The only knowledge I have on logic is due to a book I read a couple of years ago called  by Alfred Tarski. And in it he talks about free variables and sentences and quantifiers. But he never says what order logic he is talkig about or anything.  My question is this:  What is the difference between the two order logics and how can I tell if a statement is in one or the other.
describes the first-order vs. second-order logic as follows:     First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals.   It gives $\forall P\,\forall x (x \in P \lor x \notin P)$ as an SO-logic formula, which makes perfect sense to me.  However, in a , the poster claimed that $\forall x\forall y(x=y\leftrightarrow\forall z(z\in x\leftrightarrow z\in y))$ is an FO formula. I think this must not be the case, since in the above formula, $x$ and $y$ are sets of individuals, while $z$ is an individual (and therefore this must be an SO formula).  I mentioned this as a comment, but two users commented that ZF can be entirely described by FO logic, and therefore $\forall x\forall y(x=y\leftrightarrow\forall z(z\in x\leftrightarrow z\in y))$ is an FO formula.  I'm confused. Could someone explain this please?
Keep processes running after SSH session disconnects Are there any drawbacks to legacy BIOS as opposed to UEFI? Are there any drawbacks to installing Ubuntu 13.10 (and soon 14.04) in Legacy BIOS instead of UEFI?  I have installed 13.10 64bit in Legacy by mistake and was just wondering if it would hurt anything to simply leave it there as is?  EDIT  Everything seems to be running just fine in Legacy and all my drivers work. Adding a citation after paper is accepted I am a junior researcher in mathematics. I worked simultaneously on proving Theorem A and Theorem B. They both generalize Theorem X in two different directions. Theorem X was proved several years ago (not by me). The proofs of Theorem A and Theorem B use very different techniques, and so I decided ahead of time to write one paper for each.  I finished proving Theorem A before I finished proving Theorem B. Then, I submitted Paper A to a journal (and the arXiv) proving Theorem A. Paper A includes a survey of related problems in the introduction. After a short while, I managed to prove Theorem B, and two months after the submission of Paper A, I submitted Paper B (to a different journal, and also to the arXiv). Now, six months after the submission of Paper A, it is accepted with requests for minor corrections. There is no response yet regarding Paper B.     In this situation, is it a reasonable/common to add one/two sentences to the survey in introduction of Paper A, explaining and citing the arXiv prepreint of Paper B, in addition to the minor corrections the referees asked for? (and letting the editor and referees know that I did that in my answer to their reports which comes along with the corrected manuscript).   Note that I am not asking if it is ok to cite a preprint. This is something I do, and so do other people, at least in math. My question is about making a change, after acceptance, which was not asked by the referees and the editor. I know that if we solve the Maxwell equation, we will end up with the phase velocity of light being related to the permeability and the permittivity of the material. But this is not what I'm interested in - I want to go deeper than that. We know that the real speed of light is actually not changing, the decrease in speed is just apparent. Material is mostly empty, the light will still travel with $c$ in the spacing. The rare atoms will disturb the light in some way. So I am interested in how the atoms affect the light. Photon absorption-emission theory Some textbooks that I read explain it in a way kind of like this: In a material the photons are absorbed by atom and then re-emitted a short time later, then they travel a short distance to the next atom and get absorbed&amp;emitted again and so on. How quickly the atoms in a material can absorb and re-emit the photon and how dense the atoms decides the apparent speed of light in that material. So the light appears slower because it has a smaller “drift speed”. Interference theory But recently I realize an alternative explanation: Atoms respond to the light by radiating electromagnetic wave. This “new light” interferes with the “old light” in some way that results in delayed light (advanced in phase), this can easily be shown by using simple phasor diagram. Consequently effectively the light covers a smaller phase each second, which gives the impression of a lower phase velocity. However the group velocity is changing in a complicated way. I think that the first explanation does not explain the change in phase velocity of light. if we consider light travelling into a slab of negative refractive index non-dispersive material, let’s say the light is directed perpendicular to the slab. The phase velocity’s direction will be flipped, but group velocity’s direction in the material will not change. Only the second explanation can explain the flipped phase velocity direction. I guess that the velocity that we get in the first explanation is actually belongs to the group velocity. It makes sense to me that the front most of the photon stream determines the first information that the light delivers. So the question is What really cause the phase velocity of light to be decreased?  &quot;drift velocity&quot; of photons (they aren't the same photons, they are re-emitted all the time) phase difference between absorbed and emitted light something else  And also, I still don't really understand detailed explanation of the absorption-emission process for small light's wavelength (for large lambda compare to the atoms spacing, the photons will be absorbed by the phonons). The dispersion relation that we know is continuous and also some material is non-dispersive, therefore the absorption process must occur in all frequency for a certain range. So definitely it doesn't involve the atomic transition, otherwise it will be quantized. My guess is that the relevant absorption process gets smooth out by the dipole moment. What makes the spectrum continuous? EDIT: link for dispersion relation: Film where aliens looking like humans are waiting in an airport closed due to snowstorm I saw the movie 6 - 7 years back on cable tv, from the looks of it it seemed to be made in the late 1990's or early 2000's. Unfortunately I could not see the complete movie. The story goes like--   Some people are waiting at a small airport because snowstorm activity has cancelled all flights in and out. There are two children in the group as well (one boy and a girl). Somehow they discover there are aliens looking like humans among them. They decide to check passengers through baggage scanners (checking the skeletal structure via x-ray). They find and kill 2 aliens.   and the big twist:      5. They then realize the two children are also aliens.   any ideas what the name of the movie is ??? Short story with space microbes latching onto electromagnetic fields This is a sci-fi story set on Earth. From the depths of space comes these microbes that latch onto electromagnetic fields. Anything electronic is rendered useless. There's the usual stuff where governments try to solve the issue but make it ultimately worse. It's only when they realise that the microbes surround the planet that they realise it is futile. The story closes with people having returned to a simpler life, using medieval farming methods, not using electricity in any form, getting about by horse, etc. I think it was a short story. Specifying geography with cardinal directions So here in America we are in "the West".  Would this become "the Western world" or "the western world? "  I'll assume the same applies to "the East" (Asia). Debugging JavaScript in IE7 I need to debug JavaScript in Internet Explorer 7. Unfortunately, its default debugger doesn't provide me with much information. It tells me the page that the error showed up on (not the specific script) and gives me a line number.  I don't know if that is related to my problem. It'd be nice if it could narrow down the error to a line number on a specific script (like Firebug can). Is there an addon to debug JavaScript in IE7 like Firebug does in Firefox? Thank you! See also: Don't you think people need to know what they did wrong exactly so they can fix it and get unbanned from asking questions or posting answers? When learning a new subject, I would frequently use lecture notes found somewhere in the Internet. When writing a paper (or a master thesis, as in my case, but the rules should be similar, I believe) one should give some reference for used results which are not common knowledge, if I understand correctly. This make me wonder: what do I do if I want to reference a result I found in some notes?  The natural thing to do would be to just add these notes to bibliography. What format would be preferable for this? Note that there will generally not be much publishing information, perhaps not even a definite year and place. (A BibTeX template would be .)  Secondly, is it OK to cite such materials as a reference? Delete hiberfil.sys on Windows 7 Ok, so I know what hiberfil.sys is, and I want to use hibernation on my laptop.  But I have another hard drive active in my laptop, it's taken from my old one... and I want to get rid of the hiberfil.sys on that drive. But even with admin privileges I can't delete it.  Any ideas? How can I make a "Movie Clip" node start at a certain frame of the scene? How do I get list of all tables in a database using TSQL? What is the best way to get the names of all of the tables in a specific database on SQL Server? What's the best way to protect your house from Endermen stealing your walls? In a 2D top-down game, how can I create projectiles that have a height? What is Hawking radiation and how does it cause a black hole to evaporate? My understanding is that  isn't really radiated from a black hole, but rather occurs when a particle anti-particle pair spontaneously pop into existence, and before they can annihilate each other, the antiparticle gets sucked into the black hole while the particle escapes. In this way it appears that matter has escaped the black hole because it has lost some mass and that amount of mass is now zipping away from it.  Is this correct? If so, wouldn't it be equally likely that the particle be trapped in the black hole and the antiparticle go zipping away, appearing as if the black hole is spontaneously growing and emitting antimatter?  How is it that this process can become unbalanced and cause a black hole to eventually emerge from its event horizon and evaporate into cosmic soup over eons? The Wikipedia page () proves that $$f(x) = \begin{cases}  \exp(-1/x),  &amp; \mbox{if }x&gt;0 \\ 0, &amp; \mbox{if }x\le0 \end{cases}$$ is a non-analytic smooth function. But I don't understand why the fact that "the Taylor series of $f$ at the origin converges everywhere [where does "everywhere" mean here?] to $0$" implies the Taylor series converges to $0$ when $x&gt;0$? Can someone explain this in greater detail? I guess there should be a gap in my understanding of power series. Proving $\sqrt{ab} = \sqrt a\sqrt b$ I am currently in high school and we are studying radicals. I had asked my math teacher why $\sqrt{ab}=\sqrt{a}\sqrt{b}$ (for all a,b>0) and he tries to prove it by arguing that $a^{1/2}*b^{1/2}=(ab)^{1/2}$ (an exponent law). However, I find this proof problematic since $x^{1/2}$ is simply defined as $\sqrt{x}$, so the reasoning is circular.   My view is that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ because once we square both sides we get $ab=ab$. Since we're obviously referring to the positive root, and the function $f(x) = \sqrt{x}$ is injective, it necessarily follows that the original expressions $\sqrt{ab}$ and $\sqrt{a}\sqrt{b}$ are equivalent because for injective functions it is not possible to map distinct elements in the domain to the same element in the range ($ab$). Hence they are equivalent expressions.   My question therefore is, is my proof valid and/or rigorous (I find it convincing but maybe it's wrong; I just want to be clear) and secondly was my teacher's proof correct? I made changes to iptables config file in /etc/iptables/filter in Ubuntu and want to reload them. I read the man page and also googled but couldn't find the information. Any help will be appreciated. Time Expressions with "in" I have seen some examples about using "in" with time expressions. They give some examples as shown below:     I will leave for vacation in a month. (Exactly one month from now I   will go on vacation.)      We’ll be there in an hour.  (Exactly one hour will go by and then we   will arrive.)   But these comments confused me. Then I looked the usage of "in" up in a dictionary and it says about usage of "in" with time expressions :"before or at the end of a particular period". I think the comments and the definition in the dictionary contradict each other?    I have examples of mine:     1- Dinner will be ready in ten minutes. (Maybe dinner will be   ready at the 5th minute? or 8th. Does it strictly mean dinner will be   ready at the end of the ten minutes?)      2- My letter arrived in two days. ( Does that mean it took   exactly two days to arrive? or Could it be just one day because he/she   doesnt remember the exact time?)      3- Imagine you are on the first day of a month and your friend says:   I will be going outside the country in this month. (Does that mean he/she could go tomorrow or ten days later or at the end of the   month?)      4- He can run a mile in 5 minutes. ( Does that mean he has to   spent entire 5 minutes to run a mile? Could it be 4 or 4.5 minutes?)