@@TomRocksMaths hey tom, I have a question about the M1 and M2 axioms becuase they are not always true. Like in quaternions M1 isn't true and in octonions M1 and M2 aren't true.
@@ranjitsarkar3126 This is the axioms for real numbers(not all of them), and since quaternions are in a different number system, those numbers will have different axioms too.
@@nilsastrup8907 then are they really axioms, or are they just properties? Arent all of these the result of peanos axioms so they arent actually axioms? Edit: I found out that there is a difference "logical" and "non-logical" axioms these are "non-logical" axioms so they are only "axioms" (basic truths) within this mathematical field, but they could be proved using set theory and peano for example.
There are now 10 videos available in the series for you to enjoy - find them all here: ruclips.net/video/by8Mf6Lm5I8/видео.html *and yes D is meant to be a.(b+c)= a.b+a.c that's on me.
You have to start WAY before these and define what '+' and '=' mean and in which way a + b is different than (or the same as) b + a. You see, you can't say 'I take a and then add b because it means that you are DOING something and mathematics is not about DOING anything. If you say 'I have three stones, then you are not in mathematics, you are back in the real world. If, on the other hand you say 'I have three' - then what do you actually mean by that. This I think is where you should be starting from if you want to talk about axioms. What are numbers exactly??????
This comment is really old, but i want to let you know that those things are 'definitions', not axioms. Definitions and axioms are not even remotely the same thing. Therefore, he should Not have added your suggestions to the list, because they're not actually axioms. :)
Hi Tom, great video, I love your work. Thanks. Can I just check, is that last "b" just a "typo" on axiom D? You wrote "a . (b + c) = a . b + b . c". Should it actually read "a . (b + c) = a . b + a . c" ? I came to your recent talk at Oxford on Navier - Stokes. Really enjoyed it! I'm not a mathematician or engineer, but I really appreciate it when people like you take the time to explain maths and science concepts in a way that's accessible to people like me. I now have a better understanding of Navier-Stokes, and feel I can take a few more steps to learning more. Thank you. Brilliant! Keep on rocking it Tom! Cheers! Paul
Well you can't do that much math with these axioms though. If we just want to do some analysis, you also need a whole lot more set theory. Using a set of axioms like ZFC then also makes your axioms obsolete, as they can all be derived from that. Actually, this set of rules is just the necessary conditions for a set with an additive and a multiplicative map to be considered a field. That's interesting for real valued calculus for example, but for many sets these axioms are not fulfilled.
I agree with @ CARERS Training For axiom D i think it should be a . (b + c) = a.b + a.c It was a great video and I thought you were very energetic. Thanks
5:50 If you have two operations (Lets call them # and $) in a set and we assume that both operations have neutral elements e and f so that a#e = a and a$f=a and there are inverses under both operations. If you then have a distributive law: a#(b$c) = a#b $ a#c then f can't have an inverse under #.
I'm a physicist and I like to think of numbers as mathematical objects which can only interact in certain ways and as such these axioms are more the rules of interaction for a certain mathematical object: numbers. When you go to matrices, which are a different mathematical object than numbers, M1 doesn't apply. As for why, that's like asking why electrons repel each other but neutrinos don't. They're completely different objects and so interact differently. Although, that does make me wonder. Could we come up with different mathematical objects which don't interact using these rules?
Yes, there are more general structures. This is about fields, but there are also division rings, rings, monoids, etc. which drop some of these behaviors.
You say that A1 is the associative law and A2 is the distributive law. I believe A1 is the commutative law and A2 is the associative law, since D is also the distributive law.
I remember that if 2 of the STRUCTURES axioms are satisfied its a GRUPOID, with 4 it's a GROUP, with 5 a ABELIAN or ORDINARY group, with 6 a RING; with 8 a SPACE, if the space is nomed, vectorial and complete (no holes) it's a banach space. P.s. writing from my head and may be wrong, correct me if need.
Apollonios Kiqlos very true as the real numbers (about which these axioms are stated) are of course a set. Nice proof, there are several ways to prove all of the results I believe.
Love the enthusiasm, but there are a lot of little mistakes which add up to a lot. Also, I might disagree with you that we have axioms so that "math makes sense". Rather, it seems we first use math and then determine what properties are absolutely necessary for such math to work. We make sense of numbers without any need for axioms. I agree with you if by "make sense" you mean provide a rigorous axiomatic foundation for such a structure. But why start with the axioms for a field and not the natural numbers (Peano axioms)? Or maybe instead a semiring (the algebraic structure of the natural numbers).
One thing I'm curious about is with field axioms, say you have something like b -a = b +(-a) and a certain number of axioms and theorems before that, do people think like 'how do I transform b-a, to get it to the form b + (-a) using this set of statements? I was trying to prove it, and then I looked at the answer and it hit me with 'let x = b-a and y = b + (-a), we must prove x =y' and it's just like, I didn't know you could do that. But I was more interested in why they chose to do that, is it because using the single x they could plug that into previous axioms and get results? the previous axiom they then use is x + a = b. So the reason they used 'let x = b-a' is so that it becomes one encapsulated object and that they could then use that axiom? I often don't understand why they do certain things, what the thought process is like. For me, I look at prove b-a = b + (-a) and I just think, how do I start with b-a and use the axioms to make it of the form b + (-a)... which may sound redundant, but I guess I want to know how 'normal' my thinking was for a beginner, am i thinking in the wrong way. any advice.
where can if ind the proofs for the problems 3-12. I dont want to do all of them right now, but would be interested in having them around for later reference. PS. love your Navier-Stokes Tattoo( saw it on Numberphile)
There are different axioms depending on the object that you are studying, eg. a field (as in this video), or a set (see Zermelo-Fraenkel), or a group etc. You can even 'make up' your own and see what interesting maths comes out of it!
Could you explain how to use powers. In particular, how does one prove that (a+b)^2 = a^2 + b^2 + 2ab ? I have looked elsewhere about axioms, and don't even know what one can say about a^2. Obviously we know that's a times a, but it does not follow from any axiom that I can see.
You could if you created a system where all answers are defined as 42, but you couldn’t name that system “math” since the name is already taken. Also, graders are probably not going to buy that
'M1' is called the commutative law, not the associative law, and 'M2' is called the associative law, not the distributive law. You then go on to saw 'D' is the distributive law, so you have repeated yourself?
Hi i just found your channel and subscribed.I have a few questions tho.First of all, how do we know that all these axioms hold for every real numbers and not contradict each other?And second, why would we define these axioms for all the real numbers?I mean these properties are intuitive for natural numbers but when we started defined all other sets of numbers why did we keep the same properties? Thank you
these are the starting rules on which all other maths (on the real numbers) is based. we can check that they do not lead to any contradictions, eg. the identity element is unique, inverses are unique etc.
Bit late to this: I worry (about science/maths). We're using math to model the real world (at our current level of science & technology it performs well). However, let's take A4: Nowhere in the universe does this happen, you can't combine a negative cow with and +ve cow and get nothing. Even fundamental particles (say positron and electron) don't combine to give nothing, they combine to give something else entirely (ie energy). Taking a leap here (do to space constraints) thus maths has no relation to reality, thus if we're using maths to create our models of the universe, thus our models must be wrong...
Yes you can assume that but you have to then make a whole different from of mathematics which is different from our mathematics that we use in this world.and your mathematics gonna be applied only where there are 1(something/identity/unit) is equal to 0(nothing/non/non existence).or in simple words you can use that type of mathematics only in a situation where law of none contradiction doses not work.
I am currently working on a set of axioms that make dividing by zero possible. I do however have trouble with the additive Identity, which in those axioms can not be 0. Anyone smarter then me habe an Idea? I can send what I have so far on request.
You're doing mathematical axioms that define a field. That's not the most basic mathematical thing, and it's all just kinda confusing since you're constantly making viewer think these are something more than field axioms. Title of your video makes it seem like this video should be about ZFC, but if we ignore the title but follow the spirit of the title, then it would be interesting to discuss limitations of these axioms, and what they can prove, and what they cannot. If you just want to discuss fields, then this is pretty much good video with very misleading title and intro, but I'm not sure you even used the word "field" in the video.
You are probably right Joni. I based the video on the axioms that were taught in the University of Oxford Analysis course, but as I am an applied mathematician (my PhD is in fluid mechanics) then no doubt I may have used some terms incorrectly, despite my best efforts. This is only intended as a beginners look at the idea of an axiom and I felt that ZFC and set theory overcomplicated the issue. Hopefully, watching this video and then reading the helpful comments below will encourage people to look further into the idea of axioms.
@@TomRocksMaths What a lame excuse! Person with your background should know primary level mathematics. Kids in 6th, or 7th, or 8th grade study these axioms. It is like saying, hey! my grammar is bad because I am not English language major. You are not even talking about abstract nature of those axioms like Peano axioms or Zermelo Fraenkel set axioms. Wow! Oxford University. I guess, you got into Oxford because of White Supremacy.
@@karabomothupi9759 this kind of knowledge are prerequisites to study a specific type of math, in this case real analysis. And, every prerequisites starts with primary level of mathematics taught in 6th to 12th grades (at least here in India). Abstract nature of explanation changes as we take higher level classes. What he is teaching is very basic in nature. He is not even using undergrad level vocabulary. He is totally explaining it wrong. He should have done some research work before he made this video or should have used teleprompter.
These 10 articles are actually NOT axioms. They can be deduced from the Peano axioms of natural numbers. When you realize how significant these 10 articles are, you are *formally* entering the world of mathematics. Before that, you are probably understanding mathematics by instinct.
That unnecessary as 2 was not defined anywhere in those axioms. Those axioms only defined 0 and 1 and it was clearly stated that they are not equal. When you start to construct numbers under those axioms you make some assumptions and one could be that 0 is a natural number and all natural numbers have followers, but 0 is not a follower of any natural number. Then you can define 2 as the follower of 1 and automatically get that 2 is not 0.
Nothing was interesting except, your heavy British accent. A1: *Wrong* , it is called *Commutative* law of addition A2: *Wrong* , it is called *Associative* law of addition A3: *Right* , but rather call it *additive identity* A4: *Right* M1: *Wrong* , it is called *Commutative* law of multiplication M2: *Wrong* , it is called *Associative* law of multiplication M3: *Right* , but rather call it *multiplicative identity* M4: Instead of saying "you can not divide by zero", say *division by zero is undefined* D: Congrats! you played yourself by identifying two different axioms as distributive law. Only this time you were correct. Z: whatever! Namaste from India!
There's no such as a British accept. Britain is made up of England, Scotland and Wales. He has an English accent. Of course, different parts of England have different accents, but his accent is more English than Welsh or Scottish.
its as if one of the weasleys had a baby with Neville Longbottom. Just kidding. Thank you for sharing your knowledge!! i'm terrible at math and you make it interesting! cheers!
How did you get this so wrong lol. A1 is the commutative law not associative. A2 is the associative law not distributive. M1 is again the commutative law of multiplication not associative. M2 is again the associative law of multiplication not the distributive law. D which you correctly call the distributive law, despite also calling M2 the distributive law, is incorrectly written as a*(b+c)=a*b+b*c, which is false, u meant a*(b+c)=a*b+a*c.
You spelled properties wrong :p jk I know people have been giving you hell already for describing these rules as axioms just thought i'd throw a jab in myself.
Wow, you got a lot wrong, associative and distirbutive mixed up, what an axiom is and you think that is the most basic set? That is quite late. How about trying set theory instead?
Thanks for the feedback. I chose to base the answer on the axioms that were taught in the University of Oxford Analysis course. I am an applied mathematician (my PhD is in fluid mechanics) and I do not claim to be an expert in set theory. I tried to give a basic introduction in simple terms to the idea of axioms and I think set theory would overcomplicate things for beginners.
Applied? Ew :P Well to the question of "most basic axioms" you cannot go to analysis which requires set theory, you would have to go for set theory to answer it, otherwise your answer here is wrong. I agree set theory is not easy, but when the question is "What is the most basic axioms"....sorry that is where you have to go, only other place I am aware of is category theory.
Tom Rocks Maths Since you already replied to this comment, can you please address the point about mixing up the associative and distributive axioms? I thought that A1 is commutative, A2 is associative, and distributive is the second to last one, D. You said A1 is associative, A2 is distributive, and D is also distributive.
A1 and M1 are Commutative Laws; A2 and M2 are Associative Laws; and I believe D should say a.(b+c)=a.b+a.c
Indeed, thanks for pointing out the errata
I agree
"These are all of the axioms"
Gödel is typing...
Nice.
@@TomRocksMaths hey tom, I have a question about the M1 and M2 axioms becuase they are not always true. Like in quaternions M1 isn't true and in octonions M1 and M2 aren't true.
@@ranjitsarkar3126 This is the axioms for real numbers(not all of them), and since quaternions are in a different number system, those numbers will have different axioms too.
Good one
@@nilsastrup8907 then are they really axioms, or are they just properties? Arent all of these the result of peanos axioms so they arent actually axioms?
Edit:
I found out that there is a difference "logical" and "non-logical" axioms
these are "non-logical" axioms so they are only "axioms" (basic truths) within this mathematical field, but they could be proved using set theory and peano for example.
I've seen you on numberphile, didn't know you had your own channel! I've subscribed, of course
Awesome! Thank you!
D is incorrect. It should be a(b+c)=ab+ac
Thank you
BRUH that dot means multiplication stoopid
@@alessandromordasiewicz9818 someone wasn't paying attention.
yet that same someone speaks with a distinct sense of superiority.
have a nice day.
Yes, my bad. Thanks for pointing it out.
Books recomending?
math is just great, all the great stuff mathematicians have produced is just the application of logic in to these axioms. so mind blowing
There are now 10 videos available in the series for you to enjoy - find them all here: ruclips.net/video/by8Mf6Lm5I8/видео.html
*and yes D is meant to be a.(b+c)= a.b+a.c that's on me.
You have to start WAY before these and define what '+' and '=' mean and in which way a + b is different than (or the same as) b + a. You see, you can't say 'I take a and then add b because it means that you are DOING something and mathematics is not about DOING anything. If you say 'I have three stones, then you are not in mathematics, you are back in the real world. If, on the other hand you say 'I have three' - then what do you actually mean by that. This I think is where you should be starting from if you want to talk about axioms. What are numbers exactly??????
This comment is really old, but i want to let you know that those things are 'definitions', not axioms.
Definitions and axioms are not even remotely the same thing.
Therefore, he should Not have added your suggestions to the list, because they're not actually axioms.
:)
@@cututorials wow I didn't know that! That's a pretty interesting difference
@@cututorials it is stated in the video, that all maths can be derived from these axioms, but in fact it is not even enough to proof that 2+2=4
Great video! I've always find the axioms really interesting.
Hi Tom, great video, I love your work. Thanks.
Can I just check, is that last "b" just a "typo" on axiom D? You wrote "a . (b + c) = a . b + b . c". Should it actually read "a . (b + c) = a . b + a . c" ?
I came to your recent talk at Oxford on Navier - Stokes. Really enjoyed it! I'm not a mathematician or engineer, but I really appreciate it when people like you take the time to explain maths and science concepts in a way that's accessible to people like me. I now have a better understanding of Navier-Stokes, and feel I can take a few more steps to learning more. Thank you.
Brilliant! Keep on rocking it Tom!
Cheers!
Paul
Thanks Paul - and I'm glad you enjoyed the Oxford talk. And you're right about the slight mistake with D, my bad.
@@TomRocksMaths hi you mentioned that axiom two is distributive law, but isnt it associative law?
Wonderful!! Thank you for sharing this 😊. I'm reading Maths (first year), and this is very helpful. Keep it up and wish me luck!!
Good luck - how did it go?
I guess you’re a third year now 🤨🤨
Well you can't do that much math with these axioms though. If we just want to do some analysis, you also need a whole lot more set theory. Using a set of axioms like ZFC then also makes your axioms obsolete, as they can all be derived from that.
Actually, this set of rules is just the necessary conditions for a set with an additive and a multiplicative map to be considered a field. That's interesting for real valued calculus for example, but for many sets these axioms are not fulfilled.
Can I call you Navier Stoke's guy ??
with pleasure
@@TomRocksMaths Best of luck buddy . You are the reason that I am interested in fluid mechanics so much. Keep teaching us.
my linear algebra class using almost all of the axioms on this video. Basically it’s easy to understand, but kinda hard to proof. haha
Very well explained prof. Anyone did the exercises? Share your answers please.
I agree with @ CARERS Training
For axiom D i think it should be a . (b + c) = a.b + a.c
It was a great video and I thought you were very energetic.
Thanks
Glad you enjoyed it - and yes sorry for the slight mistake, these things are a nightmare to write down.
That’s alright - thanks for the acknowledgment!
5:50 If you have two operations (Lets call them # and $) in a set and we assume that both operations have neutral elements e and f so that a#e = a and a$f=a and there are inverses under both operations. If you then have a distributive law: a#(b$c) = a#b $ a#c then f can't have an inverse under #.
Good work.
Thanks!!
Well explained ❤️ from india🇮🇳
Thanks
For the second example is it valid to prove it like this?
a•0 = a•(b-b) by A3
= ab - ab by D1
=0 by A3
This guy has a navier stokes equation on his stomach
It's on the right-hand side of my ribcage to be precise...
I don't know how you can introduce completeness without the language of set theory. With this, one may define a construction of the real numbers and prove those numbers have the same properties as given by your axioms.
Axioms of propositional calculus (Jan Łukasiewicz)
ax-1 ⊢ (𝜑 → (𝜓 → 𝜑))
ax-2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-3 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
ax-mp ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓
Axioms of predicate calculus with equality (Tarski's S2)
ax-gen ⊢ 𝜑 ⇒ ⊢ ∀𝑥𝜑
ax-4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 ⊢ (𝜑 → ∀𝑥𝜑), where 𝑥 does not occur in 𝜑
ax-6 ⊢ ¬ ∀𝑥¬ 𝑥 = 𝑦
ax-7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Axioms of predicate calculus with equality (auxiliary schemes)
ax-10 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥¬ ∀𝑥𝜑)
ax-11 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Zermelo-Fraenkel Set Theory with Choice (ZFC)
ax-ext ⊢ (∀𝑧(𝑧 ∈ 𝑥 ⇔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
ax-rep ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ⇔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-pow ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-un ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-reg ⊢ (∃𝑦𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-ac ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ⇔︎ 𝑢 = 𝑣))
The Tarski-Grothendieck axiom (huge sets (inaccessible cardinals) exist)
ax-groth ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦(∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
Then completeness could be written as: A nonempty, bounded-above set of reals has a supremum
axsup ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
The above adapted from thousands of computer-verified theorems developed from these axioms at: us.metamath.org/mpeuni/mmset.html
I'm a physicist and I like to think of numbers as mathematical objects which can only interact in certain ways and as such these axioms are more the rules of interaction for a certain mathematical object: numbers. When you go to matrices, which are a different mathematical object than numbers, M1 doesn't apply. As for why, that's like asking why electrons repel each other but neutrinos don't. They're completely different objects and so interact differently. Although, that does make me wonder. Could we come up with different mathematical objects which don't interact using these rules?
Yes, there are more general structures. This is about fields, but there are also division rings, rings, monoids, etc. which drop some of these behaviors.
@@EpicMathTime Ooh that sounds interesting
You say that A1 is the associative law and A2 is the distributive law. I believe A1 is the commutative law and A2 is the associative law, since D is also the distributive law.
Thank you.
Did you say have sented?
I remember that if 2 of the STRUCTURES axioms are satisfied its a GRUPOID, with 4 it's a GROUP, with 5 a ABELIAN or ORDINARY group, with 6 a RING; with 8 a SPACE, if the space is nomed, vectorial and complete (no holes) it's a banach space.
P.s. writing from my head and may be wrong, correct me if need.
Correction:
A1 is commutative law
A2 is the associative law
These 10 statements are not axioms because they can be proved by using the axioms of set theory.
I do not doubt that you are correct Apollonios, but I have taken the axioms from the Analysis course that I was taught at Oxford University.
Apollonios Kiqlos very true as the real numbers (about which these axioms are stated) are of course a set. Nice proof, there are several ways to prove all of the results I believe.
Love the enthusiasm, but there are a lot of little mistakes which add up to a lot. Also, I might disagree with you that we have axioms so that "math makes sense". Rather, it seems we first use math and then determine what properties are absolutely necessary for such math to work. We make sense of numbers without any need for axioms. I agree with you if by "make sense" you mean provide a rigorous axiomatic foundation for such a structure. But why start with the axioms for a field and not the natural numbers (Peano axioms)? Or maybe instead a semiring (the algebraic structure of the natural numbers).
Godel: hold my beer
thank you
One thing I'm curious about is with field axioms, say you have something like b -a = b +(-a) and a certain number of axioms and theorems before that, do people think like 'how do I transform b-a, to get it to the form b + (-a) using this set of statements?
I was trying to prove it, and then I looked at the answer and it hit me with 'let x = b-a and y = b + (-a), we must prove x =y' and it's just like, I didn't know you could do that. But I was more interested in why they chose to do that, is it because using the single x they could plug that into previous axioms and get results? the previous axiom they then use is x + a = b. So the reason they used 'let x = b-a' is so that it becomes one encapsulated object and that they could then use that axiom?
I often don't understand why they do certain things, what the thought process is like. For me, I look at prove b-a = b + (-a) and I just think, how do I start with b-a and use the axioms to make it of the form b + (-a)... which may sound redundant, but I guess I want to know how 'normal' my thinking was for a beginner, am i thinking in the wrong way. any advice.
How learn and understand theorem? Please upload the video to make me understand.
Bro defined math with letters
this video impress me, thx
Good
A1 is associative law? I think it's commutative.
me too
No shit I feel like I went through a whole semester of algebra
I hope that's a good thing!
How do you prove 2+1=3 by using these axioms?
Is the "D" correct?
Isn't that supposed to be a.(b+c)= a.b+a.c instead of =a.b+b.c....???
Yes - well spotted. My apologies (these things are a nightmare to write out).
where can if ind the proofs for the problems 3-12.
I dont want to do all of them right now, but would be interested in having them around for later reference.
PS. love your Navier-Stokes Tattoo( saw it on Numberphile)
If you send them to me - www.tomrocksmaths.com/contact - I'd be happy to take a look.
How many known axioms are in math? Are they new axioms to be discover? Are there infinite axioms?
There are different axioms depending on the object that you are studying, eg. a field (as in this video), or a set (see Zermelo-Fraenkel), or a group etc. You can even 'make up' your own and see what interesting maths comes out of it!
So you can proof calculus theorems with those 10 axioms?
there are many steps in between (as taught in a real analysis course) but yes you will get to calculus eventually
Is a=a not also a fundamental axiom?
So we just have to assume these to be true and work are way up and build Math's or there are things more fundamental than these.
tom can prove the following basic mathematical axioms are true. if you can prove them please make a video of it
Do we generalise these axiom to complex numbers.
Yes, but for the real and imaginary parts separately.
@@TomRocksMaths Thanks so much.
Could you explain how to use powers. In particular, how does one prove that (a+b)^2 = a^2 + b^2 + 2ab ? I have looked elsewhere about axioms, and don't even know what one can say about a^2. Obviously we know that's a times a, but it does not follow from any axiom that I can see.
I think it just follows form the distributive law. You can write it as (a+b)(a+b) and then follow the axioms.
@@TomRocksMaths thank you.
So I can just make my own axioms and then write at the math exam 42 as the answer and claim it to be right?
Unspoken agreement to use the math axioms.
Well yes, but actually no
No. Think axioms like thoughts or sentences. Out of all the infinite sentences that can exist, some make sense and the rest dont.
You could if you created a system where all answers are defined as 42, but you couldn’t name that system “math” since the name is already taken.
Also, graders are probably not going to buy that
Isn't the distributive property a(b+c)=ab+ac instead of a(b+c)=ab+bc?
Yes, sorry that was my mistake - these things are a nightmare to write down.
Terrence Howard's axiom: 1x1=2
Nice vid Tom. Cheers!
Gilberto Urdaneta thanks dude! Hope you're good.
You should specify that these are just field axioms when the axioms of mathematics are ZFC set theory axioms
Please told me what is math Olympiad tom
This guy is like police inspector purposely leading the investigation in wrong direction
'M1' is called the commutative law, not the associative law, and 'M2' is called the associative law, not the distributive law. You then go on to saw 'D' is the distributive law, so you have repeated yourself?
Came here to say this.
I don't know english I know Russian
Some of those can be proven using the definition of addition and multiplication, I don't get why they are called axioms
nice
I'll second the Hilbert comment too..
Hi i just found your channel and subscribed.I have a few questions tho.First of all, how do we know that all these axioms hold for every real numbers and not contradict each other?And second, why would we define these axioms for all the real numbers?I mean these properties are intuitive for natural numbers but when we started defined all other sets of numbers why did we keep the same properties? Thank you
these are the starting rules on which all other maths (on the real numbers) is based. we can check that they do not lead to any contradictions, eg. the identity element is unique, inverses are unique etc.
Bit late to this: I worry (about science/maths). We're using math to model the real world (at our current level of science & technology it performs well).
However, let's take A4: Nowhere in the universe does this happen, you can't combine a negative cow with and +ve cow and get nothing. Even fundamental particles (say positron and electron) don't combine to give nothing, they combine to give something else entirely (ie energy). Taking a leap here (do to space constraints) thus maths has no relation to reality, thus if we're using maths to create our models of the universe, thus our models must be wrong...
Why axiom 0 not equal to 1?
Yes you can assume that but you have to then make a whole different from of mathematics which is different from our mathematics that we use in this world.and your mathematics gonna be applied only where there are 1(something/identity/unit) is equal to 0(nothing/non/non existence).or in simple words you can use that type of mathematics only in a situation where law of none contradiction doses not work.
@@noexception9598 They probably want to know why the other axioms don't prove this already.
a-shoe-ming
0:52 ©^ 2:24→
I am currently working on a set of axioms that make dividing by zero possible. I do however have trouble with the additive Identity, which in those axioms can not be 0. Anyone smarter then me habe an Idea? I can send what I have so far on request.
hello can you throw us an update
The idea is impossible and ambitious so I'm just curious
A1-is supposed to be commutativity not associativity and A2 is supposed to be the former
You're doing mathematical axioms that define a field. That's not the most basic mathematical thing, and it's all just kinda confusing since you're constantly making viewer think these are something more than field axioms. Title of your video makes it seem like this video should be about ZFC, but if we ignore the title but follow the spirit of the title, then it would be interesting to discuss limitations of these axioms, and what they can prove, and what they cannot. If you just want to discuss fields, then this is pretty much good video with very misleading title and intro, but I'm not sure you even used the word "field" in the video.
You are probably right Joni. I based the video on the axioms that were taught in the University of Oxford Analysis course, but as I am an applied mathematician (my PhD is in fluid mechanics) then no doubt I may have used some terms incorrectly, despite my best efforts. This is only intended as a beginners look at the idea of an axiom and I felt that ZFC and set theory overcomplicated the issue. Hopefully, watching this video and then reading the helpful comments below will encourage people to look further into the idea of axioms.
Can you plz enlighten me, why video title's should tacitly suggest/sound about ZFC and not Dedekind-Peano axioms?
@@TomRocksMaths What a lame excuse! Person with your background should know primary level mathematics. Kids in 6th, or 7th, or 8th grade study these axioms. It is like saying, hey! my grammar is bad because I am not English language major. You are not even talking about abstract nature of those axioms like Peano axioms or Zermelo Fraenkel set axioms. Wow! Oxford University. I guess, you got into Oxford because of White Supremacy.
@@mindfreakmovies9586 So primary school kids appreciate Real Analysis? Or are you drunk?
@@karabomothupi9759 this kind of knowledge are prerequisites to study a specific type of math, in this case real analysis. And, every prerequisites starts with primary level of mathematics taught in 6th to 12th grades (at least here in India). Abstract nature of explanation changes as we take higher level classes. What he is teaching is very basic in nature. He is not even using undergrad level vocabulary. He is totally explaining it wrong. He should have done some research work before he made this video or should have used teleprompter.
Axioms much? Nerd here
A2 is associative..
a(b+c) = ab+ac
Whose here because of The Map of Mathematics
please you keep on saying axiom one is associative please is it commutative or associative i stand to be corrected though
As pointed out an error and this is for reals. Half marks.
Do you listen to punk rock , emo music, metalcore music?
oh, no set theory?
These 10 articles are actually NOT axioms. They can be deduced from the Peano axioms of natural numbers. When you realize how significant these 10 articles are, you are *formally* entering the world of mathematics. Before that, you are probably understanding mathematics by instinct.
Now prove 0 is not equal to 2. ;)
That unnecessary as 2 was not defined anywhere in those axioms. Those axioms only defined 0 and 1 and it was clearly stated that they are not equal. When you start to construct numbers under those axioms you make some assumptions and one could be that 0 is a natural number and all natural numbers have followers, but 0 is not a follower of any natural number. Then you can define 2 as the follower of 1 and automatically get that 2 is not 0.
Nothing was interesting except, your heavy British accent.
A1: *Wrong* , it is called *Commutative* law of addition
A2: *Wrong* , it is called *Associative* law of addition
A3: *Right* , but rather call it *additive identity*
A4: *Right*
M1: *Wrong* , it is called *Commutative* law of multiplication
M2: *Wrong* , it is called *Associative* law of multiplication
M3: *Right* , but rather call it *multiplicative identity*
M4: Instead of saying "you can not divide by zero", say *division by zero is undefined*
D: Congrats! you played yourself by identifying two different axioms as distributive law. Only this time you were correct.
Z: whatever!
Namaste from India!
There's no such as a British accept. Britain is made up of England, Scotland and Wales. He has an English accent. Of course, different parts of England have different accents, but his accent is more English than Welsh or Scottish.
@@dr.davidkirkby1959 This is all what you have to say? Anyways, we don't care your blather.
Very cool. Have you done all the exercises?
How to prove 1/0 is infinity?????????????????? ¿??????
... it isn't??
Godel kinda ruined this btw
Classic Godel, ruining everyone's fun... (but also his work is incredible)
say what?
Am I the only one who wanted to write === on his list of 10 axioms?
its as if one of the weasleys had a baby with Neville Longbottom. Just kidding. Thank you for sharing your knowledge!! i'm terrible at math and you make it interesting! cheers!
You're very welcome.
How did you get this so wrong lol.
A1 is the commutative law not associative.
A2 is the associative law not distributive.
M1 is again the commutative law of multiplication not associative.
M2 is again the associative law of multiplication not the distributive law.
D which you correctly call the distributive law, despite also calling M2 the distributive law, is incorrectly written as a*(b+c)=a*b+b*c, which is false, u meant a*(b+c)=a*b+a*c.
Nice video, but I think "all of maths" is a bit of an overstatement. These are just the field axioms, every structure in maths is not a field.
You spelled properties wrong :p jk I know people have been giving you hell already for describing these rules as axioms just thought i'd throw a jab in myself.
Wow, you got a lot wrong, associative and distirbutive mixed up, what an axiom is and you think that is the most basic set? That is quite late. How about trying set theory instead?
Thanks for the feedback. I chose to base the answer on the axioms that were taught in the University of Oxford Analysis course. I am an applied mathematician (my PhD is in fluid mechanics) and I do not claim to be an expert in set theory. I tried to give a basic introduction in simple terms to the idea of axioms and I think set theory would overcomplicate things for beginners.
Applied? Ew :P
Well to the question of "most basic axioms" you cannot go to analysis which requires set theory, you would have to go for set theory to answer it, otherwise your answer here is wrong.
I agree set theory is not easy, but when the question is "What is the most basic axioms"....sorry that is where you have to go, only other place I am aware of is category theory.
Tom Rocks Maths
Since you already replied to this comment, can you please address the point about mixing up the associative and distributive axioms?
I thought that A1 is commutative, A2 is associative, and distributive is the second to last one, D.
You said A1 is associative, A2 is distributive, and D is also distributive.
It's math, not math"s"
Joshua:
In the states yes. But in the UK( the home of the English language ) we call it Maths.
No, it was originally mathematics in the UK, now shortened to math or maths. The uk calls it maths, but doesn't mean thats what it should be
Oh man, mistakes. Good effort but redo this you're messing with peoples lives.