After a few years of university mathematics I just realised that I do not really know the difference between an axiom and a definition. With some &quot;research&quot; on the interwebs (, ). I think that axioms are definitions that build the foundation of a Theory. E.g. we have group axioms because we build a theory on them. But they are nothing else than a definition and we can check if some object satisfies the axioms/definition to be a group. Lets look at continuity. Its a definition that says when we call a function continuous. We can check if that is the case or not. Its not called an axiom because we first need to define other things to get to continuity, like topology or metric spaces. There can be multiple equivalent definitions, which is the case with continuity. What do you think? Is that a somewhat accurate understanding of the terms?
I am wondering what the difference between a definition and an axiom.  Isn't an axiom something what we define to be true?  For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the successor of any natural number. Why can't we define 0 to be the natural number such that it isn't the successor of any natural number?  In general, why is an axiom not a definition?   Are there cases where we can formulate definitions as axioms or vice versa?
Sensor for detecting airsoft bullet I'm trying to make airsoft chronograph/chronometer (a device that measures airsoft bullet speed). The idea is to measure time in which bullet is traveling from point A to point B, and then I can calculate bullet speed. Bullet is shot into plastic pipe with diameter of 32mm. I've tried to put on one side IR LED and on the other side phototransistor (BPW 40), both at the start and end of the tube. Something like this:  Blue circles are phototransistors, red circles are IR LEDs and white circles are airsoft bullets. When bullet passes through area where bullet 2  is drawn (just in between IR LED and phototransistor), everyting works perfectly. But when the bullet passes where bullets 1 and 3 are drawn, it isn't detected. That is expected, but unwanted behavior. So my question is: how can I detect bullets regardlessly of it's position ? I've thought about putting IR LEDs and phototransistors around the pipe (not only in one place), like this:  but this solution isn't cheapest: 5 IR LEDs + 5 BPW 40 = cca $12 times 2 (because I need it on both sides of the pipe) = $24. Is there any cheaper solution ? Is BPW 40 good choice for phototransistor ? I don't know which IR LEDs I'm using (and salesman in the store also doesn't know - he told me these are generic ones for remote controllers, like TV or DVD controller). Diameter of the pipe is 32mm and it will be long about 14cm (sensors will be 10cm apart). Diameter of airsoft bullet is 6mm. EDIT: I'll go for my second idea. I just have one more question: is it better to arrange IR emitters and photo transistors this way:  or this way:  Thanks !! How to create & execute a script file How to create a separate subnet for wireless access? I need to configure my wireless router so all wireless devices are on a different subnet (192.168.2.1).  The wireless devices should be able to access the internet, but ideally not the workstations on the LAN.  Here is my network:    All suggestions welcome! What space does the spell Cloud of Daggers fill? I am trying to workout what template  (PHB, p. 222) would be on a grid. The spell states:     You fill the air with spinning daggers in a cube 5 feet on each side, centered on a point you choose within range.   So does that mean it would look like this:     Or like this:    From what I've read online, there seems to be some suggestion that it is a single square (tile) though this does stack with the spell description of "a cube 5 feet on each side".  Clearing this up for my group would be great, and I am sure our DM will be pleased as his monster were nicely diced by this spell. Thanks. Require minimum reputation on the target site for migration How can I test for a player wearing specific armor? How can I test for a player that is wearing an iron helmet named "test1" and an iron chestplate named "test2" in Minecraft Version 1.12? Which word to use when we ask for help?  Some conditions:     We know that the person asked is able to do it. We don't know if the person asked is able to do it. How does remittance work? How does it differ from direct money transfer? I have a bank account in Germany in which I receive my salary in Euros. I want to deposit a part of it in my Indian bank account for personal savings. My Indian bank account is in INR. As I see it, I have two options:   To use a remittance service that converts my Euros into INR and deposits in my account. I am not sure how it works. The service (which I know is reliable) I signed up for asks me to deposit Euros in a certain account ("funding") which will be converted to INR and deposited in my account. How does that work exactly? Use my bank's service to make a international bank transfer to my account. Is that even possible, given that my Indian account is in INR? Will I incur higher conversion fees and lower exchange rate?   In general, what are the pros and cons of each of the above methods? And are there any other methods? How to create 2D shadows I'd like to create lights and shadows in a 2D environment, like in the image below:    I did several searches but got no results. Do you have any hint? Best way to store currency values in C++ The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? I want to cd into /var/named but it gives me a permission denied error, and when I want to use sudo to do this I am not permitted. What is the technical reason for this, and is it possible to do this some other way? Finding the nth non square free number We all know what  numbers are. I was more interested in checking an efficient way to find nth non square free number. Full List of inventory items After beating the game, I still have several empty inventory spots.  I most likely missed several items in my addicted frenzy to finish the game.  What are all the items that can be found in the game? And where can they found? I came across a map and when I was playing I found some coloured and bolded item names. I thought it isn't possible as at most you can only use the custom name tag to rename things! Is there a way to colour and bold item names in vanilla Minecraft? External programs are OK. Oh, and the wiki page of formatting codes doesn't really do much. No space left on device -- but there is space I'm running an Ubuntu 12.04.1 server in a VMware virtualized environment that has a strange issue. Twice now, at seemingly random times, the machine has suddenly been unable to write to the root partition despite plenty of space seemingly available. I'm at a loss trying to figure out what's wrong.  "df -h" shows all mounted volumes with 40% or more free space. Both times this has happened, rebooting resolved it for a few minutes, and then it was back to saying "no space left on device." Restoring it from a snapshot seems to fix it for longer, but since this has now happened again, I'm not sure what to do with it.  If it helps, the machine is a network print server running CUPS 1.5.3 and Samba 3.6.3. I'm having trouble distinguishing between the concepts of scedasticity and stationarity. As I understand them, heteroscedasticity is differing variabilities in sub-populations and non-stationarity is a changing mean/variance over time.   If this is a correct (albeit simplistic) understanding, is non-stationarity simply a specific case of heteroscedasticity across time? Definite article before name of languages Should I use the definite article before name of languages?   (the?) Persian language (the?) German language (the?) Arabic language (the?) Russian language (the?) English language   Some language name may also refer to other things, so you need language word after their name. I can type:  dirname ~/home/blah/file.zip   for instance, and this works fine alone, but when I use this syntax as the parameter for some command it always reads 'dirname' as the desired directory. i.e.:  cd dirname ~/home/blah/file.zip bash: cd: dirname: No such file or directory   Basically, how do I get 'dirname file' to be read as one entity? I am looking for an adjective/noun for someone who is able to speak/ makes speeches very well and uses words effectively to influence/impress people.