The proofs of this fact I've seen all rely on appealing to AC at some point or another. But is this required? Is there a choiceless proof?
Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$  in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th constructiable set, $|x|$ is the cardinals of $x$.  I am learning a set theory, and I have read a proof by using AC. The AC is used only when $\alpha$ is a limit ordinals during the transfinite inductive method for $\alpha$ . I wonder if there is a proof without AC. 　THANKS!
Python function attributes - uses and abuses Not many are aware of this feature, but Python's functions (and methods) can have . Behold:  &gt;&gt;&gt; def foo(x): ...     pass ...      &gt;&gt;&gt; foo.score = 10 &gt;&gt;&gt; dir(foo) ['__call__', '__class__', '__delattr__', '__dict__', '__doc__', '__get__', '__getattribute__', '__hash__', '__init__', '__module__', '__name__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__str__', 'func_closure', 'func_code', 'func_defaults', 'func_dict', 'func_doc', 'func_globals', 'func_name', 'score'] &gt;&gt;&gt; foo.score 10 &gt;&gt;&gt; foo.score += 1 &gt;&gt;&gt; foo.score 11   What are the possible uses and abuses of this feature in Python ? One good use I'm aware of is 's usage of the docstring to associate a syntax rule with a method. But what about custom attributes ? Are there good reasons to use them ? How to parse a month name (string) to an integer for comparison in C#? I need to be able to compare some month names I have in an array.  It would be nice if there were some direct way like:  Month.toInt("January") &gt; Month.toInt("May")   My Google searching seems to suggest the only way is to write your own method, but this seems like a common enough problem that I would think it would have been already implemented in .Net, anyone done this before? I want that my monitor turns off when I am inactive for a while, but I do not want that I am prompted for a password when I revoke my pc. In Ubuntu 16.04 it was possible to disable the lock but still enable the screensaver as explained .  In Ubuntu 18.04 I entered the following in terminal:  dconf-editor   and disabled ubuntu-lock-on-suspend:    However, since I made this change, my monitor never suspends anymore. I tried restart but it didn't help.  Any ideas how I can disable lock but still enable suspend mode? Should I change the URLs in the XML sitemap after 301 redirecting them? I am using 301 redirects for all URLs.  Should I change sitemap URLs now?  The sitemap currently contains the following URLs:   example.com/cat/1.html example.com/cat/2.html example.com/cat/3.html   Should I change the sitemap's URLs to a new destination?   example.com/cat/year/1.html example.com/cat/year/2.html example.com/cat/year/3.html Change the height of a function being plotted Proving a distance between molecules defines a metric space. A DNA molecule can be represented as a string of symbols $A$, $C$, $G$ and $T$, such as  $$GGATAATTCTAG. . .GACCGTACCC$$ For the purposes of this question, we will assume that all DNA molecules contain the same (large!) number $N$ of symbols. Thus, a DNA molecule is an $N$-tuple $x = (x_1, . . . , x_N)$ where $x_i  ∈ \{A,C,G,T\}$ for each $i$.  Define the distance between two DNA molecules $x$, $y$ as the number of elements $i ∈ {1, . . . ,N}$ such that $x_i \neq y_i$. Prove that this defines a metric on the set of DNA molecules. Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea. What word indicates spatial data within the geographical and geological domains? In particular, what word is precise enough for a science journal article (i.e., formal prose)? Candidates might include &quot;spatial data,&quot; &quot;geodata,&quot; &quot;geospatial data,&quot; and &quot;geographic data.&quot; &quot;Spatial data&quot; seems too generic (not necessarily within the geographical and geological domains). What is the consensus, if any exists? Is Java viable for serious game development? Find all solutions to the equation $(z+1)^6=z^6$. Express the solutions in simplified rectangular form. Hint: Clearly $z=0$ is not a solution. $$(z+1)^6=z^6 $$ $$\frac{(z+1)^6}{z^6 }=1$$ $$\left(\frac{(z+1)}{z }\right)^6=1$$ $$\left(\frac{(z+1)}{z }\right)^6=\cos(2*\pi*k)+i\sin(2*\pi*k)$$ $$\frac{(z+1)}{z }=\left(\cos(2*\pi*k)+i\sin(2*\pi*k)\right)^{1\over6}$$ $$1+\frac{1}{z }=\cos({2*\pi*k\over6})+i\sin({2*\pi*k\over6})$$  $$\frac{1}{z }=(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3})$$ $$\frac{1}{(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3}) }=z$$ I dont know how else to solve this. A lot of help is appreciated. Also i know that k should go from 1-5. Because the beginning of the letter is moved a bit to the left when in italics mode, the breve also needs to move to the left,otherwise it is no longer under the letter.  \documentclass{article} \usepackage{semtrans} \begin{document} Right position: \U{h}  Wrong position: \textit{\U{h}} \end{document}     Any ideas how to correctly use a breve with italics? I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. Downloading all photos from iCloud Photo Library without Photos app Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $.  Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be unbiased.  I thought about simple estimator $ \hat{\theta} = \max \left( x_i \right) $.  Based on simulation it is not biased, yet I couldn't show it analytically.  Could anyone, please, show it is unbiased?  BTW, I could easily find another, easy to prove, unbiased estimator, $ \hat{\theta} = 2 \mathrm{mean} \left( {x}_{i} \right) $ As Scott Myers wrote, you can take advantage of a relaxation in C++'s type-system to declare clone() to return a pointer to the actual type being declared:  class Base {     virtual Base* clone() const = 0; };  class Derived : public Base {     virtual Derived* clone() const };   The compiler detects that clone() returns an pointer to the type of the object, and allows Derived to override it to return a pointer to derived.  It would desirable to have clone() return a smart pointer that implies transfer of ownership semantics, like the following:  class Base {    virtual std::auto_ptr&lt;Base&gt; clone() const = 0; };  class Derived : public Base {     virtual std::auto_ptr&lt;Derived&gt; clone() const; };   Unfortunately, the relaxation of the conventions does not apply to templated smart pointers, and the compiler will not allow the override.  So, it seems I am left with two options:   Have clone() return a "dumb" pointer, and document that clients are responsible for disposing of it. Have clone() return a smart base pointer, and have clients use dynamic_cast to save them to a Derived pointer if they need it.   Is one of these approaches preferred?  Or is there a way for me to eat my transfer of ownership semantics and have my strong type safety too? How do I use F3 to get my coordinates when my laptop makes F3 a special key? I can't get the coordinates on my computer because it keeps changing the brightness.  I have a windows 8 Samsung, and I'm playing Minecraft version 1.6.4. Is there a way so that I make make it when I left click/right click with a certain item it applies an effect? I know that you can apply attributes to wearing armor and things but I want it so when you use the item it applies an effect on your player.  For example, let's say I want something so that when I right click with a certain item (let's say an iron ingot that was given to me) I'll then have a jump boost for 10 seconds. So kinda like a potion in a way.  Is this possible with command blocks/commands/any mods? Is there any Sound enhancers/equalizer? Factoring $x^n + y^n$ over the integers How to enable automatic update in wordpress running on IIS7 and Windows 2008? I have a dedicated server using Windows 2008 and IIS7 running 5 blogs with Wordpress. I can't use the autoupdate feature of the wordpress neither the automatic plugin instalation.  I believe I must set writing permissions in some temporary folder, the question is which one. Do you have any idea?