I'm reading a book on the , and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author acknowledges that this is not a real proof of the theorem, but the way it was presented implies that it was supposed to look somewhat convincing. I, however, have absolutely no idea how the proof given even remotely relates to FLT, and would greatly appreciate it if someone could make the connection for me. Below is an almost verbatim excerpt from the book.   Theorem: There are no positive integers $x$, $y$, and $z$, and integer $n&gt;2$, such that $x^n + y^n = z^n$.  Proof. Let $R$ stand for the Russell set, the set of all things that are not members of themselves: $R= \{x : x \notin x\}$. It is straightforward to show that this set is both a member of itself and not a member of itself: $R \in R$ and $R \notin R$. So since $R \in R$, it follows that $R \in R$ or FLT. But since $R \notin R$, by disjunctive syllogism, FLT. End.
I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$)  to prove any possible statement? It has been ages since i studied logic at the university so i would prefer a simple explanation (if possible)
How to attach my repo to heroku app Javascript - alert("zoo">"house") returns true Proof that $n^2 < 2^n$ How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct? For a physics major, what are the best books/references on Greens functions for self-studying?  My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas. 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What is the difference between correlation and causation?  Pirates and Global Temperature Example  For example, how do we know when we're dealing with correlation only and not also causation here?   What Certain Philosophers Seem to Hold Regarding This  It would seem that   Hume would saycorrelation ≡ causation. Aristotle would say that in generalcorrelation ≢ causationand thatcausation ⇔ correlationonly if we are not dealing with chance events.   ("≡" means "is identical to," and "⇔" means "implies each other.") How do I pipe each command given to the shell? I'd like to edit my .bashrc so that every command executed on the shell is piped to something, for example:   $ sudo apt update   _________________ &lt; sudo apt update &gt;  -----------------     \   ^__^      \  (oo)\_______         (__)\       )\/\             ||----w |             ||     ||   I have managed something rather similar, but not entirely:  $ bash $ exec &gt; &gt;(cowsay) $ echo "Hello AU!" $ exit  _______ &lt; Hello AU! &gt;  -------     \   ^__^      \  (oo)\_______         (__)\       )\/\             ||----w |             ||     ||   It is not the desired result, as it only happens after exiting the current shell.  It's mainly for fun/learning purposes. I have a typical scientific manuscript in a LaTeX .tex file, and I need to convert it to MS Word .doc file.  The reason for having to convert to MS Word is I'm submitting the manuscript to an academic journal and they only accept MS Word (I know...)  The manuscript includes title page, figures, tables, equations (inline and in their own align environment), footnotes, bibliography, and an annex.  The tables are in their own separate tables.tex file, which I include using the \include{tables} command.  Most tables take up a whole landscape page, and were generated sing the package pdflscape.  I am using Windows 7 Professional.    My plan is to use pandoc to go from .tex to .odt, open the latter in Libre Office, and convert to .doc.  I have read a related  but it is too general.  Similarly the examples in the Pandoc  are too simple.  I have played around but I am unable to accomplish what I want.  This is surprising since converting a scientific manuscript is probably the most common use case for Pandoc. Here are some sample failures:  Example 1  I open a command line in the project folder, and execute the following:  pandoc -s document.tex -o document.odt   I get this error message:  pandoc: figure1: openFile: does not exist &lt;no such file or directory&gt;   where figure1 is the name of a figure file (e.g. figure1.png) in the project folder referenced in a line as \includegraphics[width=5.8in]{figure1}.  I suspect pandoc expects a .png extension but not sure how to provide it.  Example 2  Next I try .html, and excute the following:  pandoc -s document.tex -o document.html   The program executes fine.  I open HTML file.  Footnotes are there but figures are missing, tables are displayed as LaTeX, bibliography is missing, in-line math displays well, but math in align environment does not, section labels are displayed, and some other minor issues.  So given that mine is probably a typical use case scenario, my question is this: What commands should I use to get the .odt file I want?  I could not find a fully worked out example on the web.    Here is a specific list of errors.  I'll update how I corrected them based on community suggestions:   Figures not rendering.  Solved by adding .png extension to .tex file in \includegraphics command.  Now figures are included but they are huge, with half of each figure outside the page. No bibliography.  Solved. First, I have one huge consolidated Latex .bib file where I keep all my citations.  I manage it using .  This was giving me problems as I do not keep the cleanest .bib file in town.  So I reduced the problem by using a neat trick in JabRef that allows you to subset your master .bib file using the .aux file generated by Latex when compiling your manuscript.  In JabRef click on Tools > New Subdatabase based on AUX file.  This way I generated a much smaller biblio.bib file with only the articles referenced in my manuscript.  Running pandoc -s document.tex -o document.odt --bibliography=biblio.bib did the trick. Display math. Math in \begin{align} environment displayed in verbatim \latex;  (A partial solution is to use the  Libre Office extension.  Copy and paste the latex math code in the .odt file created by Pandoc into the equation editor, and so on.  Surely this could be built into a macro that can post-process all remaining math.)  UPDATE: Display math works very well using --mathjax extension. Inline math. Inline equation do not always render properly.  Bold math is a problem.  E.g. $\Sigma=\sigma^2\bm{I}$ displays as $\Sigma=\sigma^2\bm{I}$; Labels are displayed (e.g. section labels show as [sec:empirical] blah blah]; All tables display as raw latex. Why do I get a segmentation fault when writing to a "char *s" initialized with a string literal, but not "char s[]"? How can I `alias sudo !!`? How can I decode HTML characters in C#? I have email addresses encoded with HTML character entities. Is there anything in .NET that can convert them to plain strings? 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