I have learned that ZFC has not been proven consistent, and that further more if one were to start from ZFC and prove ZFC consistent, this would imply that ZFC is not consistent, due to Gödel.  A few questions about what I just said: Does this mean there is a possibility that ZFC is inconsistent? Are there some really good (mathematical) arguments for why, even though ZFC might be inconsistent, it probably is consistent? (or at least that we should still use it even if we aren't sure it's consistent?) What would happen if somebody were to prove that ZFC were inconsistent? Would we have to throw out all of our old proofs? And lastly, does Gödel's work imply that, in order to prove ZFC consistent (without automatically proving ZFC inconsistent in the same stroke) we would need to make more assumptions outside of ZFC? In other words, is there hypothetically a way of proving ZFC consistent without making more assumptions?  I would appreciate answers to even one of these questions.
If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur?
Can't preview smoke simulation on Blender 2.82 Whenever I add a Quick Smoke effect, I can't preview it on the viewport unless I bake it.   And Yes, I added the Smoke Domain on the object.   I've seen tutorials like this  and this  that clearly shows the possibility of previewing the smoke before baking it.   I'm using Blender on a Mac with AMD GPU. I would guess that this is something related to OpenGL, as thoroughly explained here -  but that's for Windows.   Does anyone have any idea how to fix this? Probability of two uniform random numbers being more than $\frac{1}{2}$ apart Two numbers are chosen randomly and independently from the unit interval $(0,1)$.  What is the probability that they differ by more than $\frac{1}{2}$? Hint: Think of the $2$ numbers chosen as coordinates in the Cartesian plane. Draw a square with vertices at $(0,0)$,$(1,0)$,$(1,1)$, and $(0,1)$. Allowable choices for the $2$ points lie below the line $y=x-\frac{1}{2}$ or the above $y=x+\frac{1}{2}$. Find the combined area of these regions.  I need help on this. I know what $y=x-\frac{1}{2}$ and $y=x+\frac{1}{2}$ looks like but I am not understanding the problem as a whole Is there a way to delete all stack exchange accounts at the same time?  I also want to remove all traces of my posts on the site. Is that possible?   Will I have to delete all of my posts manually?  Can I use the stack exchange API for this? I looked at MyAnimeList, there is a classic series, it's Devilman, had an anime adaptation in January 5th, 2018. I am interested to watching this series. Quick search for these series, it has a lot of adaptations since 1970's.   Do I have to watch all of those serieses before started to watch Devilman: Crybaby? Are these series unrelated to the past series?  Note: I have never watched or tried to watch this series, so I have no idea about this franchise. Strong Induction Proof: Fibonacci number even if and only if 3 divides index The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \&amp; \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$   Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green}{n = 1 }}$ $2 \mid F_1$ is false. Also, $3 \mid 1$ is false. The implication [False $\iff$ False] is vacuously true.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Induction Hypothesis}}}$ Assume that $2 \mid F_i \iff 3 \mid n$ for every integer $i$ with $1 ≤ i ≤ k$.  $\bbox[5px,border:1px solid green]{\color{green}{k + 1 \text{th Case}}} \;$  To prove:  $\quad 2 \mid F_{k + 1} \iff 3 \mid k + 1.$   $\bbox[5px,border:1px solid green]{\color{green}{n = k + 1 = 2}} \;$ $2 \mid F_2$ is false. Also, $3 \mid 2$ is false. So [False $\iff$ False] is vacuously true.  Hence assume that $k + 1 ≥ 3.$ We now consider three cases:  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 1: } k + 1 = 3q}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 \require{cancel}\cancel{\mid} (k − 1)$. By the ind hyp, $3 \require{cancel}\cancel{\mid} k    \iff F_k$ odd &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Since $F_{k+1} = F_k + F_{k−1}$, thus $F_{k+1}$ = odd + odd = even.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 2: } k + 1 = 3q + 1}}$  Thus $3 | k$ and $3 \require{cancel}\cancel{\mid} (k − 1).$ By the ind hyp, $3 | k \iff F_k$ even &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Thus $F_{k+1}$ odd.  $\bbox[5px,border:1px solid green]{\color{green}{{\text{Case 3: }} k + 1 = 3q + 2}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 | (k −1).$ By the ind hyp, $3 \require{cancel}\cancel{\mid} k \iff F_k$ odd and $3 \mid (k − 1) \iff F_{k - 1}$ even. Thus $F_{k+1}$ odd. $\blacksquare$  $\Large{1.}$ Does the proof clinch the $(\Leftarrow)$ of the $(k + 1)$th case?   $\Large{2.}$ Since the recursion contains $n, n - 1, n - 2$, thus the recursion "time lag" is $3$ here. So shouldn't $3$ base cases be checked?   $\Large{3.}$ Further to #2, shouldn't "assume $k + 1 \geq \cancel{3} 4$" instead?   $\Large{4.}$ Shouldn't the $n = k + 1 = 2$ case precede the induction hypothesis?  I referenced . Source: Exercise 6.35, P152 of Mathematical Proofs, 2nd ed. by Chartrand et al    Supplement to peterwhy's Answer:  $\Large{1.1.}$ I wrongly believed that all 3 Cases proved the $\Leftarrow$. I now see that Case 1 is $\Leftarrow$ via a Direct Proof. Cases 2 and 3 are $\Rightarrow$ via a Proof by Contraposition. Nonetheless, how would one foreknow/prevision to start from $3 \mid n$  for both directions of the proof? How do you mark a struct template as friend? What is a "locked" post? A post can be locked.   What does it mean to lock a post? Who can lock a post? When should a post be locked? Partial adb restore ICS devices support creating backups using adb backup, and restoring them using adb restore. Using adb backup, one can select that everything should be backed up, or only a certain package.  However, adb restore lacks all the options of adb backup. When I call adb restore backup.ab, where backup.ab is only the backup of a single package, the phone shows me a warning dialog saying that all settings are going to be lost after restoring the backup. Is it true that restoring a single package backup wipes all data on the phone? If yes, is it somehow possible to restore a single package using adb restore? Which equals operator (== vs ===) should be used in JavaScript comparisons? ubuntu 14.04 login loop problem 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 Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $ Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... Why use a dedicated light meter instead of the one built into the camera? Sorting an Array of Objects in PHP In a Specific Order I have two arrays in PHP. The first array ($author_array) is comprised of user_ids in a particular order, like so: (8, 1, 6)  The second array ($user_results) is comprised of an array of objects like so:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         ) )   I'd like to "sort" the second array so it's in this order, which matches the order of the values in the first array of (8, 1, 6). So it'd look like this:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         ) )   I'm weak on data structures. How could I do this? :-)  Thanks in advance for your help!  -Bob What is the emergency phone number in different countries? A worker carries two bags. Each of the bags initially contains N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point the worker picks an empty bag. Find the probability that the other bag contains exactly m nails.  My reasoning:  The desired probability: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$, where A, B - sets of nails in each bag. Then we can "divide" this probability (I suppose that I deal with independent events):  $P(|A| = 0\ and\ |B| = m) = P(|A| = 0) \cdot P(|B| = m);$  $P(|A| = 0) = (\frac{1}{2})^N$, where $|A| = |B| = N$ (initial condition).  $P(|B| = m) = (\frac{1}{2})^{N - m}$  Result: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$ = $C_2^1 \cdot (\frac{1}{2})^{2N - m}$  Am I wrong or it is correct? I was asked to evaluate the following sum: $$\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$$  I'm trying to use $$\int_{0}^{\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ However, it doesn't seem to work. Any help is greatly appreciated. We know that the needle that is used in a compass is a permanently magnetized ferromagnetic material and commonly steel is used.  If we used an unmagnetized iron needle instead, would it still align with Earth's magnetic field lines? If yes, how? How can I convert a string to boolean in JavaScript? Can I convert a string representing a boolean value (e.g., 'true', 'false') into a intrinsic type in JavaScript?  I have a hidden form in HTML that is updated based upon a user's selection within a list. This form contains some fields which represent boolean values and are dynamically populated with an intrinsic boolean value. However, once this value is placed into the hidden input field it becomes a string.  The only way I could find to determine the field's boolean value, once it was converted into a string, was to depend upon the literal value of its string representation.  var myValue = document.myForm.IS_TRUE.value; var isTrueSet = myValue == 'true';   Is there a better way to accomplish this? As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a prime, $x^{p-1}+x^{p-2}+\dots+x+1$ is irreducible over $\mathbb{Q}$?