Zorn's lemma states that, if X is a partially ordered set such that every chain in X has an upper bound, then X contains a maximal element.  I think that $(Z^+,\le)$ satisfies the conditions of Zorn's lemma. The order is partial and every totally ordered subset has an upper bound in $Z^+$. Yet, since $Z^+$ is infinite, it has no maximal element.  Where is the fault in my reasoning?
Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.  Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?  There's something I don't get in Zorn's Lemma statement.
I have installed QGIS3 from the AUR package, as well as all optional dependencies:   However, when launching, I receive the following error:  Couldn't load plugin 'processing'   ModuleNotFoundError: No module named 'osgeo'  Traceback (most recent call last):   File "/usr/share/qgis/python/qgis/utils.py", line 311, in loadPlugin     __import__(packageName)   File "/usr/share/qgis/python/qgis/utils.py", line 664, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/usr/share/qgis/python/plugins/processing/__init__.py", line 28, in      from processing.tools.dataobjects import *          # NOQA   File "/usr/share/qgis/python/qgis/utils.py", line 664, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/usr/share/qgis/python/plugins/processing/tools/dataobjects.py", line 51, in      from processing.algs.gdal.GdalUtils import GdalUtils   File "/usr/share/qgis/python/qgis/utils.py", line 664, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/usr/share/qgis/python/plugins/processing/algs/gdal/GdalUtils.py", line 35, in      from osgeo import gdal   File "/usr/share/qgis/python/qgis/utils.py", line 664, in _import     mod = _builtin_import(name, globals, locals, fromlist, level) ModuleNotFoundError: No module named 'osgeo'   Python version: 3.6.4 (default, Jan  5 2018, 02:35:40) [GCC 7.2.1 20171224]  QGIS version: 3.0.0-Girona Girona, exported   Python Path: /usr/share/qgis/python /home/thiago/.local/share/QGIS/QGIS3/profiles/default/python /home/thiago/.local/share/QGIS/QGIS3/profiles/default/python/plugins /usr/share/qgis/python/plugins /usr/lib/python36.zip /usr/lib/python3.6 /usr/lib/python3.6/lib-dynload /usr/lib/python3.6/site-packages /home/thiago/.local/share/QGIS/QGIS3/profiles/default/python   It seems to be a path issue, but I am not sure how to solve it. 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I tried working with Dupliverts but that didn’t work out well. Why do all SSH-RSA Keys begin with "AAAAB3NzaC1yc"? Overriding some DNS entries in BIND for internal networks Was the "Eye of Sauron" Sauron's actual physical form? During the time frame of the Lord of the Rings, was the Eye of Sauron Sauron's actual physical form? If not, what form did he have? Recovering accidentally deleted files I accidentally deleted a file from my laptop. I'm using Fedora. Is it possible to recover the file? Rendered image looks pixelated When I hit F12 to render an image and zoom in it is quickly pixelated. How can I render a higher res display? I have looked through the settings and couldn't find a solution. By default after a new user signs up, a redirect to the home page happens.  But in such case I want to redirect such user to another page like node/35.  In this node/35 I then want to display some content. My machine has a SSD (fairly small) and a Hard disk (large). I want to have windows and a couple of programs on the SSD but I want to have the documents folders (such as my documents and my downloads) on the HD. I also want the default install directory pointing at the larger HD. Can anyone advise or point me to a relevent guide? Thanks. Sponsor Gallery, Page, or Something Like That? Why does the compiler say "a constant value is required" for the first case...the second case works fine...    switch (definingGroup) {     case Properties.Settings.Default.OU_HomeOffice:           //do something           break;     case "OU=Home Office":           //do something         break;     default:         break;  }   also tried...  switch (definingGroup) {     case Properties.Settings.Default.OU_HomeOffice.ToString():           //do something         break;     case "OU=Home Office":         //do something         break;     default:         break;  }   ...same error  Here's the Properties.Setting code  [global::System.Configuration.ApplicationScopedSettingAttribute()] [global::System.Diagnostics.DebuggerNonUserCodeAttribute()] [global::System.Configuration.DefaultSettingValueAttribute("OU=Home Office")] public string OU_HomeOffice {     get {         return ((string)(this["OU_HomeOffice"]));     } } Answer list only displays first 30 answers The answer list only displays first 30 answers, for questions with lots of answers - no pagination or lazy-loading once you reach the bottom, to view the other answers.  Personally counted every answer twice. Gravitational slingshot of light using a black hole/massive object Wikipedia has this page on  using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very oversimplified manner.   That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? What limitations are stopping it from doing this?  Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this. Note: If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school.