I know by intuition that $\omega^{2}\cong\{k-\frac{1}{n}\mid k&gt;0,n&lt;\omega\}$ How can I find a set which is isomorphic to $\omega^3$? I would love to see an example If possible, I would like to see an example for  $\omega^n$ or $\omega^\omega$ aswell.
I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, &lt;) is isomorphic to ($\alpha, \in$).  To do it I am supposed to show the following stronger statement by induction on ordinals $\alpha$ &lt; $\omega_{1}$:  Let P($\alpha$) be the statement "For every interval (a, b) with rational endpoints, there is an A $\subseteq$ (a,b) such that (A, &lt;) is isomorphic to ($\alpha, \in$)."  I think that the purpose of breaking $\mathbb{Q}$ into intervals (a,b) is to make $\mathbb{Q}$ a countable set, but I don't see the significance of doing this or how it would make the proof easier.  I feel like I need to define a well-ordered set in (a,b) but I don't know how to do this since if a and b are both negative then there is no least element, so no well-ordering.  I believe that I am supposed to be using transfinite induction? But I don't know how P($\alpha$) implies P(S($\alpha$)) for S($\alpha$) successor of $\alpha$.  Any help would be greatly appreciated... thanks in advance.
Showing $\gcd(2^m-1,2^n+1)=1$ Why would spacetime curvature cause gravity? Possible to use draws from two distributions to get draw from distribution with density their product? Windows 10 won't let me set default apps for file extensions How many permanent SPECIAL points are available in Fallout 4? I tried control+shift+power button as well as command+shift+power button. Neither seem to work. Rebel flight suits include an ejection harness, Biggs tells Porkins to eject. This seems to suggest that Rebel fighters have an ejection system.  How would they operate for different fighters? Does the whole cockpit separate? Do they function like current aircraft ejection systems - if so what protects the pilot from the vacuum of space? How to troubleshoot unwanted white spaces smartly? Why isn't there a continuously differentiable surjection from $I \to I \times I$? I was asked this question recently in an interview. Why can't there be a a continuously differentiable map $f \colon I \to I \times I$, which is also surjective? In contrast to just continuous, where we have examples of space filling curves.  I have one proof which is direct via Sard's theorem. But the statement seems simple enough to have a proof using at most implicit (and inverse) function theorem. How to create a bootable USB stick? Is it normal for BibTeX to replace similar author names with "------"? In my .bib file, I have two entries with exactly the same author names:  @article{Seshadrinathan2010A-Subjective-St,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....  @article{Seshadrinathan2009Study-of-Subjec,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....   When cited after each other, and typeset with the IEEEtran bibliography style, the .bbl file reads:  \bibitem{Seshadrinathan2010A-Subjective-St} K.~Seshadrinathan, R.~Soundararajan, A.~C. Bovik, and L.~K. Cormack, ``A   subjective study to evaluate video quality assessment algorithms,''   \emph{SPIE Proceedings Human Vision and Electronic Imaging}, 2010.  \bibitem{Seshadrinathan2009Study-of-Subjec} ------, ``Study of subjective and objective quality assessment of video,''   \emph{IEEE Transactions on Image Processing}, 2009.   As you can see, the author names have been replaced with ------. It looks like this in the final PDF:    Is this normal behavior? Are there any expressions that describe going from a bad to a worse situation? Are there idioms or expressions in English that describe going from one bad situation to one that's even worse?  I heard "between a rock and hard place" but this describes a dilemma not really a transition. I am looking for an expression that describes someone trying to get out of a bad situation but after much effort, the situation only got worse. Detecting absolute motion inside a box This is not a contradiction and I know it is impossible but still consider a thought experiment by me and point out if something is wrong. See the following picture and then the explanation follows.    Rest frames are easy to understand. They are just for clarification. Lets move on to the moving frames. The velocity of the box is $0.1\ c$. Now a photon is emitted (I am not taking a light ray to avoid complication in discussion). After emission the box also moves a certain distance ahead. So the photon takes more than one second to reach the wall. But even the source moves that much distance ahead. When the light is reflected back, the source and the wall again move forward as the box is moving. But after reflection the wall has no role to play. We are concerned about the source. The source moves a certain distance ahead and therefore light takes less than one second to reach back. But the difference of the first case and the second case is not the same. To explain, I will give some equations.  Moving frame 1:  Velocity of the box: $0.1\ c = 30,000km/s$  Time taken by light to reach the wall $=\frac{330,000\ \text{km}}{300,000\ \text{kms}^{-1}}= 1.1s$  Distance moved ahead by the box from original position: $30,000\ \text{km}$  Moving frame 2:  distance moved ahead by the box from original position  $&gt;30,000\ \text{km}$ as the box also moves ahead after reflection.  let $d$ be the distance moved ahead by the box after reflection.  Time taken by light to reach the source: $\frac{270,000\ \text{km}\ -\ d}{300,000\ \text{kms}^{-1}} &lt;0.9s$  Therefore we see that this could possibly determine the absolute motion.  So in total we see that the light will take lesser time to reach the source. Please correct me if I am wrong and give your opinion why is it so.  EDIT 2 for clarification  The box is the compartment in space in which you are moving and I am calculating the time the light takes to reach back the source. I am not even adding velocities. Please correct me and tell me where did I add velocities. For clarification just let me give a simple example. Suppose person A is standing still and B is running towards a light source. Obviously he will see the light before person A even though the speed would be the same. So, what I am doing over here is not increasing the speed of light, but instead decreasing the distance to be covered by it. How old is the word "prolly"? Prolly is given this definition at :     Clipped pronunciation of probably.   I was reading an interesting article today that  from 1947 and that surprised me. Unfortunately, I wasn't able to find a listed source for the claim. So how old is prolly? How to fix slow wireless on machines with Intel wireless cards? Localization of ideals at all primes I need to refer to a model described elsewhere, from a section title, like  \section{Analysis of model X}    One way of referring to a model is to create a new counter, and then write  \section{Analysis of \cref{grandmodelofeverything}}   But now hyperref will complain because of invalid token, and it does not look good in the TOC either, especially since the titles of the sections the describe the models are more descriptive than  \section{Model 1}   They are currently like  \section{Adding the effect from Y processes}   Do I need to invent semi-descriptive long names like "The model by Authors with this modification" or can I live with these quirks? What's the difference between "established" and "founded" In this sentence: "CARE was ...... in 1945 to help people after World War II."    A. founded  B. built C. established D. constructed    The answer is C. But why don't we use "founded" in this case, I think they have the same meaning. Can you help me to give a reason for that? I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. When is "Y" a vowel? In school we are taught the vowels: A, E, I, O, U, and sometimes Y.    got me thinking about when the letter Y is considered to be a vowel.  I understand (perhaps incorrectly) that in words like bicycle and why it is a vowel. What about the word voyeur (as mentioned in the XKCD alt-text)?   If I've got this backwards, and Y is almost always a vowel, how can I tell when it is a consonant? Thinking back, I don't think my education ever covered the difference between them, we just memorized which letters were which.