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Consent ordered field

okay so in the last video I introduced the ordered field axiom so there's the field axioms the ordering axioms and then the sort of special axioms that combine them all together you can refer to all of those things as the ordered field axioms right and so there are basically two theorems in this section in the book which basically prove some familiar properties of the real numbers or the rational numbers directly from those axioms right and I want to emphasize that the part of the purpose of this is to teach you that in general it's bad to take things for granted even if they seem obvious it's good to get in the habit of sort of mentally justifying why everything is true and this hopefully will be a good exercise for you in thinking very carefully about how those seemingly obvious familiar properties of these operations and everything really follow from the axioms that we listed down ok so let's start let's just go through some of the proofs I'm only going to do a couple of them so let's look at theorem 3 point 1 which is about field axioms so part 1 of this is about cancellation of addition so they just say a plus C equals B plus C implies a equals B this is of course implicitly for all a B and C and R or Q or whatever we haven't really defined R yet but well you know I'll get there we're not really going to define it fully but the axioms that we list will basically be everything we need to know about it so let's look at the proof of this so basically if you look at the equation here you know to try to get from here to here your gut instinct is to say well let's just subtract off C right the problem is you can't exactly do that I mean the accidents we listed down don't even really find subtraction but what you can do is you could add the the additive inverse of C to both sides and then you have to be a little careful about how it goes so so if we suppose a plus C equals B plus C then let's add negative C so a plus C plus negative C equals B plus C plus negative C notice how I'm being very careful with the parentheses here this this is just saying you know we're not using an axiom to actually get to this point you're always allowed to do the same operation to both sides of an equation that's just sort of philosophically true but what you have to remember is that when you do an operation to one side of the equation you have to treat that entire thing that side of the equation at least you know a priori before we've done all this stuff proving all these things you can't have to treat that side of the equation as this like atomic unit you can't break it apart or anything like that right so it's really because we have the associative law right so then by a 1 we can reassociate the parentheses here so we get a plus C plus negative C equals B plus C plus negative C right so then by now we can cancel these things so by a 4a plus zero zero and remember that actually or sorry so up here we are also actually implicitly using a 4 to define the symbol negative C right a 4 actually tells us this is just supposed to normal a so a 4 actually tells us that negative C exists and has this property right so and so then using the additive identity property so then by a three I believe a equals B so that's all we wanted to show we're done now just you know take stock of what happened here we started with the sort of assumption here and then we derived the conclusion and each step of the way we only ever used like one axiom basically right so absolutely nothing was left to the imagination here this is sort of the ultimate level of rigor that you could possibly ask for in a proof it's just completely laid out everything is justified in terms of the axioms okay so obviously that is not the standard of proof that I'm asking for on the homeworks or anything you know this is a question that comes up a lot and I'll be happy to address it on Piazza or in the discussion sections I don't want to waste too much time on it here but basically when it comes to your work you should just try to ask yourself the question like does it seem obvious to me if you try to use like if you're trying to make a leap of logic from one state to another and you're not sure whether you need to justify it just ask yourself well does it seem obvious how it should be justified and like also I guess ask yourself am i skipping a large fraction of the proof by doing this so but anyway I'll talk more about that in other places so let's just do one more of these I'm gonna erase I wish I knew how to erase erase better but yeah I'm gonna erase this proof here so we're gonna say we've proved that now let's look at the next one so this one is about multiplying by zero so it's saying a times zero equals zero for all a you know in R or whatever and again it's one of those things that seems obvious but actually it takes a little bit of work to justify so here's the proof of that one so by the additive identity axiom eight times zero equals a times 0 plus 0 right 0 plus 0 equals 0 that's by the additive identity axiom so now using the distributive law I'll just continue this equation here and kind of like put the justifications on this side it's sort of a common way of formatting these things so I have 8 times 0 plus 8 times 0 right so actually let me kind of this looks confusing normally you know like I'm asking for you guys so this comes from that by the distributive law so normally like on homeworks and stuff I'm asking for you guys to really try to use complete sentences to explain your argument okay this is I'm doing this type of formatting like for lecture purposes because it's more efficient right but please please try to make everything you say on homeworks be phrased in terms of complete sentences ok it's sounds silly but it's actually really important in helping you think about math not just helping you like express your ideas it actually helps you think better about math if you learn how to do that okay so okay then now that we have that then right you can add 0 [Music] to the left side so so we can say zero plus a times zero right because that equals a times zero just by so I'm not I don't have to do anything to this side because I know that a times zero equals zero plus a times zero by a three so this is by a three again and then and then I can cancel right the the right term from both sides so zero equals a times zero that's by the previous part which we already proved okay so you can kind of like once you prove something then you can use that in the future and that way your proofs can get more and more complicated because you'll already have done a lot of complicated stuff kind of behind the scenes okay so that's so that's a proof with the field axioms now I want to do a similar thing for some of the ordering axioms but I'm actually going to make another video for that one