I'm new to set theory and ordinals and such things and currently have a question.  I have a proposition which says that the axiom of choice is equivalent to this: "the power set of every well-ordered set is well-ordered."  It's trivial that the statement is true by well-ordering (which is equivalent to axiom of choice). but I can't find the proof in the opposite way. I also have a proof which I can't understand. It seems that the author tried to prove well-ordering (for any sets) using the mentioned statement. but I can't get it.  Note that I need to prove well-ordering using the mentioned statement not proving the statement with or without AC!  This is the proof:
Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.  Question: can we show the stronger result that if $X$ is well-orderable, then so too is $2^X$?  Proof of Proposition. Pick any well-ordering of $X$. Then the lexicographic order totally orders $2^X$.  More explicitly: for any two $f,g \in 2^X$, define $f &lt; g$ iff   there exists $x \in X$ such that $f(x) \neq g(x)$, and if $x \in X$ is minimal such that $f(x) \neq g(x)$, then $f(x)=0$ and $g(x)=1$.   It can be shown that $&lt;$ totally-orders $2^X$.
How can I search an Excel spreadsheet in SharePoint? Looking to be able to search an Excel spreadsheet from the SharePoint mobile app.  I have a spreadsheet with Postal Codes/Zip Codes with corresponding information. I would like to be able to search as you would in Excel.  When I upload a test spreadsheet, it opens as like a PDF. User has "Read" permissions for the document library the spreadsheet is located in.  Thanks in advance, I'm quite confused about attacking with two weapons in both hands and wondering what is required to be able to do to (in Pathfinder).  1. Base Attack Bonus Looking at the base attack bonus tables for different classes (e.g. the ) it seems like up to +5 you get only one attack per full action but starting from base-attack-bonus of +6 you get a second attack with +1.  2. Feats There is a feat . The existence of the feat sounds like you could not do two-weapon-fighting without it. However, in the description of "normal" it says:     If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. When fighting in this way you suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand. If your off-hand weapon is light, the penalties are reduced by 2 each.   3. Core Rulebook Chapter about combat If I understand the  it says that you need a full-round action do attack with both weapons in two hands.  So how do these things interact? Can I only attack with my off-hand weapon if my base-attack-bonus is high enough that I get second one with +1?  If not, how does the "multiple attacks from high base-attack-bonus" interact with dual-wielding? Assuming I reached a state of base-attack-bonus with +11/+6/+1 and have a full-round action to attack. Does that mean I can either attack with my primary weapon three times or use the primary weapon and off-hand weapon to attack only twice? Why would I then ever choose dual-wielding? Or can I attack three times with both weapons (leading to a total of 6 attacks with heavy malus) Or would I get 3 attacks with the primary weapon + 1 attack with the off-hand weapon? If so, what attack bonus do I have to use for each of the attacks? How to enable or disable services? Lightning QuickAction (API) without modal I am attempting to replace JS buttons with quick actions. These button simply call a web service method and alert the result and need no user interaction.  With quick action a modal always seems to appear and this is not desirable.   I am calling the Webservice method in the doInit function and  firing a toast event with the results and have no need for the modal popup.  Seems an oversight maybe? But is there a way to accomplish this without the modal to closely mimic the JS button functionality?  Ideally we could show the spinner without the modal   Right now I am simply calling   $A.get("e.force:closeQuickAction").fire();   at the end of the doInit but it is not the most ideal in way of a UI to do this IMHO as it flashes the modal on the screen briefly. Blade of the Ruined King Vs. The Bloodthirster Effects:  +25 attack damage +15% life steal +40% attack speed Passive  Unique: Your attacks deal 5% of the target's current health in physical damage (60 max vs minions and monsters). Active   Unique: Drains target champion, dealing 15% of the champion's maximum Health in physical damage and healing you by the same amount. Additionally you steal 30% of their movement speed for 4 seconds - 60 second cooldown. (500 range)  Cost: 3200g VS.   Effects:  +70 attack damage +12% life steal Passive: Gain +1 attack damage and +0.2% life steal per kill (maximum +30 attack damage and +6% life steal). Half of the bonuses are lost upon death.  Cost: 3200g What is better for what situations, and when should you build BT instead of BotRK and Vice Versa. Which provides more lifesteal, the only variable being the stats of the weapon? I find myself constantly questioning my decision on which weapon to get and would really love some in depth answers. Would be great if any examples given were given from the ADC side of the house. I was wondering if someone knows if there is a foot/feet symbol available in any of latex packages that I could use in my text? I would appreciate any hint.  I do really mean a foot/feet icon as in the picture below. Gandalf (as with all the Istari) did not have a corporeal form before coming to Middle-earth and was essentially ageless. We also know that Gandalf did age, albeit very slowly, while on Middle Earth. By the end of the Third Age (before battling Durin's Bane), he had the appearance of an old human male (non-Númenórean, etc) in his 70's or 80's. This leads me to wonder:  Is there any indication in Tolkien's writings that describes Gandalf's appearance (in human-looking years) when he arrived in Middle Earth? In other words, was Gandalf the Grey originally "Gandalf the Tall Dark &amp; Handsome in his 20's"?  I'm not talking about his actual age, but rather his appearance in human years. I just updated my iPhone 7 Plus to iOS 13 tonight and noticed that the post menu in the Stack Exchange app became close to unreadable, because the font/text is too large to be properly displayed. Luckily we can still recognize the icons...    (Yes, app development has been frozen for quite a while but the app continues to serve me well.) Can Jupiter be ignited? Our solar system itself contains two candidate "Earths"  One is Jupiter's moon Europa and another is Saturn's moon Titan. Both of them have the problem of having at low temperature as Sun's heat cannot reach them.    Jupiter is made up mostly of Hydrogen so is our Sun. Can we ignite Jupiter so that it will produce enough heat to warm these two earth like planets/satellites? What can be consequences of any of such attempt? I am using an APEX batch to process and insert large amounts of data. Essentially I would like it to process every row to catch any errors with the data and then rollback the entire batch if a single error is found. Is there any simple way to do this?  I have tried using a Save Point, but because they are not serialisable, it doesn't work across the entire batch.   Just to clarify, by error I mean an error that I have defined, error with validation, etc. Not an Exception thrown by the system. Good reasons to prohibit inheritance in Java? Switching to HTTPS - redirect question Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. What color would you see if you place 2 mirrors in opposit when one is a one way mirror How do I sort a std::vector by the values of a different std::vector? I tried getting a good measurement with my caliper, but the flexibility of the tire doesn't help much. Let's say I have a 26&quot;x2.125&quot; tire. I know this fits on an ISO 559mm bead seat diameter rim, and 26&quot; = 660.4mm. 26&quot;:  The OD of the tire including the tread? The OD of the tire excluding the tread? When aired up or deflated? Means it fits a 559mm ISO BSD None of the above?  2.125&quot;:  Width of tread? Width of sidewall? Inflated or deflated?  Essentially, I'm converting the tire sizes to &quot;height off rim in mm&quot; and drawing them with a wheel in a web application, so I need to know how &quot;tire size&quot; translates to this measurement. I'm somewhat assuming that the fractional sizes can be converted to the same measurement, but I'm aware that they have certain differences - so any help there would be of value as well.  has been helpful, but still leaves me with the questions above. How to Uninstall Blender Software I installed Blender for a project working on and it's way more than I need. I'd like to uninstall it, but can't figure out how to do so. It's not showing up in Programs (using Windows 9) and I can't find instructions on how to remove it in the Blender documentation. Any assistance would be greatly appreciated. Thanks. I had an article, but I lost it.  It showed and described a couple of C/C++ tricks that people should be careful. One of them interested me but now that I am trying to replicate it I'm not being able to put it to compile.  The concept was that it is possible to change by accident the value of a const in C/C++  It was something like this:  const int a = 3;          // I promise I won't change a const int *ptr_to_a = &amp;a; // I still promise I won't change a int *ptr; ptr = ptr_to_a;  (*ptr) = 5;               // I'm a liar; a is now 5   I wanted to show this to a friend but now I'm missing a step.  Does anyone know what's missing for it to start compiling and working?  ATM I'm getting invalid conversion from 'const int*' to 'int*' but when I read the article I tried and it worked great. Move item to folder when added/edited (SP2013 or O365) Creating point features with exact coordinates in QGIS?