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Recover ordered field

so welcome to this video this is the first video in a playlist on real analysis now we are trying to go towards a definition of the real numbers which is a starting point for all of real analysis in order to do that as part of the definition of the real numbers we are going to need to understand what is meant by an ordered field and that is the reason for this video now i warn you i personally do not find this a particularly thrilling topic however it is an extremely important part of the definition of the real numbers so it is necessary to cover this so first a warning this video is really going to be on orderings and what is meant by an ordered field it is not intended to explain what is meant by a field i'm going to assume that as prerequisite knowledge so when i say something like a field is a commutative ring where all elements bar the additive identity has a multiplicative inverse i'm hoping that you understand that statement i'm also hoping that you've potentially seen the construction of the field of rational numbers from the integral domain of the integers if you're not familiar with that content then may i suggest that you leave this video and instead go and learn some algebra because understanding algebra is more foundational than understanding real analysis and really i think it's not a good idea to try and rigorously understand the foundations of the real numbers when you don't even rigorously understand the foundations of the rational numbers so now that we've said that let's begin discussing ordering and what is meant by putting order properties on a set so let's illustrate this with an example so we have here a set that we've called the set s and this bubble here represents our set and then inside the set we have three symbols and these symbols i just pulled them out of thin air so they are of course some of math's favorite symbols so we have the symbol x we have the symbol phi and we have the symbol theta so this set consists of three elements x phi and theta now if you want to put an order on this set to make it an ordered set that is exactly what you would ensure to be imagine it to be it means that we are going to specify which elements are less than other elements we're going to put the elements in a line where uh the more leftwood you are on the line it means you're lower down the more rightward you are on the line it means you're higher up so here are some examples of orderings that i can put of these three elements so here we have x is the smallest one then we have phi in the middle and then we have feature at the top here's another example of an ordering where x again has been put at the bottom and then phi this time is at the top and then theta is in the middle so now the question is how can you capture or encode the information of this ordering for the set without the picture well what you can do is you can create a table which is what i've done here for this top ordering so you can create a table where you're going to have all of the elements of the set in the column here and also in the row here and then we're going to have inside this table answers that are either ticks or crosses for true or false if you were a complete keen computer scientist you could ask to use ones and zeros instead so you could use one third a tick and zero for a cross and what this table is going to say for each two elements is whether it's true that the element here in the column is less than or equal to the element in the row here so if we look at an entry x with x what we are asking is it true that x is less than or equal to x well of course that's true because they're equal to one another then this next entry here is it true that x is less than or equal to five well we're encoding this ordering here yes that is true x is less than phi then the next one is it true that x is less than theta yes it's true so there's a tick there of course there's going to be ticks in all of these entries here because you're asking is the element equal to itself so you're going to have ticks all along that diagonal there and then these opposite entries down here go are now going to be crosses because they're asking the opposite question so here we're asking is phi less than or equal to x but of course that's not true because actually phi is greater than x and then likewise for these two entries here and this this table is abstractly encoding all of the information of this picture here so actually you could now get rid of the picture and you could just have your set with this abstract table of information here which you could actually view as being a function from the cartesian product of the set with itself onto a set containing tick and cross or one and zero so the set with that structure that structure with all that information in is actually encoding the same thing as the picture it's describing the same thing as the picture so this with this is an ordered set just to explain what i meant about the cartesian product in a little bit more detail so the cartesian product of the set with itself is all cartesian pairs of elements from the sets it's all things of the form x y where x and y are both elements of the set so every single one of the nine combinations in this table is a cartesian pair within this set so x with x is in there x with phi is in there phi with x is in there all of the nine entries here are going to have a cartesian pair a related cartesian pair over here so actually you could view the information of this table being a function from that set of cartesian products or from the set of all the entries here to the set that contains just tick and cross which are the answers in the table here so that's what i was meaning when i said you can view this table if you like as a function from the cartesian product to the set of tick and cross or the set of one and zero if you wanted to use one for tick and zero for cross so now an interesting question to ask is what properties does a table like this have to obey in order for it to actually describe a proper order a picture like this if i just make up a table with random ticks and crosses all over the place would that actually be sensible would it mean anything would it relate to a picture like this well here we go here is one that i've made a random table where i've made up the answers and you can see quite clearly that this is absolute rubbish because i've got you know that x is not less than or equal to x and phi is not less than or equal to phi and i've got that phi is less than or equal to x but also x is less than or equal to phi this is obviously rubbish so this structure this property that we've put on the set this order property has to obey certain rules in order to actually be sensible to mean anything so there are three axioms then that order properties must obey and we'll go through these now so i've written the first one down here so axiom number one is for all x is an element of the set so now x is not just the symbol we're using here x is a general member of the set it must be true that that element is less than or equal to itself what that is saying is that all the diagonal entries in your table must be ticks they must be true so rule number one actually has a fancy name it's called reflexivity rule number two then is also has a fancy name and it's called anti-symmetry and it again is very intuitive so what it says is that if you take any two elements in the set so for all x and y in the set where x is not equal to y so we want two distinct elements in the set don't pick the same element twice then either x must be less than y or y must be less than x so one of them has to be lower than the other one but not both so whichever one is true the other one then is automatically untrue what that is saying is that if the table is going to mean something it has to be the case that it's anti-symmetric and that's the reason this property has its name so if you have these tips on this side of this diagonal down here then the opposite entries the symmetric entries have to be the opposite answer so it has to be an anti-symmetric table now it's very easy to create a table that obeys those first two properties you could do it in your sleep however with just those two properties alone it won't be the case that your table necessarily has a meaning it won't be the case that you can necessarily use it to create an order on your set of elements a picture like this the final property is the one that it's much more difficult to get your table to a bay and indeed if you were just naively making your table up it almost certainly is not going to obey this property but this is the one that is absolutely crucial in order to make it mean something in order to make it the case that you can go from the table to a picture like this so this final third property is called transitivity and it says that if you have three elements within your set which we'll call x y and z then if it's true that x is less than or equal to y and it's true that y is in turn less than or equal to z then it must be true that the relationship between x and z in your table it must be true that x is less than or equal to z so this takes two points in the table and then there's a third entry point in the table that has to be related to the answer in these two so this is much more complicated to make your table obey this property but this is obviously absolutely crucial clearly if you've got a picture like this it's going to obey this property so hopefully it's fairly obvious to you how if you have a set with a picture like this i.e a intuitive order in a picture form like this hopefully it's obvious that if you turn that into the abstract table like so then obviously it's going to obey these three properties we now need to go the other way i need to convince you that if you have a set with one of these abstract tables that obeys these three properties that that can go the other way that you can use that to make an intuitive picture of the order like so because then i've truly shown that they're interchangeable that they are equivalent so i'm going to do this now with this picture that i've drawn here so we're now going to imagine that we've got a much more complicated set s so it's going to have more than three elements here so i've drawn the table here so we've got all the elements of s up here and all the elements of s up here down here and then all the entries here are going to be filled in with ticks and crosses and we're going to say that it obeys these three properties and i want to convince you that from that i can go to having a picture of all of the elements in the great big line so let me explain how so if you wanted to put the elements in the line using this information how would you start well you'd start by just picking one of the elements of the set so pick a random one that you're going to start with and then put that on your line and then you go to element number two so you pick another element in the set and then all you need to know is does it need to go on the right hand side of this first element or does it need to go on the left hand side of this first element so you just need to ask which one's less than the other and the anti-symmetry property ensures that that is well defined without this you're not going to know because if it were true both ways then you don't know which way rounds put them but the anti-symmetry means that it's only going to be true one way so it's obvious whether you need to put it on this side or this side so if it's bigger than it you put it on this side like so and we're assuming for this picture that it was bigger than it so we put it on the right hand side here if it's less than it you put it on this side then you go to element number three and now there are three positions that element number three could go in it could go up here it could go in between the two or it could go down here so just compare it to this element element number two here and ask is it bigger than element number two and if it's bigger than element number two you're going to put it here and the fact that transitivity is true means that then element number one is forced to be smaller than element number three because element number three we've already said is greater than element number two and we knew already that element number two was greater than element number one so if our table obeyed this property which we began by saying it's going to then it's well defined element number three has to go up here then the other option is it could be smaller than element number two but bigger than element number one in which case it will go in here or it could be smaller than element number one uh in which case it goes down here and again that's well defined because transitivity ensures that if it's smaller than element number one then it has to also be smaller than element number two because element number one was smaller than element number two so transitivity ensures that this decision process is um well defined that you're not getting to situations where there could be more than one answer it forces that there's only one answer for where your next point you're adding in is going to be so now if we imagine that we've done this loads of times and we've got now loads of points in a line if we imagine the nth point so let's now imagine adding a point on here so we've only got six on our picture so we're adding in element number seven from our set s where does it go again it's got all these possible positions it could go here it could go here it could go here it could go here it could go here it could go here it could go here you need to make that decision by comparing it to all of these points and transitivity ensures that it's well defined because it ensures that wherever its position is going to be it forces it to be less than or equal to all of the elements above it so if if its position is going to be here for instance that means it's going to be less than or equal to this element here because transitivity is true and this element is smaller than these two elements that forces that this element is indeed going to be smaller than these two elements because of transitivity so transitivity ensures it's well defined likewise the other way it's here so it's less than or equal to this element by transitivity it forces it to also be less than or equal to these elements in the table which means that this is pictorially well defined imagine the situation where if you had a crap table that didn't obey this property then you could end up with a situation where this element is you know you're adding in this seventh element and in the table it says it's less than or equal to this element but greater than or equal to this element so you want to put it there but what if when you compare it to this element down here this element is actually greater than this in the table that would be a nightmare it would mean that this you don't know what to do the picture isn't well defined and the fact that transitivity is true in the table means that this is going to be well defined and you can therefore create a picture in this way from a table that obeys these three properties and i hope i've convinced you of that there so a set with an order is called an ordered set now onto the main topic of the video what is meant by an ordered field well the first thing to say it is it is not repeat it is not just a field with an ordering on it it is a field with an ordering on it that interacts nicely with the algebraic properties of the field the addition property and the multiplication property so an ordered field is a field f such that it obeys these two properties so these are the axioms the additional axioms that this structure which already obeys all of the field axioms already obeys all of these axioms of the ordering in addition it has to obey these if it's going to be called an ordered field and we're going to spend the rest of the video trying to understand what these mean so firstly you will be familiar with these sort of properties because throughout your naive maths life as a child in school you will have been using these properties when you used either the rational numbers or the real numbers which are both ordered fields so property number one is but they are not they're not given you know they have to be defined they have to be added on you have to insist that they are true uh so we have to make sure that this ordering that we're defining on our field is nice in order for these to be true if you just took a random ordering on your field it wouldn't obey these you have to pick a very nice ordering to make it the babies and of course you've been using that beautiful ordering on the rational numbers and the real numbers when you've been doing maths in school which is why i obeyed these two properties so property number one is that if you have an element x that is less than or equal to an element y then if you take some a and you add it onto each of these elements you take get the new element which is x plus a within your field and the new element which is y plus a in the field then it remains true that x plus a is less than or equal to y plus a and that holds true for all x y and a's in the field f the second property is that if you have again two elements x and y and x is less than or equal to y and then you take another element which we'll call alpha i don't know why people suddenly switch to using the greek letter here rather than the latin symbol but that is conventions if we take some alpha which is greater than zero which is an element of the field then if you multiply alpha with x and y so you get the new element in your field which is alpha x and you get the new element within your field which is alpha y uh then it again remains true that alpha x is less than or equal to alpha y so the rationals with their normal ordering that you've all intuitively known from school i.e that normal line that you put them on in school level maths that ordering that obeys these two properties however if you were to mess it up if you were to put them in some totally random order and accept that as the order that you want to have on the field of rational numbers then it's still got all of its algebraic properties but those algebraic properties are not going to interact nicely with the order anymore they're not going to obey these two properties and what i'm actually going to now show to you is that there is only one ordering that you can put on the rational numbers such that they will obey these two properties such that they will be an ordered field there's no other order that you can put on the rational numbers it would be cool if there was but it doesn't exist and we're going to explore the reasoning for that now so um let's start really abstractly so we're not going to talk specifically about the rational numbers we're just going to deal with an abstract general field f now we know that this field f is going to contain the element 0 and it's going to contain the element 1 the additive identity and the multiplicative identity so the first thing that i want to show you is that if it's going to obey these two properties then 1 must be greater than 0 in this ordering so the way we can do that is by proof by contradiction so assume the opposite is true assume that 1 is less than 0 so the multiplicative identity is lower than the additive identity then what we can do is it must be the case then if this is true that -1 is greater than zero the reason is that you can add on to both sides of this inequality here minus one so we can use this property here so we're going to take a as minus one so if you add on the inverse the additive inverse of one onto both sides which is called minus one uh then this is going to happen so one plus minus one of course by definition that's going to equal zero and on the other side we get zero plus minus one which is just going to give us minus one so we've then got the statement that minus one is greater than zero so if one is less than zero then instantly by property one you can get to the fact that minus one has to be greater than zero now this is a problem you can't have minus one greater than zero and have property number two true because if minus one is now greater than zero then we can take minus one as the alpha here and we can just multiply both sides of this by minus one and it should the inequality should still hold true so multiplying zero by minus one of course you still get zero but multiply minus one by minus one and you get one so you'll get then that zero is less than one so one is greater than zero but we started off by saying that one was less than zero so you can't then have that one is less than zero it must be the case that one is greater than zero just to add a little bit of background why is minus one times minus one equal to one well that comes from field theory so if you want it you know when you insist that fields obey distributivity in fact you don't even it doesn't even need to be a field a ring you insist obeys distributivity um it forces it to be the case that the additive inverse of the multiplicative identity multiplied by itself makes the multiplicative identity so we can demonstrate that here so if you imagine one plus minus one that of course is zero if you imagine timesing that by minus one of course the answer is going to be zero because zero times anything in a field is zero so we know this is zero but we can now apply the distributive law onto this thing that we've created here and we can say that this is one times minus one which is going to be just minus one because this is the multiplicative identity and multiplies by anything to give itself plus minus one times minus one so whatever the answer to this is it has to be something that adds to minus one to give 0 i it has to be the additive inverse of minus 1 the additive inverse of the additive inverse of 1 but that of course is just 1. so there's your little proof from field theory from algebra that minus 1 times -1 is equal to one so if you if one was to be less than zero you arrive at a contradiction here so it can't be less than zero it must be the case that one is above zero in an ordering that obeys these properties so let's go further with our derivation of meaning from this so we now have that zero is less than one we can now apply property number one again and we can say let's add 1 onto both sides and this inequality must still hold true so if you add 1 onto both sides you get 0 plus 1 must be less than 1 plus 1. so 1 must be less than and what is the name for one plus one well we call that number two the multiplicative identity added to itself in any field we call that two so we now have that one must be less than two and we can continue this on so adding one on both sides again we get that two must be less than or equal sorry must be less than three so we're building an order here for the elements in our field f and in fact what this shows is that if you have an ordered field that field must have characteristic zero i it cannot be one of these finite fields where it loops back on itself the reason being is let's say it looped back on itself at the number four so if you went on with this and added one onto both sides so you took two added one you get three but then three plus one that number that is one plus one plus one plus one four times let's say now that suddenly comes back to the element zero it's equal to zero i the field has characteristic four that it doesn't work with these axioms at all in fact it doesn't work even with the order axioms because suddenly we're going to get them that three is less than zero but that disobeys transitivity because three was less than two two was less than one sorry three was greater than two two was greater than one and one was greater than zero but suddenly you've got that three is less than zero that disobeys this law here it breaks transitivity so you can't have it the case that the field has a non-zero characteristic all of these numbers all of these elements of the field that you construct by adding one to itself a certain number of times they all have to equal different elements in the field you can't loop background again as you do in the finite fields it has to have characteristic equal to zero and we know that any field or hopefully you know from your from algebra that any field with characteristic zero is always going to contain as a subfield a structure that is isomorphic to the rational numbers and that's fairly intuitive if it's the case that has to contain separate elements for all of these ones added to themselves a certain number of times then of course it has to contain all of the positive integers and because it contains all of them it has to contain all of their additive inverses so all of the negative integers so a structure that is isomorphic the ring of integers has to exist within your field and because it's a field it then has to contain all of their multiplicative inverses which means that then it has to contain the whole of the rational numbers because after all the rational numbers is what you get from turning the integral domain of integers into a field by putting on the minimum number of elements you could in order to get an inverse a multiplicative inverse for every element so any ordered field therefore is going to have to have a subfield inside of it that is isomorphic ie equal to the rational numbers and that's certainly true for the real numbers that are going to be an ordered field they have within them a subfield that is the rational numbers so now we know that our ordered field has to contain all of the rational numbers let's continue working out what order all of those rational numbers have to be in so at present what we've got is where all of the non-negative integers are we've got this line zero one two three four five etc and we could continue going on and we did that by just using property number one and adding one onto each side of our uh inequality and we know that all of those elements have to be separate elements in the field because if it ever looped back around it would suddenly disobey transitivity so it couldn't possibly be an ordered field none of the finite fields can have orders put on them such that it obeys these properties and is an ordered field you can certainly put orders on them you can take the finite fields and put orders on them but those orders are not going to obey these two axioms here so they don't qualify as ordered fields they would literally just be filled with an order property defined on them fields with an order but not ordered fields now let's carry on let's do all of the negative integers now so we go back to our inequality zero is less than one this time we're just going to add negative one onto both sides so negative one added on to both sides negative one plus zero is negative one one plus negative one is zero so then we get that negative one is less than zero so we now know that negative one needs to go here and then we can continue on adding negative one onto both sides so then we get minus one plus minus one is going to equal minus two zero plus minus one is equal to minus 1 so we then get negative 2 must be less than negative 1 and you go on like this and of course the reason we're using these symbols minus 1 plus minus 1 we use this symbol minus two fret because it is going to be the additive inverse of the symbol that we called two which was defined to be one plus one so of course negative one is the additive inverse one so if you have negative one plus negative one and you add that to one plus one associativity and commutativity that not even communicativity just associativity comes into play there and of course they cancel out and they're going to give you the additive identity so that's why we use this nice symbol here because it is truly the additive inverse of negative two so you then end up of course with the fact that all of the integers are in this line like so which of course is the intuitive line that you've used all your life so now let's try and add the rest of the rationales into our line here so let's take for example a half so how do we work out where a half is going to be here well just ask some questions could we put a half less than zero could we have an ordering on the rational numbers where half somehow was ordered below zero and it's still obey the axis of an ordered field well the answer is no and the reason is that we can if this were true if this inequality were true and it's true that it also is going to obey the axioms of an order field then we can multiply both sides by two because we've already established that two is greater than zero so using axiom number two of an order field we can multiply both sides by two and the inequality should still hold true and that of course gives a half times two is one and zero times two is zero so that would tell us that one is less than zero which is nonsense we've already established that that's nonsense one is here already in the order so this is contradiction here so this cannot be true so the other thing must be true that the half is greater than zero but now the other half is greater than zero we know that we can multiply both sides of an inequality by a half and the inequality still has to hold true so we in particular can take the inequality one is less than two multiply both sides by half and we then get a half is less than one and we've now got exactly the position where a half should be if we're going to add it in here it should be between zero and one now finally to show you that the position of any rational is instantly fixed in this line as soon as you assume those ordered field properties so how do we do this so let's say we have our generic rational a over b and we want to know where it needs to be in the great line well in order to do that what you need to do is potentially compare it to absolutely every single element that is already in the line and ask is this rational that we're trying to position is it less than or equal to this element this element this element and that's how you'll then find its position in the line so i therefore need to tell you how you can find out whether it is true that it is less than or equal to another any other element that is potentially in this line or any other element that's potentially in this line will be another rational it will be something of the form c over d now of course if it is one of these integers then d will just equal one uh but more generally we might have put loads of rationals already in this line that you're also going to have to compare this element to so more generally it'll be something of the form c over d and the way that you can find out whether this inequality is true so there's a question mark here you're asking this question is it true that this is equal to this well what you can do is assume that these denominators are both positive integers and the reason you can do that is even if the rationals themselves are negative you can always make sure that the negative is absorbed up here at the top so the integer in the top here is going to be negative oh by the way i should have specified that when we write rationals like this we're meaning one of them the numerator is an integer and the denominator is a non-negative sorry non-zero integer so you can always make it so that the if the rational is negative that the negative is in the numerator rather than the denominator if we're talking going back to the actual construction of the rationals you can pick an element within the equivalence class of fractions that are all the same when you cancel them down you can pick an element such that the denominator is a positive integer rather than a negative integer so do that in both of these cases make sure that your denominator is not negative um then if this inequality were true we can multiply both sides through by b and through by d so multiplying both sides through by b it cancels here and then you get b times c over d multiplying both sides through by d you then get a d and it cancels here so it should be the case that this is true a d is less than or equal to b c now all of these are integers so when you multiply the two of them together you're going to get an integer answer so now this is simpler because you're just asking is this integer less than or equal to this integer and we've already positioned all of the integers so if this were true this would have to be true so all you need to do is ask is this true and if it's the case that it is true then yes this is true if it's not true then this is not true and it's the other way around so that's the way that you can when you're trying to position this new rational in your line you can actually go through every single element that's already in the line and ask is it less than or equal to each of those elements and that's how you can get the answer of where it has to be so what have we learned here we actually have learned something quite incredible so if you set your field equal to the rational numbers and you want to put an ordering on that rational numbers such that it's going to obey the axioms of an ordered field there is only one way you can do that everything is set in stone the moment you want it to obey the axioms of an ordered field instantly the positions of all of the integers within this field are set in stone and then moreover all of the other rationals all of the things of the form a over b their position is also determined because i've shown you you can work out its position if you're assuming this axiom of an order field which says that the inequality must hold true if you multiply through by a positive element then that instantly fixes where every single rational must go from the fact that we've already know that where all the integers go is fixed because of what i've just shown you if this is true this has to be true so the positioning of the integers fixes the positioning of all the rationals if you're requiring it to obey the axioms of an ordered field so there is only one ordering that you can put on the rationals that is going to obey these axioms and that's the ordering that you've been using for years the intuitive ordering and hopefully it's obvious to you that that intuitive ordering does indeed obey these properties indeed you've been using them for years you've been picturing them in your head you can you have a pictorial understanding of what these mean in your head we've also discussed that any larger ordered field must contain as a subfield the rational numbers so the rational numbers is the smallest of all of the ordered fields that potentially exist so the real numbers which we will move on to in the next video which was concocted to be a mathematical model of one-dimensional reality it is going to be an ordered field and we'll see why but what that instantly means is that inside of it as a subfield is going to be the rational numbers with the ordering that we all know and love from years of school level maths we'll end there thank you for watching this video i hope you haven't found it too boring and i hope you've learned something