Actual Proof 1+1=2
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- Опубликовано: 4 окт 2024
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This video presents a clear and concise proof of why 1+1 equals 2, a fundamental concept in mathematics. It breaks down the logic and reasoning behind this basic equation, making it understandable for anyone interested in math.
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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information. Viewers should always verify the information provided in this video by consulting other reliable sources.
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First
Actually we can prove 1+1=2 by proving lim x->1 x+1=2, using delta epsilon
You used addition in your definition of addition
And a bit more in, a giga condensed form, on my website
@@krzysztofszczepanik8380true
That hit hard when he said "If you really want to up your math game..." at the end of the video where he explains why 1 + 1 = 2.
What do you mean? Are you being ironic? I hope so, because this Bri, so called "The Math Guy", is the most intellectually dish0n3st math youtuber out there.
Bri's Mozart!
Hit hard? Because is s1lly af, right?
I have spent the last 40 years of my life trying to disprove this postulate. I graciously accept defeat at your hands, sir. SIGH.
You just wasted 40 years of your life, doing nothing 😊
0(1+1)=0(2+1)
Cancel the zeros out!
@@LeoV6502 Illegal! I shall report you to the Math Police!
@@LeoV6502uhhh that’s not how it works
Mua ha ha ha ha haaah!!!
This is what the teacher expects you to do when they say "show your work"
I always enjoyed randomly inserting the phrase "and then a miracle occurred here..."
@@throughthoroughthought8064 lmao, i always did 90% of the stuff in my head, not even on tests just in schoolwork and the teacher would conplain
No, it's really not hard at all.
It was revealed to me in a dream
@@Allyfyn This comment was obviously a joke, in case you didn’t know that.
Meanwhile john hush proving 1=2 in every single way possible 🗿
That looks like a rather immediate task: there are no such ways
@@ilmionomenonlosothey’re all division by 0
@@ayuballena8217 of course they are…
@@tri99er_ exept…
@@ayuballena8217 except if we assume that every solid in 3 (or more) dimentions are mesureable :)
This "logic" proof could be written in a computer language called Prolog:
_% Base case: Adding 0 to any number N results in N._
plus(0, N, N).
_% Recursive case: To add A and the successor of B, we first add A and B, then find the successor of the result._
plus(s(A), B, s(C)) :- plus(A, B, C).
?- plus(s(0), s(0), s(s(0))).
And that is basically how arithmetic is defined in logic programming
And here's a similar thing but for the programming language and proof assistant, Idris:
data Nat : Type where
Z : Nat
S : Nat -> Nat
one, two : Nat
one = S Z
two = S one
plus : (a : Nat) -> (b : Nat) -> Nat
plus a Z = a
plus a (S b') = S (plus a b')
data Equals : (a : ty) -> (b : ty) -> Type where
EqualityIsReflexive : (x : ty) -> Equals x x
theProof : Equals (plus one one) two
theProof = EqualityIsReflexive two
This doesn't need any sophisticated reasoning because `plus one one` simply evaluates to `S (S Z)` and so is already the same thing. The type | proof checker can verify it mechanically.
Also, if I were to use the standard library just these two lines would do the trick:
theProof : 1 + 1 = 2
theProof = Refl
@@mskiptr Excellent. Yes. Question Idris or Agda?
@@gheffz To be honest I don't have much opinion on Agda because I haven't really tried it (yet). It seems to have a bit more active community around it and I would guess the tooling should be less janky. Also, from what I've seen it's more focused on maths while I'm overall more interested in applying type theory to CS concepts. Oh, and then there are the fancy symbols used everywhere. I kinda like the idea but it has a massive potential to be really annoying and ugly.
@@gheffz Idris for the programmers, Agda for the scientists :)
My Second-grader says "nu-uh." And I can't disagree with them.
The name of the video is miseading, in my opinion. This is not really a proof that 1+1=2, but a proof that the definition of addition presented here is consistent with the result "1+1=2". If it wasn't consistent, the conclusion had not been that 1+1 does not equal 2, but that our definition for addition is faulty. This kind of proof put the result as something we want to achieve, because we assume it is true intuitively but lack the formal tools to describe it: we try to build a system of axioms and definitions that will lead to the result we wanted it to lead, and than we prove we succeded. Therefore it is meaningless to say we "proved that 1+1=2". What we did was finding an extremely elegant definition for addition that is consistent with the intuitive idea we already had and just wanted a formal set of axioms that will lead to it. This definition, or an extension of it, might become interesting and useful when it comes to adding things we don't have intuition about, such as infinity.
Agreed. Axioms are supposed to be the basic self-evident truths, but there is no way that Peano axioms are more self-evident than the fact that 1+1=2.
The video proved that "1+1=2 under the Peano axioms", and mentioned that the Peano axioms give us natural numbers, that are identical to the natural numbers we intuitively know. Thus 1+1=2. Proving the equivalency of the two groups would be a different proof.
@@b3kstudio This video heavily implies that the Peano axioms are in some sense more fundamental that basic properties of natural numbers, without really explaining why. That's why it has left me a bit frustrated.
@@catbertsis I think it depends on how you look at it. You could argue, that there is nothing more fundamental than taking a pile of one sticks, and another pile of one sticks, and putting them together. But that happens to be the same as the Peano axioms. Which one is more fundamental? Piles of sticks, for which you need sticks, which are arguably quite complicated by themselves, or 4 laws that can be used to represent the same thing.
In the end, I think saying "this pile is 1, and this pile is 2" then taking two piles of 1 and putting them together is not a worse proof in any way, than the one in the video. Because really, 1+1=2, because we defined 1 and 2 that way.
@@b3kstudio formally. you are right, of course. But the motivation of such a proof is not to prove that 1+1=2, but to prove that Peano Axioms are worth something and not leading to results we don't want. From that point of view, "1+1=2" was the result we wanted to have and the Peano Axioms were the axioms needed to achieve that - they are the starting point from the point of view of Logic, but not from the point of you of what we really wanted to find.
0 is very often included as a Natural Number when you are using set theory as the underlying basis since the Naturals are then defined as being the set of all possible finite cardinalities, and since the cardinality of the Empty Set is 0 that makes it a Natural Number.
Where 0 isn’t usually included as a Natural Number is when you’re working in Number Theory since 0 is an annoying exception in a lot of theorems involving factorization. It’s simply more convenient to define the Naturals as starting at 1 in that context so you don’t have to keep dealing with 0 as a special case.
1:40 how can we use addition to define addition? How can that make sense?
look up what an inductive proof looks like.
He's not using addition to define itself, he's using the base case over and over again as a function of counting to define addition.
It's as said before an Inductive definition such as for the factorial: if a (different) 0, then a!=(a-1)! x a ,Else (a=0) a!=1. You don't use the operation to define it. You define it by an operation that you already know (multiplication for the factorial,successor for the addition) on the previous term. And by having defined the first term, you have define Ur operation for every terme, since the first term is always reach
You just repeat applying the definition if it’s not the base case until you get to the base case, where there is no such holdup.
Addition itself is an axiom, so it is not being used to define itself. It is just being defined.
We do not define the abstract essence of 'addition', but rather each specific case. Each case makes sense because it is reduced to the basic one, which is defined as true.
we got proof that 1+1=2 before gta 6
💀
Wish these videos were out 12 years ago. Been wishing for an understandable explanation since high school, thank you so much!!!
You define addition with addition?? how is that proof bruh.
I'm with you, he makes no sense
It's a recursive definition. It wouldn't make sense without the base case a+0 = a. If you think about it and look for it on the internet, you'll see that it was a great choice to define things that way since you can prove a lot of things with the induction principle
If I give you a iPhone at 7:39 and another iPhone at 7:39 how many iPhones would I have given you?
It will eventually go to another definition
Recursion
It is interesting how often in science and math the most tangible things are often the hardest to define in abstract terms.
So 0 or 1 being the initial natural is actually a big split in conventions, in ny experience German speaking areas were more likely to start with 1, French and English speaking with 0, though English was the most mixed of the bunch
And this split goes back even before Peano pubished his formalization, he was actually beaten to the punch by Dedekind (his formalization is equivalent, but also harder to state and closer to second order logic than first order)
In Dedekind's initial manuscripts he started at 0, but somewhere in the process he began starting it at 1, he never wrote down why he changed it, but if I were to guess, the way he was approaching proofs became more elegant and simpler to write after the change, others went with 0 because their approaches had the opposite side for elegance
0 indeed is not always considered a natural number, but it can be included, if needed. Also this way of addition extends to every whole number, not just naturals, so it is accepted here for convenience.
0:20 at this point the 2 circles and the loading icon came together perfectly lol
upd: maybe only in my phone
We invented 1+1=2 to describe one apple and one apple equals two apple. And now we go full circle to prove what we define
That's true, but some mathematical experts disagree with us .
Imagine writing this all down while solving a complicated math problem.
Can you explain the set theory behind the definition of addition?
You can define the successor of a set by adding {set} to it. So you have a set X that represents some number in Arithmetic. The successor will be {X, {X}} where {X} is the only set whose unique member is the set X and X and {X} cannot be equal by the foundation axiom of Set Theory.
This proof is beautiful. Thanks
The way you ended the video is sorta funny 😅 The way "If you really want to up your math game" just abruptly came in
I'm a bit confused with the definition of addition for b0 at 1:35 :
a + b = S(a + c), with b = S(c)
Aren't we introducing an addition in the definition of addition? How are we going to know what the successor of (a + c) is, if we haven't yet defined what an addition is?
"you 'probably' agree with me"
well, probably is a little like maybe.
For a moment i thought you were going to talk abt the few hundred paged proof but at least this was a little more meaningful to think about.
This is essentially the same proof. For the one that takes hundreds of pages, most of the pages are used to define the rest of the Peano axioms that were mentioned in the video, so it's not actually hundreds of pages just to do 1 + 1 = 2. For that you only need successor and addition.
@@ZachAttack6089 The other Peano axioms are just that the successor function is injective and the axiom of induction. How do you need hundreds of pages for that? I think if one really writes hundreds of pages to prove that 1+1 = 2 that's not a proof that's an exercise in obfuscation.
It took 379 pages to prove this in Principia Mathematica.
This is because the concepts weren't well understood. Nowadays it's much simpler. Because we have solid foundations for both Peano Arithmetic and models in Set Theory (so whatever method you choose to prove it with is straightforward).
it didn't take 400 pages to prove 1 + 1 = 2; the authors used that space to establish an entire new foundation of mathematics notation and definitions, and then they just happened to prove 1 + 1 = 2
I thought they didn't quite finish it.
Yes but can you prove that 6 was in fact scared of 7?
1:34 even the definition of addition has addition
How and why are you using the + in the definition of +? S(a + c) includes a + but it is in the definition of +
Do you just need recursion?
@@fatfurry Yeah, you use recursion until you have a + S(0)
@@ViktorLoR_MainuDo you just need recursion?
@@mcr9822 Yes, you use recursion until you have a + S(0)
@@ianmccurdy1223 Do you just need recursion?
It's funny it's actually common sense if you think about it. Another way we could write the equation is more expansive. A=N B=M and C=B-1. Then writing A+B=1+(A+(B-1)) that plus one moves a few spots but if you can sub S for one you will always get that the left and right sides are equal.
I should clarify it is simplistic the way you explain it however it is still a strange concept if not approached right.
Was taught growing up to always be skeptical and think critically. Wasnt sure about this one. Had to cover all my bases.
This is the math equivalent of having to meet the minimum 1000-word count in your essay
I love this proof so much! X3
Gonna become a preschool teacher to show the kids this thing so they are even more terrified for the next 10 years. :3
I’m no mathematician, but I don’t think this is possible. Math is a form of representation. Assigning numbers to things is a uniform way to represent otherwise dissimilar and discrete things with one symbol. If I have a sheet of paper and a cup on my desk, I say there are two things on the desk not because the objects share some material property, but because I choose to represent them abstractly. This eliminates the need for me to be able to perceive a difference in the material objects, so long as I reason that they are indeed not the same object. So if I individually call those objects ‘one,’ I can declare that one ‘one’ and another ‘one’ makes ‘two.’ This verdict of nomenclature isn’t a mathematical law-it’s arbitrary. There’s nothing about the cup that makes it ‘one.’ In other words, the logic that outlines the definition of a number and affirms that 1+1=2 isn’t axiomatic. The validity of the statement is implicitly asserted to make mathematics’ epistemological laws possible, which all ultimately collapse when tasked to explain themselves, much like the paradox of trying to explain language using language.
o (1 circle) + o (1 circle) = o o (2 circles)
Edit: In hindsight, this is technically visual proof and is not reliable for calculations.
if only youtube existed when Russell tried to proof the same thing! ;)
I am still waiting for a math video on RUclips titled "The man who ALMOST broke math"
I recommend y all checking out type theory, you can actually implement addition in prog languages like agda
This could be generalized by adding the fact that 1+1=2 is e true iff it's belong to ordered ring(if a ≤ b then a + c ≤ b + c.
if 0 ≤ a and 0 ≤ b then 0 ≤ ab. In fact one can check by replacing usual addition with modulo addition and simply 1+1 mod 2=0 and so the statement is false) and it's multiplicative identity which is 1 and additive identity which is 0 don't coincide i.e 1=0 and 1+1=0+0=0 false otherwise its true and the ring which 0 and 1 belong to it it's trivial
Descartes malding rn
I like to define ℕ as an initial semiring, or simpler, when only addition is concerned, as an initial pointed monoid. It really feels more natural than Peano for me. No successor function to talk about.
Ah I forgot to add. Then 2 is defined as 1 + 1. It really is no less unfair than defining 1 = S0 and 2 = S1.
What _is_ more interesting to prove based on any definition of ℕ, though, is 2 + 2 = 2 ⋅ 2. In the monoid case, we’ll need to define multiplication to do this, but it’s simple via endomorphisms: like endomorphisms of an abelian group comprise a ring, endomorphisms of an abelian monoid comprise a semiring, which will be that same initial semiring talked about at the start; we only need to show that an initial pointed monoid is abelian first.
Also we can show the principle of induction from just a monoid definition (all Peano axioms, really; likewise with the initial semiring-initiality is that strong). Cool exercise, won’t spoil.
how did you type the fancy N
@@cloverisfan818 I have a custom AutoHotkey script which replaces strings like \bbN␣ with ℕ but also for many other characters (like \times␣ with × or \A0␣ with Å - my keyboard layout doesn’t have × or Å on its own). Without that, I’d open a character map and searched by name (hoping that I remotely remember how it’s called in Unicode).
@@cloverisfan818I think he is using some kind of 3rd party maths keyboard.
How did you type black board N in RUclips comment?
@@cloverisfan818 Sorry, for some reason my comment didn’t post two days ago. I use a custom AutoHotkey script that replaces certain strings with various characters like ℕ, ×, Å, °, «» which I usually first take from the basic Character Map in Windows, but there is a lot of other software for that, including on the web. I won’t post links because sometimes it blocks posting, unfortunately, just search for “unicode character map”.
Using those by itself might be tedious but I don’t have a good source for advanced keyboard layouts that contain esoteric characters like ℕ, or scripts like mine (mine is a mess, I don’t think it would be useful to anyone except me).
thanks bro now i can troll my entire class using this complex equation
Math can sometimes be a convoluted way to express intentions and inuitions.
Oh, so it's like a recursion. If the numbers include something more than one, you have to translate the numbers inside S() with the same rules until you reach a point where 0 is one of the numbers, hence a recursion proof. If you're adding more than one number, you can do two of them and then do another addition.
You are an inspiration sir ✨🙇
imagine a universe where 1+1 is a hard problem and people have to study to understand it.
I'm not a mathematician, but you use an addition in the proof of adding?! S(1+0). Is that valid?
math just make things up
this is why I love math👍
That one friend on a Tuesday afternoon :
I need proof that this proof is legitimate before coming to the conclusion that 1+1=2
Terrence Howard: hold my beer
😂😂😂😂😂😂
You know, if you don't pay a university $100.000 dollars, you can just prove 1+1=2 by adding 1+1 together.
Exam paper: 1a.What is the following calculation euqal to?
1+1= [1 mark]
b. Prove why. [6 marks]
Do not write under this QR code
most life-changing moment in my life.
Same
If you question 1 + 1, then you should question a + b, i.e. the `plus` operator itself. It means that `plus` is not defined and you are trying to define it, but at the same time you are using `plus` freely (as if already defined) in arguments for S... e.g. S(a+c). You are not allowed to use it since `+` is not defined.
No, it's just defined inductively.
Going to write this in my resume
Real analysis flashbacks!
Is this the rube goldberg machine of arithmetic?
Now do a video on the proof that 1+1=2 from Russell's and Whitehead's book, Principia Mathematica!😁
1+1=2 is not self-evident. Because it´s not always the case that one thing plus another thing just equals two of them. See, one idiot is just an idiot. But one idiot plus another idiot does not just give you two idiots, it gives you a problem.
When a simple equation is solved with algebra
Yes but why S(1)=2 ? Because S(n)=n+1 ? But why ? And what is n+1 if we use S to define addition. And if u dont use addition, I feel like at the end of the day you use « intuition » (the one we avoided at the beginning) to define what is a Successor
That's how naturals numbers are defined via Peano axioms. So, 1 is defined as S(0), 2 is defined as S(1)=S(S(0)), 3 is defined as S(2), and so on. S is not "plus one", it's "next", so 0 is the first, 1 is next, then goes 2 etc. There is no intrinsic addition or multiplication, just a sequence of things. You can define an addition over them, the video shows you how. You can also define multiplication: a * 0 = 0; a * S(b) = a + (a * b). Funny enough, you don't even have equality out of the box. Let's define it: 0 == 0 = True; S(a) == S(b) = a == b; _ == _ = False
S(1)=2 because 2 is just a symbol that means nothing . I mean we can define set of natural numbers as N = {0, S(0), S(S(0)), S(S(S(0))), and so on …}. Numbers are just a symbols.
@@daniiltonkonog186 right, we can also talk about bijections between {Z,S(Z),S(S(Z)),S(S(S(Z))), ...}, {0,1,2,3,...}, {zero,one,two,three,...}, and other sets that enumerate natural numbers
Ngl, I clocked out after the 2 stars.
This looks like something b russel would try and failed
Use a numberline and get two ones they take up area one when "added together" (one put in front of the other) they take up an area of 2 so 1+1 must be 2
This is some terrance Howard level stuff
Somebody send this to Terrence Howard
1+1=10
Terrence Howard does not like this video 🤣🤣🤣
you haven't shown that n = m implies S(n) = S(m) which you use when you say S(1+0)=S(1). You also haven't shown why 1≠0. you could use 1=S(0) and S(n)≠n but that hasn't been shown either
S(n) is a function which is by definition right-definite; that is the property you're looking for.
The second claim can't be proven from the axioms given in the video. The peano axioms have a statement for this: "zero isn't the successor of any number" or in first order logic, forall n: Nat, S(n) ≠ 0.
Bertrand Russell approves. Now, prove that 1 + 2 = 3
question: if b=1 is the successor of 0 then wouldn't a=1 also be a successor of 0 the the equivalence of a+b would be S(0+0) which is 1. please help to explain
Prove "1" = "one" and that "2" = "two."
For that matter, prove "1" or "one" = the concept to what they refer. You can't. These things are only true because we stipulated them as such. We could just as easily have stilulated the numeral "1" to correspond to the word "eight," and for both of those to correspond to the idea of 962.
The whole assumption of the video is that we have already defined the sequence of natural numbers. You are completely correct in that we have *defined* that 2 is the successor of 1, and that is completely arbitrary. It could have been 3, or 4, or X, or any other scribble. But that's not the point of the video. We assume that 2 is the successor of 1, and prove that 1 + 1 = 2. More precisely, we define the semantics of the addition operator using inductive reasoning, and then apply it to the case where the left- and right hand side of the addition operator are 1, which turns out to be 2, which in turn proves 1 + 1 = 2.
Proving without definitions is impossible. Otherwise, what are you proving? It could be anything.
*Counting* is considered more basic or fundamental than addition, just like addition is more fundamental than multiplication and multiplication is more fundamental than power.
Forward counting is more fundamental than backward counting. Incrementing by 1 is more fundamental and addition can be defined on top of incrementing more than once.
In a nutshell, 1+1=2 is more fundamental than several assumptions that you have made in the video (which of course are indeed true and can be proved after assuming that some symbol "2" comes next after we define the symbol "1" to be the very first symbol and the start of counting).
1+1 is not more fundamental than the Peano axioms. It seems that way because it is easier to comprehend and/or because the foundations of math are complicated (at least to most people). But it definitely is not "more fundamental"
Counting is exactly what the successor function formalised. The successor of a number is simply the number that comes next. That you can represent the successor by adding 1 is nontrivial and exactly one of the addition axioms.
Some mathematicians prefer to use set theory to define the natural numbers. So that
0 = {} is the empty set and 1={{}} is the set containing the empty whereas 2={{}, {{}}} and 3={{}, {{}},{{}, {{}}}}
By that definition sets theory is more fundamental than numbers.
Thought it's by definition; i.e,. 2 is defined as being the successor of 1. What suggests thr operation isn't well defined?
What about fields? 1+1 = 0 in a field of characteristic 2, or, 1+1 = 0 (mod 2).
I think it’s more accurate to say that 2 is 1+1. 2 is, by definition, the number that comes immediately after 1, and is thus the successor of 1 (which is 1 + 1)
1️+1️=11
Finally… the correct answer 🤪
I put one apple in a basket and another apple in the basket and now I have two apples❤
The equation 1 + 1 = 2 is one of the most basic concepts in mathematics. It’s based on the idea that the number "1" represents a single unit or object. When you add another "1" to it, you’re combining two separate units. This combination gives you "2," which represents two units together.
Mathematically, this can be explained using the Peano axioms, which are basic rules that define how numbers work. These rules tell us that if you have 1 and you add 1 more, the result is the next number, which is 2.
This idea is not just about counting; it’s fundamental to all of mathematics. Whether you’re adding numbers in a simple equation or using math in complex science and technology, the concept that 1 + 1 = 2 is a building block that everything else relies on.
In essence, 1 + 1 = 2 is true because of how we define and understand numbers, and it’s a principle that applies universally, no matter what you’re counting or calculating.
Now prove that addition is commutative!
Next he's gonna prove how every 60 seconds in Africa, a minute passes
except for minutes with leap seconds...
I thought it was by definition
It is, this proof is wrong. This proves that 1+1=S(1) but of course the last step S(1)=2 is by definition... So not a proof, just a definition.
@@gabrielbarrantes6946 I think you could theoretically prove it by using set theory…
How succesor of 1 can be 0 if there is no way to get 1 by adding 0?
You have it backwards, the successor of 0 is 1.
@@ianmccurdy1223 k
Or just stick to the take one object and then take another of that, you'll have two. I'm convinced.
Just get 1 cookie and another cookie and you get 2
Nat = Z : Nat | S : Nat -> Nat
1. a + Z = a
2. a + S(b) = S(a + b)
S(Z) + S(Z) =[#2]=> S(S(Z) + Z) =[congruence #1 over S]=> S(S(Z)) Q.E.D.
It is the definition of '2' to be the succesor of 1 ..
when the teacher asks you to show your work ahhhh video
CGP Beige
A book called Principia Mathematica has 360 pages of theorems to get to 1+1=2
everyone knows 1 + 1 = 10
we all need to get on the binary train
So now there are three ones?? This is madness.
I don't understand why you'd need to "prove" this. Addition is an operator, a function. It is defined. It is defined as two when inputing 1 and 1. 1•1 -> 2
Well it’s not just 1+1. You also need to prove that for the set of natural numbers, addition and multiplication are distributive, associative, and commutative. A proof of just 1+1=2 is just one line long but proving that multiplication is associative is a bit more challenging. Do you know how to prove multiplication is associative and together with addition being distributive? It’s kinda fun.
I'm not sure if S(1+0) is equal to S(1), I need proof
He showed that, by definition, a+0 = a
We know by definition that "x+0=x" for an arbitrary "x".
So, we have that "1+0=1" is true, and by definition of "=" we have that given an arbitrary property "φ", then "φ[1+0] φ[1]" is tautology.
So, particularizing "φ[x]" as "1+1=S(x)" for an arbitrary "x", as he's already proven in the video that "φ[1+0]" is true, by modus ponens we have "φ[1]" is true, and therefore "1+1=S(1)" is true.
So, by transitivity and symmetry law, we have: "x=y" and "y=z" implies "x=z", and "x=y" implies "y=x" for "x", "y" and "z" arbitrary. Therefore, as we have "1+1=S(1+0)" implies "S(1+0)=1+1" and "1+1=S(1)" is true (as we've already proven), we conclude that "S(1+0)=S(1)". □
@@GabriTellNicely done!
@@GabriTellNicely done
Easy, S(S(0)+0) =[congruence of a+0=a over S]=> S(S(0))
Please try to integrate 1/x^5+1
Actually its a window if you combine it all
2 is just a symbol to represent what 1+1 will be
We can call it anything
Yes that's the truth but so called mathematical experts and genius believes that 1+1 can be proven mathematically .
Thats ridiculus you answered the question with question itself so why 1+0=1 dont need these weird things tp prove that you have fingers in one hand open 1 finger also in another hand and approach them so there is 2 fingers thats all
fingers are not proof
Before symbols (conventions) there were fingers.
I might not understand something but didn't you use an addition operation while defining the addition operation? (When you said, "a + b = S(a + c)...")
Terrance Howard : hold my crack pipe
but this is all predicated on the understanding that we're incrementing by 1. like you still need to prove that the number line increases in that order, right? is the natural number axiomatically, objectively true, or is that also something the can be proven
To prove means to deduce WITHIN a THEORY
And a THEORY is a structured system with a number of premises (axioms) and rules of inference (logic)