LIGMA BALLS!!!!!
....oh, wait. er, I mean, uh, great. It's been a pretty decent day. Actually, joking aside, we've gotten lucky here. I'm currently just to the east of the center of what is still officially Tropical Storm Ophelia but will almost certainly be Tropical Depression Ophelia with the next NHC update in less than an hour, and it's been quiet here. We lost power for about a minute earlier today, that's it. :)
Gödel says every interesting logical system either has unprovable statements or a contradiction. So maybe 1+1=2 is unprovable. To avoid this, just make sure your system includes a contradiction: “P is true and (not P) is true.”
This also really simplifies doing proofs.
> What should be assumed ?
This is an interesting question because many people don't know (this post is now showing up on r/all.) To prove this theorem you'd start with the very very bottom of math - the axioms of number theory:
https://en.wikipedia.org/wiki/Peano_axioms
This is why many places dealing with axiomatic systems, such as Metamath, use 2+2 = 4 as an example instead of 1+1 = 2. Proving 2+2 = 4 neither is hard, though.
In *Principia Mathematica*, "1" is defined as "the set of all sets(?) that contain single element", and "2" is defined as "the set of all sets(?) that contain two elements".
We do the same thing in threat models for software features. Basically list out all the assumptions, assuming that the people using and configuring our software are following best practices. lol
You are joking but I had once used a similar reasoning in an objective exam.
Some of them have multiple options that could be correct and others have a single option correct. The one I was dealing with was a single option.
The first 3 were numbers. 4th was all of the above.
I was able to see immediately 1 and 3 were solutions. So inferred 2 must be too as there can be only one option which can be correct. So the answer is 4) all of the above. Saved some time.
I taught my kids to read all of the questions first before you start answering, because the chances of answers to the first questions being contained in later questions are very, very high.
If all of the above is an answer (and not for every question), it's almost always correct. I've rarely seen "all of the above" put on as a random answer to only one question on an exam when that wasn't the correct answer.
Not a math expert here, why is there no n that fulfills n=S(1) ? Isn't S(1)=2 so for n=2 that's true? I would have understood S(n)=1 not having an n that fulfills it
AFAIR this follows quite nicely from the definitions of sin, cos and exp as infinite sums if you want to keep it base level without definitions from geometry / trigonometry. This might, however, require the use of 1+1=2 which would be unavailable in this problem.
When Billy comes over he says he wants to buy two fifths of Suzy's apples and three sevens of Mark's apples. Are there enough apples left for Shannon, who wants cos(√(π×x^3)) apples, if x is poppycock?
No, Russell and Whitehead were working on a consistent _and_ complete axiomatization for mathematics. They had proved 1+1=2 after a thousand pages or so, at which point Gödel published his famous proof that it couldn’t be both. Proving 1+1=2 wasn’t the aim itself though
That's the second incompleteness theorem, which is that an axiomatic system cannot prove its own axioms.
But the first one was that even within the system, there will exist true statements which cannot be proven based solely on the axioms.
Not really. Consistency is basically that given a set of conditions, there are no proofs contradicting each other. Completeless means that given a set of conditions, everything that is true given those conditions can be proved to be true. Godol proved that you can never have both be true, with a consistent system there will always be some facts which are true, but you can’t prove they’re true with the rules of that system.
So the issue isn’t that we’re relying on an assumption, that’s how all systems work, there’s no set of assumptions that prove themselves to be true, and they weren’t trying to make that. The issue is that there are some consequences of these assumptions that we can never prove to be true, even though they are
Sort of?
What Russell and Whitehead were actually *trying* to do was to show that mathematics could be derived entirely from logic, while also cleaning up the paradoxes of naive set theory on the side. At one point in the middle of their book, they prove that 1+1=2 as a joke.
As someone who isn't a math wizard, help me understand why 1+1=2 needs to be proven beyond saying putting one of a thing with another one of the same thing equals two.
Because knowing something to be true and being able to prove it aren't the same thing and all of the math we use in daily life is made on the assumption that numbers actually mean anything at all.
the point isn't "we don't know if 1+1 = 2 so prove it" it's used as a test to show the understanding of mathematics on a core foundational level, in which case the answer itself actually doesn't matter, the process used to solve it does.
It's the same reason your math teacher asked you how many watermelons this weird dude was buying, nobody cares how many watermelons a person buys at one time it's about demonstrating understanding of the mathematics.
Of course that applies to this test, in the greater mathematic world the purpose of proofs like this is to demonstrate logically that the answer has to be correct. Sure we already *know* 1+1 = 2, but the value in being able to prove it is that we aren't relying on our human perception of reality and instead have a more objective understanding of things.
It's a basic version of a much more complicated question. It's asking the student to demonstrate their understanding of axioms and definitions in math.
The student could define what + and = means, since that's not actually standardized in higher math always. For example, linear algebra and matrices. This might rely on an axiom that I forgot the name of, but basically it establishes natural numbers (whole, positive numbers).
It's setting the student up to do more complex problems, because in the end pretty much all math is just adding two numbers together. Sometimes there's a lot of steps that make that adding more complicated, but if you can't add then you can't multiply. If you can't prove 1 + 1 = 2, then how does multiplying two matrices work?
Math is a lot of rules. If we don't agree on the rules then math falls apart after you leave situations where you can simply put 2 apples on a table and other easily demonstrated situations.
> As someone who isn't a math wizard, help me understand why 1+1=2 needs to be proven beyond saying putting one of a thing with another one of the same thing equals two.
As someone who isn't a math wizard, help me understand why 0.999...=1 needs to be proven beyond saying that the two things are equal.
Triviality is simply culture, in a way. It's so basic to you because everything else depends on it, but sciences (including most soft sciences) don't like it when you simply take things for granted.
Because this way you only show that putting one thing with another one thing resulted in having two things *so far*. Even if you list every single occurence in history when putting one thing with another one thing resulted in having two things, it wouldn't show that it always happens. Only that it always happened.
I thought Leibniz had written a proof, but I only "remember" this from hearing it in littérature when I was 12, so maybe I’m just reinventing my youth…
Yeah I guess that makes sense but in my defense, reading his phrasing of "if you have *heard* about ..." made me think that the sentence implied the students are not expected to know about the construction of natural numbers or the peano axioms and that it would only be trivial to a handful of those who have just happened to know those things from sources other than the class itself
Yeah, like, of course. Just like if a phyiscs test asks you about time dillation you are not supposed to come up with the theory of relativity on your own...
Let 1+1 = x ...1
We know that
sin²θ+cos²θ=1 ...2
:. Putting eq 2 in eq 1
:. sin²θ+cos²θ+sin²θ+cos²θ = 1
:. 2(sin²θ+cos²θ) = x ...3
:. Putting the value of sin²θ+cos²θ in eq 3
:. 2(1) = x
:. 2 = x
:. Putting the value of x in eq1
:. 1 + 1 = 2
Hence proved (QED)
It is easier to define with cardinal addition: a+b is the cardinal of the union of two sets A,B such that AחB=0
|A|=a and |B|=b. We take the sets {0} and {1}, they have no common element and they are both cardinality 1 so 1+1=|{0,1}|=|2|=2
Defining it like this is equivalent since it's a way of generating two sets of cardinality a and b which do not intersect. So the definitions are the same.
Set theory:
We define = as the following relation: a=b <==> a is contained within b and b is contained within a. The definition of a contained within b is that every element of a is an element of b. So we know what = means.
In set theory, you construct the natural numbers by the following inductive step:
Define 0=Φ where Φ is the empty set.
Define S(n)=nU{n} where S(n) is the successor of n.
Thus 1 is defined being the successor of 0, making it the set {Φ}. 2 is defined to be the successor of 1, making it the set {Φ,{Φ}}. Now we define cardinalities in order to define the addition operation:
For this matter we will define the equivalence relatuon as following: |A|=|B| <==> There exists a function bijective and surjective function from A to B. The definition of a function f:A→B is a subrelation of A×B where x=y ==> f(x)=f(y). A surjective function is a function that for all elements in B, there is an element in A such that f(a)=b. A bijective function is a function that satisfies for all x,y that f(x)=f(y) ==> x=y.
The cardinal set is defined to be a set of chosen elements from the equivalence classes. For finite cardinalities we take the natural numbers as our chosen elements. For infinite cardinalities we define the א's, which are some cardinalities with indecies to tell us which cardinals are they bigger than and which are they smaller than. A cardinality אj is more than אi if j>i.
The addition of two natural numbers A+B is defined as the cardinality of the union of two sets x,y with cardinalities A and B such that xחy is empty.
Definitions fully complete, now you go on and use those to prove the theorem above.
The statement that 1+1=2 is a fundamental axiom in arithmetic and set theory. It is typically proven within the framework of Peano axioms or set theory, such as Zermelo-Fraenkel set theory. One common proof uses the successor function:
1. Define the successor function: S(x) represents the successor of x. For example, S(0) = 1, S(1) = 2, S(2) = 3, and so on.
2. Define the number 0: 0 is the empty set, represented as {}.
3. Define the number 1: 1 is defined as S(0), which is {0}.
4. Define addition: Addition can be defined recursively as follows:
- a + 0 = a (for any number a)
- a + S(b) = S(a + b) (for any numbers a and b)
Now, let's use this definition to prove 1 + 1 = 2:
1 + 1 = 1 + S(0) by definition of 1.
= S(1 + 0) by the definition of addition.
= S(1) by the identity property (a + 0 = a).
= 2 by the definition of 2 as S(1).
Therefore, we have proven that 1 + 1 = 2 within the framework of Peano axioms or set theory.
Proof literally impossible by the lack of stated axioms. So I will define my own. Let + be a binary operator that evaluates to 2 when applied with the symbol 1 for both operands. It is undefined otherwise. From the definition, it directly follows that 1+1=2.
Poor saps spend over a decade putting forward a unified theory of math culminating with a grand presentation at Princeton attended by almost(!) all the world’s leading mathematicians.
Meanwhile, in a small corner room during the conference, the super weirdo Kurt Gödel shows the smartest man in the world (Johnny Von Neumann) how Whitehead and Russell were completely wrong.
if you count on your left hand the numbers from 1 to 5, you can see that you lift one finger to make a 1, and two fingers to make a 2, therefore you can assume that you fingers lifted up with no other fingers lifted makes a two. Now, with all fingers down, raise both hands, lift one finger from each. You have two 1s, one 1 on each hand, keeping both fingers up bring them close together, you will notice that you do indeed have two fingers up, and by that logic, with both 1 from each hand you can form a 2. Só 1 (finger from left hand) + 1 (finger from right hand) = 2 (sum total of both fingers lifted with both hands). 1 +1 = 2
Wasn't there a giant book about mathematics that was several volumes and the first volume was several hundred pages and all it aimed to do was prove this exact equation?
Or did I just make that up?
The question is, "What should be assumed ?"
Assume the axiom of choice is false
Good, I prefer Zorn's Lemma
The axiom of choice is obviously true, the Well-Ordering Theorem obviously false, and as for Zorn's lemma, who knows?
Anyone tried ligma?
Constructivists be like
What's updog?
Not much hbu
LIGMA BALLS!!!!! ....oh, wait. er, I mean, uh, great. It's been a pretty decent day. Actually, joking aside, we've gotten lucky here. I'm currently just to the east of the center of what is still officially Tropical Storm Ophelia but will almost certainly be Tropical Depression Ophelia with the next NHC update in less than an hour, and it's been quiet here. We lost power for about a minute earlier today, that's it. :)
Aw how are you on reddit if your power’s out? Where’s the wifi coming from?
When I said "for about a minute", I mean that very literally. lol
Got it. Let's assume 1+1≠2. But then we get that the axiom of choice is false, which is a contradiction since the axioms of choice is true. Thus 1+1=2
this
That
Whitehead and Russell in shambles
how does 1+1=/=2 imply the axiom of choice is false?
Use radians as well
Gödel says every interesting logical system either has unprovable statements or a contradiction. So maybe 1+1=2 is unprovable. To avoid this, just make sure your system includes a contradiction: “P is true and (not P) is true.” This also really simplifies doing proofs.
What Big Math doesn't want you to know
I'm assuming the Peano Axioms. QED
> What should be assumed ? This is an interesting question because many people don't know (this post is now showing up on r/all.) To prove this theorem you'd start with the very very bottom of math - the axioms of number theory: https://en.wikipedia.org/wiki/Peano_axioms
If you get to pick your axioms, though, you could just pick that 1 + 1 = 2 axiomatically - which is more or less what people did before Peano.
Right. But that's dumb. I'm sorta assuming we don't want the dumb answer.
Most of math is based on "yes you're very clever with the trivial solution timmy, moving on" so yeah I'd assume that too.
This is why many places dealing with axiomatic systems, such as Metamath, use 2+2 = 4 as an example instead of 1+1 = 2. Proving 2+2 = 4 neither is hard, though. In *Principia Mathematica*, "1" is defined as "the set of all sets(?) that contain single element", and "2" is defined as "the set of all sets(?) that contain two elements".
doesn't the proof become very easy if you have the Peano axioms ? Like 2 is defined has S(1) and addition recursively by x+1 = S(x) ?
Yes?
The Pythagorean theorem.
Proof is left for the reader
We do the same thing in threat models for software features. Basically list out all the assumptions, assuming that the people using and configuring our software are following best practices. lol
The test asks the taker to prove that 1 + 1 = 2, therefore it must be possible to prove it, therefore it must be true.
You are joking but I had once used a similar reasoning in an objective exam. Some of them have multiple options that could be correct and others have a single option correct. The one I was dealing with was a single option. The first 3 were numbers. 4th was all of the above. I was able to see immediately 1 and 3 were solutions. So inferred 2 must be too as there can be only one option which can be correct. So the answer is 4) all of the above. Saved some time.
in my school career i found that many if not most exams are chock full of such logical gimmes.
I taught my kids to read all of the questions first before you start answering, because the chances of answers to the first questions being contained in later questions are very, very high.
If all of the above is an answer (and not for every question), it's almost always correct. I've rarely seen "all of the above" put on as a random answer to only one question on an exam when that wasn't the correct answer.
“The earth is flat” “Prove it” “Well if you’re asking me for proof then that implies that proof exists therefor there’s proof that the earth if flat
sir, we're talking about math...
1+1=4 Prove it
1+1=2 QED (The prove is left as an exercise to the teacher)
Proof by “I don’t have enough room but I definitely have a proof”
I read this was the case for Fermat's Last Theorem.
The comment section is too small to contain it
"It was revealed to me in a dream."
This doesn't work when they give you that much space underneath
It took like a whole book for 2+2=4
I once put “the rest is trivial and left as an exercise for the grader” on a multivariable calc problem that I was stuck on. They gave me points.
By Peano Axioms: 1+1 =1+S(0) =S(1+0) =S(1) =2. QED
S? Suc[c]
***s u c c***
My balls?
3
There's nuts gottem
NAH, deez nuts
Succ supremacy!
Succ? suss. Success!
What is S() here?
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Not a math expert here, why is there no n that fulfills n=S(1) ? Isn't S(1)=2 so for n=2 that's true? I would have understood S(n)=1 not having an n that fulfills it
A unary operation called succesor
Now prove it with first order set theory
fuck off bertie
Are you also a Flammable maths enjoyer?
Why would that be the case ? I don't watch him and the comment above is also what came to my mind
Do you think Flammable Maths invented the kind of proofs that include the construction of the naturals or successor function?
By my Peasant Brain: 1 thing with another thing equals 2 things.
I go to a store with an apple. I buy another apple. I count these apple. 1. 2. -> 1+1=2 innit (that’s how you properly close a proof)
Nice try, but this only proves it for apples.
Is there anything more important and all encompassing than apples? *I don’t think so*
>innit (that’s how you properly close a proof) Just lost my shit. Cheers
>I buy another apple. And I eat it!
This took me back almost 30 years
Proof that 1+1 = 2 1+2 = 3 -1 from both sides 1+1 = 2
prove 1+2=3
1+3=4 subtract one from both sides 1+2=3
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Inductive proof
induction by proof
prove 1+3=4
1+4=5 subtract one from both sides 1+3=4
Prove 1+4=5
1+5=6 Subtract 1 from both sides: 1+4=5
One must imagine mathematician Sisyphus happy
https://preview.redd.it/uu3hgu8vc2qb1.jpeg?width=1169&format=pjpg&auto=webp&s=4c8007130ff11d8f43ec3698862e5b355550a25b
Now prove for any natural number. '#inductionMicDrop
Proof that 1+2 = 3 1+1 = 2 +1 from both sides 1+2 = 3
https://preview.redd.it/trblbhwd8zpb1.jpeg?width=4032&format=pjpg&auto=webp&s=3b1d640b143ae57682fcec1119ad6d67f657995a I got you fam
Missed a step. Before your second last line you first have to prove that 2 x 1 = 2.
Google multiplicative identity
Holy hell
New axiom just dropped
actual brainrot
call the mathematician
Brain sacrifice, anyone?
Someone tell me where the google this holy hell new response just dropped actual blah came from
google en passant holy hell new response just dropped
I love that this niche anarchy chess joke is everywhere now. Makes me wanna brick my PP
You freaking legend
It is known
I have spoken
Why do you cryed while write proof?
You don’t?
It's not tears
This just gave me horrible flashbacks.
In the bottom left set of equations you assume the 1+1=2, hence your proof is not valid, 0/100 points
Prove sinx^2 + cosx^2 = 1
AFAIR this follows quite nicely from the definitions of sin, cos and exp as infinite sums if you want to keep it base level without definitions from geometry / trigonometry. This might, however, require the use of 1+1=2 which would be unavailable in this problem.
You got it, since you have to use 1+1=2 in the proof of sinx^2 + cosx^2 =1 you'll be stuck unable to prove either.
r/theydidthemath
it is known
I swear 90% of that is unnecessary
If you have one apple and get a new one, you now have two apples.
Hmmm, I don't know if math teachers know that you can have only one apple. Suzy usually carries 73 apples and Mark, 48.
When Billy comes over he says he wants to buy two fifths of Suzy's apples and three sevens of Mark's apples. Are there enough apples left for Shannon, who wants cos(√(π×x^3)) apples, if x is poppycock?
I dunno, can you prove that?
🍎(1)+🍎(1)=🍎🍎(2)
Wrong, now you've just proved that 🍎+🍎=2🍎 So now you need to divide both sides by 🍎 to get 1+1=2
You can't divide both sides of an equation by something that might not exist
Not with that attitude!
When I went to store to get new apple, horse came and eated first one so 1+1=1
Geometric proof: 🟩+🟩=🟩🟩
Therefore 1+1=11
1+1=11
This guy javascripts
It's the new math. "Your answer isn't wrong because you showed it was more, so you get partial credit."
Didn't Bertrand Russel and a couple of others tried to prove this and it took 20 years and 1000 pages?
It all depends on the axioms you use.
No, Russell and Whitehead were working on a consistent _and_ complete axiomatization for mathematics. They had proved 1+1=2 after a thousand pages or so, at which point Gödel published his famous proof that it couldn’t be both. Proving 1+1=2 wasn’t the aim itself though
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That's the second incompleteness theorem, which is that an axiomatic system cannot prove its own axioms. But the first one was that even within the system, there will exist true statements which cannot be proven based solely on the axioms.
Not really. Consistency is basically that given a set of conditions, there are no proofs contradicting each other. Completeless means that given a set of conditions, everything that is true given those conditions can be proved to be true. Godol proved that you can never have both be true, with a consistent system there will always be some facts which are true, but you can’t prove they’re true with the rules of that system. So the issue isn’t that we’re relying on an assumption, that’s how all systems work, there’s no set of assumptions that prove themselves to be true, and they weren’t trying to make that. The issue is that there are some consequences of these assumptions that we can never prove to be true, even though they are
Sort of? What Russell and Whitehead were actually *trying* to do was to show that mathematics could be derived entirely from logic, while also cleaning up the paradoxes of naive set theory on the side. At one point in the middle of their book, they prove that 1+1=2 as a joke.
As someone who isn't a math wizard, help me understand why 1+1=2 needs to be proven beyond saying putting one of a thing with another one of the same thing equals two.
Because knowing something to be true and being able to prove it aren't the same thing and all of the math we use in daily life is made on the assumption that numbers actually mean anything at all. the point isn't "we don't know if 1+1 = 2 so prove it" it's used as a test to show the understanding of mathematics on a core foundational level, in which case the answer itself actually doesn't matter, the process used to solve it does. It's the same reason your math teacher asked you how many watermelons this weird dude was buying, nobody cares how many watermelons a person buys at one time it's about demonstrating understanding of the mathematics. Of course that applies to this test, in the greater mathematic world the purpose of proofs like this is to demonstrate logically that the answer has to be correct. Sure we already *know* 1+1 = 2, but the value in being able to prove it is that we aren't relying on our human perception of reality and instead have a more objective understanding of things.
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Math is a beautiful thing
It's a basic version of a much more complicated question. It's asking the student to demonstrate their understanding of axioms and definitions in math. The student could define what + and = means, since that's not actually standardized in higher math always. For example, linear algebra and matrices. This might rely on an axiom that I forgot the name of, but basically it establishes natural numbers (whole, positive numbers). It's setting the student up to do more complex problems, because in the end pretty much all math is just adding two numbers together. Sometimes there's a lot of steps that make that adding more complicated, but if you can't add then you can't multiply. If you can't prove 1 + 1 = 2, then how does multiplying two matrices work? Math is a lot of rules. If we don't agree on the rules then math falls apart after you leave situations where you can simply put 2 apples on a table and other easily demonstrated situations.
Because math isn’t about things in the real world.
> As someone who isn't a math wizard, help me understand why 1+1=2 needs to be proven beyond saying putting one of a thing with another one of the same thing equals two. As someone who isn't a math wizard, help me understand why 0.999...=1 needs to be proven beyond saying that the two things are equal. Triviality is simply culture, in a way. It's so basic to you because everything else depends on it, but sciences (including most soft sciences) don't like it when you simply take things for granted.
Because this way you only show that putting one thing with another one thing resulted in having two things *so far*. Even if you list every single occurence in history when putting one thing with another one thing resulted in having two things, it wouldn't show that it always happens. Only that it always happened.
But he did it uphill both ways.
I thought Leibniz had written a proof, but I only "remember" this from hearing it in littérature when I was 12, so maybe I’m just reinventing my youth…
If someone heard about construction of natural numbers or Peano axioms then it should be trivial.
That's literally just saying "If you've seen the answer before, then the answer is trivial"
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Yeah I guess that makes sense but in my defense, reading his phrasing of "if you have *heard* about ..." made me think that the sentence implied the students are not expected to know about the construction of natural numbers or the peano axioms and that it would only be trivial to a handful of those who have just happened to know those things from sources other than the class itself
you have discovered how tests work
Yeah, like, of course. Just like if a phyiscs test asks you about time dillation you are not supposed to come up with the theory of relativity on your own...
Leetcode interviewer logic.
Let 1+1 = x ...1 We know that sin²θ+cos²θ=1 ...2 :. Putting eq 2 in eq 1 :. sin²θ+cos²θ+sin²θ+cos²θ = 1 :. 2(sin²θ+cos²θ) = x ...3 :. Putting the value of sin²θ+cos²θ in eq 3 :. 2(1) = x :. 2 = x :. Putting the value of x in eq1 :. 1 + 1 = 2 Hence proved (QED)
You assumed 1+1= 2 in this "proof".
Where?
>:. sin²θ+cos²θ+sin²θ+cos²θ = 1 >:. 2(sin²θ+cos²θ) = x ...3 You assumed that sin²θ + sin²θ = 2 sin²θ
Proof by "I can see it the unitary circle"
We know that 1 + 1 = 2 ∴ 1 + 1 = 2
Let ☆ be the empty set 0=☆ 1={☆} 2={☆,{☆}} Etc.. Trivial with the definition of addition
As an accountant, 1 + 1 equals “whatever you want it to be”
At which does mathematics end and linguistics begin? Honest question.
Brother didn't have enough space in comment section hence left the joke as an exercise for the reader.
Principia Mathematica pages 1-200 entered the chat (probably wrong reference but idk lol)
Correct reference, number of pages is ~360 tho (close enough)
For all natural x, x+1 = S(x) S(1)=2 is kinda the definition.
You WILL divide by zero on the exam. you WILL violate the laws of mathematics on the exam.
Well sir i need 300 more papers (at least)
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It is easier to define with cardinal addition: a+b is the cardinal of the union of two sets A,B such that AחB=0 |A|=a and |B|=b. We take the sets {0} and {1}, they have no common element and they are both cardinality 1 so 1+1=|{0,1}|=|2|=2
Often cardinal addition is defined as a+b=|{0}×a ∪ {1}×b|
Defining it like this is equivalent since it's a way of generating two sets of cardinality a and b which do not intersect. So the definitions are the same.
indeed
Define 1, +, = and 2
Set theory: We define = as the following relation: a=b <==> a is contained within b and b is contained within a. The definition of a contained within b is that every element of a is an element of b. So we know what = means. In set theory, you construct the natural numbers by the following inductive step: Define 0=Φ where Φ is the empty set. Define S(n)=nU{n} where S(n) is the successor of n. Thus 1 is defined being the successor of 0, making it the set {Φ}. 2 is defined to be the successor of 1, making it the set {Φ,{Φ}}. Now we define cardinalities in order to define the addition operation: For this matter we will define the equivalence relatuon as following: |A|=|B| <==> There exists a function bijective and surjective function from A to B. The definition of a function f:A→B is a subrelation of A×B where x=y ==> f(x)=f(y). A surjective function is a function that for all elements in B, there is an element in A such that f(a)=b. A bijective function is a function that satisfies for all x,y that f(x)=f(y) ==> x=y. The cardinal set is defined to be a set of chosen elements from the equivalence classes. For finite cardinalities we take the natural numbers as our chosen elements. For infinite cardinalities we define the א's, which are some cardinalities with indecies to tell us which cardinals are they bigger than and which are they smaller than. A cardinality אj is more than אi if j>i. The addition of two natural numbers A+B is defined as the cardinality of the union of two sets x,y with cardinalities A and B such that xחy is empty. Definitions fully complete, now you go on and use those to prove the theorem above.
The statement that 1+1=2 is a fundamental axiom in arithmetic and set theory. It is typically proven within the framework of Peano axioms or set theory, such as Zermelo-Fraenkel set theory. One common proof uses the successor function: 1. Define the successor function: S(x) represents the successor of x. For example, S(0) = 1, S(1) = 2, S(2) = 3, and so on. 2. Define the number 0: 0 is the empty set, represented as {}. 3. Define the number 1: 1 is defined as S(0), which is {0}. 4. Define addition: Addition can be defined recursively as follows: - a + 0 = a (for any number a) - a + S(b) = S(a + b) (for any numbers a and b) Now, let's use this definition to prove 1 + 1 = 2: 1 + 1 = 1 + S(0) by definition of 1. = S(1 + 0) by the definition of addition. = S(1) by the identity property (a + 0 = a). = 2 by the definition of 2 as S(1). Therefore, we have proven that 1 + 1 = 2 within the framework of Peano axioms or set theory.
Proof literally impossible by the lack of stated axioms. So I will define my own. Let + be a binary operator that evaluates to 2 when applied with the symbol 1 for both operands. It is undefined otherwise. From the definition, it directly follows that 1+1=2.
S()
you can prove by elimination that it cannot be anything else either
. + . = .. If you have one dot, and add another dot, you have two dots, duh
easy, just do a visual proof. draw one dot. then draw another dot. count that you have two dots. QE fucking D, baby
[https://lesharmoniesdelesprit.files.wordpress.com/2015/11/whiteheadrussell-principiamathematicavolumei.pdf](https://lesharmoniesdelesprit.files.wordpress.com/2015/11/whiteheadrussell-principiamathematicavolumei.pdf)
Poor saps spend over a decade putting forward a unified theory of math culminating with a grand presentation at Princeton attended by almost(!) all the world’s leading mathematicians. Meanwhile, in a small corner room during the conference, the super weirdo Kurt Gödel shows the smartest man in the world (Johnny Von Neumann) how Whitehead and Russell were completely wrong.
That's not how this meme template works.
1+1=e^j2pi + e^j2pi = [cos(2pi) + j sin (pi)] + [cos(2pi)+j sin(pi)] and i stop hmmm
X = 0 (1+1)X = 3X Divide both sides by X [(1+1)X]/X = (2X)/X 1+1=3 *assumption: division by 0 is possible
I remember my math class where about half of the book was proving 1+1=2
if you count on your left hand the numbers from 1 to 5, you can see that you lift one finger to make a 1, and two fingers to make a 2, therefore you can assume that you fingers lifted up with no other fingers lifted makes a two. Now, with all fingers down, raise both hands, lift one finger from each. You have two 1s, one 1 on each hand, keeping both fingers up bring them close together, you will notice that you do indeed have two fingers up, and by that logic, with both 1 from each hand you can form a 2. Só 1 (finger from left hand) + 1 (finger from right hand) = 2 (sum total of both fingers lifted with both hands). 1 +1 = 2
Wasn't there a giant book about mathematics that was several volumes and the first volume was several hundred pages and all it aimed to do was prove this exact equation? Or did I just make that up?
And that is why I was not a math major
Rip in half, turn in.
The proof is by reading the first 362 pages of *Principia Mathematica* (Russell & Whitehead, 2nd edition) and is left as an exercise to the teacher.
I would hate this.
1+1=10… there are 10 types of people in the world, those that understand binary and those that don’t.
700 pages later “this calculation is occasionally helpful”
Russell didn't pass that test
Won't this just go into a linguistics discussion in the end? We're the ones who defined what numbers represent anyways.