1$ is a limit, then it is a limit of countably many well-orderings which inject in an order-preserving manner into $\mathbb R$, but you cannot expect these to be "compatible" in a way obvious to me.  Is there a way to make this (or another argument) work? Or are there countable ordinals with no order-preserving injections into $\mathbb R$?
Yesterday, Asaf Karagila's  sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use the Axiom of Choice.  Since the arguments used to prove this can be shady on their use of Choice when formulated tersely, fully fleshed out answers are appreciated.  To be concrete, this question asks for proofs of the following fact:     All countable ordinals $\alpha &lt; \omega_1$ are order-embeddable in the rationals $\Bbb Q$.   Especially answers avoiding the Axiom of Choice are appreciated.    Of relevance and related interest is  (but it's not a duplicate, since that question asks for explicit embeddings).
In my flagging summary under the statistics, there is 1 flag listed as "disputed".  This is the first time I've ever seen a disputed flag in my summary, so I'm just curious, what is a disputed flag? How would you even dispute a flag? Is this the same as voting a flag as invalid? Express js - can't redirect Powershell Parameters Origin of the meaning of joe How do you report percentage accuracy for glmnet logistic regression? How do I obtain the "final" standard deviation from a series of values containing individual (also null) SD values? I have a list of values, say:  AVGprice SDprice count year 10          3      5   1999 20          2      3   2000 30          8      20  2001  40        undef    1   2002   where AVGprice is the average of a certain price and SD its respective standard deviation for that particular year. Count represents the number of occurrences.  What's the best way to calculate the final AVGprice and SDprice for the whole period 1999-2002? For the former, I'd say a weighted average will do, but I am not sure about the final standard deviation - specially because of the undefined value. Standard error of a ratio I have a linear regression and two estimates, say A and B and their standard errors. I need to find the standard error of a ratio A/B [or A/(1-B)].  I guess the main problem is that I don't know the correlation between A and B. I also guess that makes this unsolvable, since the correlation plays a major role. Is this correct?  If I knew it, what would be the way to calculate the standard error? I'm using python\pyodbc and would like to access the second result set of a stored procedure. As near as I can tell, pyodbc does not support multiple result sets. Additionally, I can't modify the stored procedure. Are there any options to access the second result set using SQL or some other work-around? Perhaps create a second stored procedure that only returns the second result set of the first? MacBook Pro 13" Touch bar permanently blank I'm on a new 2019 MacBook pro 13”. The TouchID fingerprint sensor is working. But my touch bar has gone completely blank. I tried restarting my MacBook Pro but it did not help.    I've also done a factory reset, SMC, NVRAM/PRAM and an Apple diagnostics, hoping to fix the problem but it did not work.  After re-install macOs, I was able to see touch bar configuration on System Preferences -> Keyboard menu:    I am not able to see touch bar option and it is completely dark black.  I tried resetting the touchbar through the terminal by using the below syntax:   sudo pkill “TouchBarServer” sudo pkill "Touch Bar Agent" sudo pkill ControlStripAgent   I have tried to verify whether they are running by using below command and could not find any process running.   ps -A | grep -i "touch bar agent" ps -A | grep -i "controlstrip”   I can see the control strip and touch bar server is running in activity monitor:      I could see this is a common issues across mac book pro and customer don’t have any proper fix for it. Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct. Change \headheight for a \pagestyle I have a document class that auto generates header from some user input. The problem is the first page should have a different page header than the others and the headline can be multi line. So \headheight is too small. Here is a demonstration of the problem:  \documentclass[twoside]{scrartcl} \newcommand{\longfirstpagehead}{% \begin{minipage}{10cm} some\\long\\text\\text\\text\\text\\text\\text \end{minipage}} \newcommand{\normalhead}{normal} \usepackage{scrpage2}  \newpagestyle{firstpage}{%     {\longfirstpagehead\hfill} %twoside links     {\longfirstpagehead\hfill} %twoside rechts     {\longfirstpagehead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %  \newpagestyle{normal}{%     {\normalhead\hfill} %twoside links     {\normalhead\hfill} %twoside rechts     {\normalhead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %    \pagestyle{normal} \AtBeginDocument{     \thispagestyle{firstpage}}  \usepackage{blindtext} \begin{document}     %\vspace*{1cm}     \blindtext      \blindtext     \pagebreak \end{document}   I tried to read the height with \settototalheight and:   add a \vspace at beginning of document (unnice hack):  still the bad box warning (of cause) first section has different alignment  manually change \headheight in pagestyle  text height and page foot is wrong (can't change this because some scoping?)  use Komascript in pagestyle  \recalctypearea warning  geometry in pagestyle  pretty much the same problem like the manual attemp    So I would like to change the \headheight for the pagestyle firstpage, but the page should still have the same page margins, etc. The problem is that this has to work with \thispagestyle. (Better Ideas are also welcome) Something like \thispagegeometry in geometry would be great. ;) Simple property of a valuation on a field Refreshing open attribute table in ArcMap using ArcPy rather than ArcObjects? I am running the Calculate Field GP tool from the arcpy in ArcMap. When a row value is updated, it does not show up untill I use the Reload Cache tool (Table Options > Reload Cache). How do I refresh the attribute table automatically without using ArcObjects?  I have tried updating the definition query as @blah238 suggested , but this doesn't help. Resetting the data source is not an option because the feature class might be in the editing session and I don't want to lose the connection to the source data. Autodesk Maya symbol lookup error on start after installing AMD proprietary driver on Ubuntu 18.04 I've installed Maya 2018 on Ubuntu 18.04 and tested it with the amdgpu driver (open source) and it launched without problems, but some components were not working with the open source driver so I tried amd gpu pro 18.20 (amd closed-source driver). I ran  ./amdgpu-pro-install   but since then Maya won't work and gives this warning message:  ./maya.bin: symbol lookup error: /usr/autodesk/maya2018/bin/.. /lib/libOGSDeviceOGL4-16.so: undefined symbol: EVP_CIPHER_CTX_init   My card is hd7770 and it's supported by the amd closed driver, so what am I doing wrong? A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy I edited my mesh changing the number of vertices after adding shapekeys, and it get totally messed up when the shapekeys are on. Is there a way to solve this problem without remake the shapekeys all again!? In C++, sizeof('a') == sizeof(char) == 1. This makes intuitive sense, since 'a' is a character literal, and sizeof(char) == 1 as defined by the standard.  In C however, sizeof('a') == sizeof(int). That is, it appears that C character literals are actually integers. Does anyone know why? I can find plenty of mentions of this C quirk but no explanation for why it exists. How to get exact camera view? I'm currently a beginner when it comes to 3D modelling but doing okay so far. I'm struggling a bit when it comes to the final renders of my scene, specifically on the camera view I want. Sometimes its to zoomed in and sometimes to zoomed out that it's gone out of the environment of the blender. I realized the settings for the camera on blender are for actual cameras / photography and I'm no expert on that too. A tip on how to control them precisely would be nice. Thank you. Plotting recursive function using pgfplots \addplot Is it possible to plot a recursive function using pgfplots \addplot command?  I tried like this;  \documentclass{article} \usepackage{pgfplots}  \pgfmathdeclarefunction{fac}{1}{   \pgfmathparse{(#1&lt;=1)+                 (#1&gt;1)*#1*fac(#1-1)   } }  \begin{document} fac(5) is \pgfmathparse{fac(5)}\pgfmathresult  \end{document}   But the result seems to be an infinite recursion, as the entire function is being evaluated. "Looking forward to hearing from you" - May I omit "I am" at the beginning?