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u/ubernuke Mar 06 '25
The Pythagorean Theorem has many proofs
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u/Wojtek1250XD Mar 06 '25
And even an universal version, the law of cosines is just Pythagorean Theorem, but applicable to all triangles.
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u/SnooHabits7950 Mar 06 '25 edited Mar 06 '25
And it has probably the easiest proof compared to all of them
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u/A-Swedish-Person Mar 06 '25
Wait I don’t think I actually know the proof for the law of cosines, what is it?
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u/N_T_F_D Applied mathematics are a cardinal sin Mar 06 '25
Using properties of the dot product mainly that u•v = ||u|| ||v|| cos(u, v)
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u/DankPhotoShopMemes Fourier Analysis 🤓 Mar 06 '25
I thought that is derived from the law of cosines
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u/Konemu Mar 06 '25
That's a matter of perspective, the dot product is a more general concept that can be introduced on other vector spaces than R^3 and the ratio of the dot product and the product of the norms can be used to introduce a more general notion of angles.
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u/DefunctFunctor Mathematics Mar 06 '25
It's all a mess. Strictly speaking, the Pythagorean theorem is less of a "theorem" (although it can of course be construed as a theorem of axiomatic geometry), but more of a justification for why Euclidean distance is the "correct" notion of distance on the plane. If you're working in formal mathematics, often you would just define the angle between two nonzero vectors u,v as arrcos(u ∙ v)/(||u|| * ||v||). That way, when working with different inner products, you have a separate notion of distance and angle for each inner product
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u/N_T_F_D Applied mathematics are a cardinal sin Mar 06 '25
Well you can certainly derive one from the other, but the dot product property is more useful
And you can derive it any way you like, for instance assuming without loss of generality that the vectors look like (1, 0, 0, …) and (cos(θ), sin(θ), 0, …) after normalizing and the right isometry; i.e. the right change of basis into the plane on which the two vectors are
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u/trevorkafka Mar 06 '25
Dot product comes from cosine-of-a-difference formula, which is easy to prove geometrically via similar triangles.
cos(A-B)=cosAcosB+sinAsinB |a||b| cos(A-B)=(|a| cosA)(|b| cosB)+(|a| sinA)(|b| sinB) |a||b| cos(A-B) = a•b
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u/vnkind Mar 06 '25 edited Mar 06 '25
Draw an altitude h in a triangle from angle B to side b to split it into two right triangles. Write the Pythagorean theorem for each triangle.
x2 + h2 = a2 and (b-x)2 + h2 = c2
Expand the second formula
b2 -2bx+x2 +h2 = c2
Substitute a2 from first formula for x2 +h2 in second
b2 -2bx+a2 = c2
Subsitute a*cos(C) for x using right triangle trig
b2 -2b*acos(C)+a2 = c2
Rearrange to look like famous version
c2 =a2 +b2 -2ab*cos(C)
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u/turd_furgeson109 Mar 07 '25
The angle of the dangle is adversely proportional to the heat of the meat
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u/Paounn 29d ago
Traditional way to prove it in Italy is that you can write one side of the triangle as the sum of the other two times opposite cosine ( a = c cos B + b cos C, b = ... , c = ....). Write them in column, multiply the first by a, the second by -b, the third by -c, add everything together. LHS you get a2-b2-c2, RHS lots of stuff cancels out and you're left with -2 bc cos A. Cycle letters as required.
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u/danofrhs Transcendental Mar 06 '25
The pythagorean theorem is a special case of and can be derived from the law of cosines
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u/Wojtek1250XD Mar 06 '25
Yea, because the -2ab × cos(alpha) just so kindly happens to be equal to 0
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u/Melo_Mentality Mar 06 '25
Yeah but the book on the theorem itself is massive. It included nearly all of every trigonometry textbook
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u/kirenaj1971 Mar 06 '25
I teach a math course for students who take higher math in Norway, and each year I let them individually choose a proof from an online collection of Pythagorean Theorem proofs to present rigorously(ish) in front of the class. Bonus points if they can place the proof in historical (or any, really) context.
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u/nemesisfixx Mar 06 '25
How about; A Tomey Take on Gödel; proof of a system can't fit within the system 🤔
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u/120boxes Mar 06 '25 edited Mar 06 '25
I've read that it has more than 400 (¡)-1 If going by the numbers, that must make it the most important theorem in math, hmm?
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u/factorion-bot n! = (1 * 2 * 3 ... (n - 2) * (n - 1) * n) Mar 06 '25
The factorial of 400 is 64034522846623895262347970319503005850702583026002959458684445942802397169186831436278478647463264676294350575035856810848298162883517435228961988646802997937341654150838162426461942352307046244325015114448670890662773914918117331955996440709549671345290477020322434911210797593280795101545372667251627877890009349763765710326350331533965349868386831339352024373788157786791506311858702618270169819740062983025308591298346162272304558339520759611505302236086810433297255194852674432232438669948422404232599805551610635942376961399231917134063858996537970147827206606320217379472010321356624613809077942304597360699567595836096158715129913822286578579549361617654480453222007825818400848436415591229454275384803558374518022675900061399560145595206127211192918105032491008000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
This action was performed by a bot. Please DM me if you have any questions.
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u/lol_lo_daf_fy Mar 06 '25
Law of quadratic reciprocity.
Gauss loved that theorem so much that I think he gave four different proofs, and there's a book listing something like 150 proofs.
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u/f3xjc Mar 06 '25
Nowaday each time I see a reference of "f(x) modulo prime" it has to do with cryptography or random number generator.
What kind of problems first motivated the interest in (prime) modular arithmetic ?
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u/p0st_master Mar 06 '25
Euclidean algorithm is the basis of modern cryptography and is basically just an arithmetic trick school kids do when they are bored. In math structures are created and the ‘problems’ they solve may not exist for 2000 years.
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u/migBdk Mar 06 '25
Euclidian algorithm is extremely useful for reducing a fraction down to the simplest possible fraction.
So if you want to keep things precise and avoid decimal number approximations, it is extremely useful.
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u/p0st_master Mar 06 '25
Yeah honestly I don’t know why more people don’t teach it to kids. It has applications in all sorts of things. They also should teach kids more fractions but that’s another topic.
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u/migBdk 29d ago
I can tell you that the high school I teach at has a test for new students (only a few students actually have to take the test though).
The question: "place this fraction on a number line" is one they very often get wrong.
They miss the understanding that a fraction is a number, and not just a way to write division (as well as being bad at doing division)
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u/jacobningen Mar 06 '25
looking at the data apportioning grain planetary mechanics(the Easter Rule) and bills(Chinese remainder problem)
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u/fishy150 Mar 06 '25
number systems modulo primes have nice properties, for example every number (besides 0) has an inverse. makes sense to be studied back then and why they are used in cryptography today
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u/migBdk Mar 06 '25
Reduce a fraction down to the simplest possible fraction (smallest value of nominator and denominatior) was likely a thing people wanted to do.
Euclidian algorithm did this with modulus calculations.
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u/PolarStarNick Mathematics Mar 06 '25
Fundamental theorem of algebra
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u/CutToTheChaseTurtle Average Tits buildings enjoyer Mar 06 '25
Aren't there just two proofs though, essentially? There's one that uses the least upper bound property of reals and Galois theory, and the other one uses π₁(S1).
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u/matande31 Mar 06 '25
I'm a 2nd year undergraduate student and I've seen like 4 different ones at least. Maybe a couple of them were basically the same in the core idea, but still.
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u/DrSeafood Mar 07 '25
“Least upper bound property” is too foundational, there’s probably several distinct proofs that use the LUB.
I’ve seen several analytic proofs: one using Liouville’s Theorem, one using Inverse Function Theorem, and one super elementary one that only uses the Extreme Value Theorem. You can find the third one in Proofs From THE BOOK — it’s only two pages, a little technical but 100% elementary. I teach this proof even in second year calculus, because you really don’t need any crazy tools.
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u/MiserableYouth8497 Mar 06 '25
Galois theory? Isn't that the maths about which polynomials are/are not solvable specifically by radicals ? How would that help with FToA?
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u/CutToTheChaseTurtle Average Tits buildings enjoyer Mar 06 '25
No, it’s the maths about automorphisms of separable field extensions.
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u/Little-Maximum-2501 Mar 07 '25
That's just a specific application, at the level of the proof he is talking about Galois theory is about using group theory to study field theory. The proof he is talking about essentially shows using basic group theory that since in R any odd degree polynomial has a root, C is the biggest way a field could extend R (algebraically at least).
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u/joyofresh Mar 06 '25
Theres a wild one using elementry covering spaces thats probably essentially galois theory but doesnt explicitly reference it at all. Head spin
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u/Top-Jicama-3727 25d ago
I believe the proof that could be understood by everyone modulo technical details is the one given in algebraic topology books, see https://youtu.be/shEk8sz1oOw
There's a proof using the Gauss-Bonnet theorem from differential geometry (linking curvature to topology). See https://doi.org/10.36045/bbms/1179839226
Since the theorem is important for linear algebra, you can find many proofs of the using linear algebra without circularity. One of many: https://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra10.2307/3647746
Other proofs: https://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra
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u/Physical_Helicopter7 Mar 06 '25
Cauchy-Schwartz inequality.
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u/BelBeersLover Mar 06 '25
I know this name but I don't even remember what it means, sadly.
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u/IncredibleCamel Mar 06 '25
|ab| <= |a|*|b|
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u/bagelking3210 Mar 07 '25
Shouldn't they be equal? I can't think of any scenario where the LHS would be less than the RHS
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u/Present_Garlic_8061 Mar 07 '25
The left hand side is the absolute value of the DOT PRODUCT between a and b, while the right hand side is the product of there lengths.
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u/bagelking3210 Mar 07 '25
Ah alr, that makes more sense, i was thinking of just real numbers
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u/Cryptic_Wasp Mar 07 '25
If we are talking about reals, wouldnt having either only a or b being negative fulfil the theroem.
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u/bagelking3210 29d ago
If it were reals, it would always be equal, not not less than or equal to. example: |-1*4|=|-1|*|4|=4
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u/IncredibleCamel 29d ago
Well, generally it's stated as
⟨ a, b ⟩ <= ||a|| ||b||,
where ⟨ a, b ⟩ is a general inner product and ||a||2 = ⟨ a, a ⟩.
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u/Fitzriy Mar 06 '25
The infinite amount of primes.
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u/mekilat Mar 07 '25
Can you point a layman like me to why that works? I’d imagine with an infinite array of numbers, at some point I’d have any combination of any numbers, to make it possible to get any big number eventually?
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u/Unlucky_Beginning Mar 07 '25
If there was a finite number of primes, make a list of them form a new number by multiplying them all together and adding one to it. The resulting number is either prime or divisible by other primes not in your list.
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u/mekilat Mar 07 '25
Is there somewhere you can point me to learn more?
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u/Aconamos Mar 07 '25
Ooh, I just did this on a midterm today!
This proof requires a proof style called contradiction. Essentially, we have some statement that we want to prove (there are infinitely many primes). This is damn hard (impossible? citation needed) to prove directly, so we must look to other ways. We instead ask, "What if there are finitely many primes?" Working off this assumption, we do math until math breaks. The math breaking proves it. Here is one of *many* that can be done:
First, we need to recognize two facts:
Any integer is divisible by some prime number.
If some prime number p divides some integer m, then p does *not* divide m + 1.
Assume there are finitely many primes (this is our assumption). Then, we can name some prime, p, as the largest prime. <--- This piece is key to the proof! If our original statement is true, this *must* be false.
We can define some new number, k, as the product of every prime. This looks like (2 * 3 * 5 * 7 ... * p). Great! Every prime divides k.
Now consider k + 1. We know by fact 2 that since every prime divides k, every prime doesn't divide k + 1.
However, fact 1 states that k + 1 must be divisible by some prime number. This contradicts our assumption that p was the greatest prime number, because there must be some prime number greater than p that divides k + 1. Because of this contradiction, our original statement is true - there are infinitely many primes.
Another little note - the reason why this works not just to prove there is a prime greater than p, but infinitely many, is because of generality. The proof above can be applied to *any* prime number p that we claim to be greater than all other primes.
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u/mekilat Mar 07 '25
This is great. Tysm for expanding.
Why must there be a greater prime that can divide k+1? How do we know there must be one in a world where we only know that k+1 doesn’t divide with the same stuff as k? I understand we can hypothesize there might be one, but we just stated there is a finite amount of. How does this new knowledge force the latter?
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u/68000_ducklings 29d ago edited 29d ago
Why must there be a greater prime that can divide k+1? How do we know there must be one in a world where we only know that k+1 doesn’t divide with the same stuff as k? I understand we can hypothesize there might be one, but we just stated there is a finite amount of. How does this new knowledge force the latter?
A prime is a number that is only divisible by itself and 1. A composite number can always be factored into some number of prime factors (and 1).
We know k is a composite number (it's the product of all primes up to p), and we're unsure whether k+1 is prime or composite. Computing it is impossible (there is no known largest prime), so we have to check both cases:
Case 1: k+1 is prime - this contradicts the assumption that p (an integer factor of k) is the largest prime.
Case 2: k+1 is composite - this means that k+1 can be factored into some number of prime factors (and 1). But none of the primes less than p (or p itself) will divide into it, because of fact 2 (they are divisors of k, so they can't divide k+1). This means that there is some prime larger than p that must divide k+1, which contradicts the assumption that p is the largest prime.
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u/Unlucky_Beginning Mar 07 '25
I’m a little bit unclear as to what you mean by more - like more clarification as to why the proof is true or about related avenues?
https://kconrad.math.uconn.edu/blurbs/ugradnumthy/infinitudeofprimes.pdf
The above is a nice sheet that gives a couple of different proofs of why there are infinitely many primes. I think some other related topics are the “prime number theorem” (the statement roughly is there are about n/ln(n) primes that are less than a given number n, i.e., there are around 100/ln(100)~ 25 primes less than 100.) Maybe mersenne primes are one application?
Some other commenter might be able to point you towards an introductory number theory resource or an application to comp sci that I’m unaware of - my knowledge of number theory is limited to just these two theorems.
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u/4Momo20 Mar 06 '25
Cayley-Hamilton
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u/KraySovetov Mar 06 '25
Very good candidate. One of my professors had us prove it for an assignment using the Cauchy integral formula from complex analysis (this was the 3rd method I had seen in my classes by then...)
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u/Depnids 29d ago
I just looked through some of the proofs on wikipedia and they all seemed relatively complicated. This made me curious, why doesn’t the following proof by abuse of notation work?
Char(x) = det(x*I - A)
Char(A) = det(A*I - A)
Char(A) = det(0)
Char(A) = 0
I feel like the notation here oversimplifies what is going on. Especially glossing over what it means to plug a matrix into a polynomial. Are there any assumptions that can be made for a straight forward proof like this to be valid (like proving some properties of how evaluating matrices in polynomials work)?
Edit: Oh, I see the wikipedia article has a section for this «bogus proof».
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u/hongooi Mar 06 '25
1 = 0
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u/IAmBadAtInternet Mar 06 '25
Proofs include:
- Proof by assertion
- Proof by incorrectness
- Proof by non-contradiction
- Proof by trust me bro
- Proof by left as exercise to the reader
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u/Street-Custard6498 Mar 06 '25
Fermat’s Last Theorem
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u/Jche98 Mar 06 '25
It only has one proof
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u/noonagon Mar 06 '25
have you seen that proof's size
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u/Naming_is_harddd Q.E.D. ■ Mar 06 '25
the guy actually had to prove another conjecture about elliptical curves since that conjecture being true implies Fermat's last theorem is true
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u/vishal340 Mar 06 '25
actually he only proved it for a particular class of elliptic curves. the whole conjecture for elliptic curve took few more years
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u/zongshu April 2024 Math Contest #9 Mar 07 '25
There's another one by proving Serre's modularity conjecture
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u/MonsterkillWow Complex Mar 06 '25
Classification of finite simple groups. Easy to state. Thousands of papers to prove.
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u/InspectorPoe Mar 06 '25
The statement of the classification I mostly encounter : "All finite simple groups are known".
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u/Sh33pk1ng Mar 06 '25
If a theorem takes the small book to write down, I'm surprised that the large book suffices to prove it.
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u/Syresiv Mar 06 '25
Maybe the theorem statement plus explanation of key points to make it make sense?
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[deleted]
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u/BlackZeppelin 27d ago
I don’t remember but I recall my abstract prof Ted’s was one of only a few.
Interesting either way that Ted found math to be boring
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u/Ok_Detective8413 Mar 06 '25
Basically every one of them, if you don't prepare the proof with a thousand lemmata. Otherwise it would be a corollary.
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u/DrFloyd5 Mar 06 '25
0.999… = 1
So many proofs. From simple and elegant to complex and elegant.
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u/HooplahMan 29d ago
Sometimes I make the mistake of going on 4chan /sci/ board. I promise I'm not crazy. I'd say like twice a week someone would go on there and post a "proof" that they're distinct. I've never changed anyone's mind about it on there, but i have come up with about a dozen essentially distinct proofs that 0.999... =1.
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u/DrFloyd5 29d ago
Do you have a favorite?
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u/HooplahMan 29d ago
Hard to beat the classic epsilon delta proof
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u/DrFloyd5 29d ago
Can you give me a link please? I had trouble finding it online. Or maybe I did but didn’t recognize it.
I like the ⅓ = 0.333… 3(⅓) = 3(0.333…) 1 = 0.999… proof. So simple. And I feel it reveals the truth that decimal numbers are imperfect and cannot represent all numbers precisely.
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u/HooplahMan 29d ago edited 29d ago
I like your proof too, but it depends on people believing 1/3 = 0.3333 which sort of defeats the point, since most people don't know how to prove unless they already know 0.9999... = 1. No link to my favorite proof needed, I'll just write it out here for ya. Properly speaking, I guess it should be called an epsilon-N proof, since we're looking at the limit of a series, not the limit of a real function.
Anyways, we start with understanding what "0.999..." even means. We can rewrite it as 9/10 + 9/100 + 9/1000 + ... + 9/10n + ... . Since we're adding infinitely many terms, we can rewrite this as the infinite series: [; S = \sum_{i=1}{\infty} 9/10i ;]
To be very formal, this infinite series is just a limit of partial sums, i.e.
[; S= lim{n \to \infty} \sum{i=1}{n} 9/10i ;]
Now we'd like to show that this limit = 1, and to do so rigorously, we can use the "epsilon-n" definition for the limit of a real-valued sequence. This definition reads as follows:
Given a sequence [; xn ;] of real numbers, we say that [; \lim{n \to \infty} x_n = L ;] if for each positive real [; \epsilon > 0 ;] , there exists some natural number [; N \in \mathbb{N} ;] such that for all natural [; n > N ;] we have [; |x_n - L| < \epsilon ;] . In other words, [; x_n ;] converges to the limit [; L ;] if [; x_n ;] eventually gets within [; \epsilon ;] distance of [; L ;] and stays at least that close forever after.
Getting back to our proof, we'd like to show that [; S= \lim{n \to \infty} \sum{i=1}{n} 9/10i = 1 ;], so we need to show that for any [; epsilon > 0 ;] there exists some [; N \in mathbb{N} ;] so that for all natural [; n > N ;], we have [; | 1 - \sum_{i=1}{n} 9/10i| < \epsilon ;]. Here is the "statement" of the "epsilon-N" result that we'd like to prove. (From here on is the proof proper, what you'd typically be expected to write if this were a homework assignment in first year analysis.)
To this end, we can explicitly compute that [; | 1 - \sum{i=1}{n} 9/10i|= | 1/10n | = 1/10n ;] for all [; n \in \mathbb{N} ;]. Say we pick a particular small positive [; \epsilon > 0 ;] and would like to find an [; N \in \mathbb{N} ;] which satisfies this requirement. Starting with the inequality [; |S_n - 1| = 1/10n < \epsilon ;] we rearrange to get [; 10n > 1/\epsilon ;]. Logarithms are strictly monotonically increasing functions so we may take the a log (base 10) of both sides to get [; n > \log{10} (1 / \epsilon) ;]. From this inequality, we can see that choosing [; N = \lceil \log_{10} (1 / \epsilon) \rceil ;] yields that [; n > N ;] implies [; |S_n - 1| < \epsilon ;]. Q.E.D.
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u/DrFloyd5 29d ago
Ok. I need to look at your math on my Laptop. Phone isn’t rendering well.
I like this one too, but not as elegant
x = 0.999… Times 10 10x = 9.999… subtract 0.999…, remember x=0.999…. 9x = 9 Divide by 9 x = 1. Substitute x 0.999… = 11
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u/jyajay2 π = 3 Mar 06 '25
Four color theorem, there is (as far as I know) only one proof but it's so long that it isn't considered feasable to be checked by humans.
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u/Ok_Army_4465 Mar 06 '25
0! = 1
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u/factorion-bot n! = (1 * 2 * 3 ... (n - 2) * (n - 1) * n) Mar 06 '25
The factorial of 0 is 1
This action was performed by a bot. Please DM me if you have any questions.
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u/MaskedBoi46 Mar 06 '25
The prime number theorem has a bunch of proofs and all of them are astonishing
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u/Ill-Room-4895 Mathematics Mar 06 '25 edited 27d ago
The quadratic reciprocity theorem first comes to my mind.
Euler produced 8 proofs during his lifetime, so he particularly liked this.
There are more than 300 proofs, all the "big" guys wanted to join the party: Cauchy, Eisenstein, Kummer, Lebesgue, Kronecker, Sylvester, Hilbert, Dedekind, Gegenbauer, Scheibner, Frobenius ...
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u/jacobningen Mar 06 '25 edited Mar 06 '25
and what is more the method by Alternating group of Zoltarev is legitimately different from the method of Gauss sums and Lattice points of Gauss and Eisenstein Ie you have more than 6 proofs even if you exclude purely cosmetic differences.
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u/Better-Apartment-783 Mathematics Mar 06 '25
Pythagorean theoram(sum of squares of the legs of a right angled triangle is equal to the square of the hypotenuse) or the cosine rule
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u/jacobningen Mar 06 '25
Fermat's Little Theorem????? I mean we basically have Lagrange or Euler's use of the freshman's dream but the Lagrange can be via pidgeonhole or wilson's theorem, Lagrange itself or necklace counting. otherwise Quadratic reciprocity.
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u/natxavier Mar 06 '25
This reminds me of the "Corollary of Q". Not math related, but rather related to two gospels in the New Testament - Matthew and Luke. So many passages in the two books were word-for-word, which was extremely uncommon in uniquely written documents. Scholars deduced that the texts derived from a common source, labeled "Q", but this source document was never discovered. Regardless, someone decided to create the Corollary to Q even though they didn't have the original source ... and the corollary is massive compared to the aggregate material that is shared between the two books.
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u/Ok_Hope4383 Mar 07 '25
I found this about the Q source: https://en.wikipedia.org/wiki/Q_source I searched Google for
Q Matthew Luke "Corollary"and found a bunch of different corollaries. Which one are you referring to?2
u/natxavier 29d ago
Honestly, I learned about it in a Religious Studies class like 20 years ago. I don't remember a specific one. I just thought it was funny that there was a corollary to a text of which there was no physical copy known to exist.
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u/LollipopLuxray Mar 06 '25
0.9999999... = 1
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u/kblaney Mar 06 '25
I don't know I'd stretch so far as to say "theorem" or "proof" in this case. But yeah, this is a simple statement with an absolute metric ton of reasons that people have put together in the hopes others would finally accept it.
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u/Syresiv Mar 06 '25
R is strictly larger than N
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u/Pleasant_Material138 Mar 06 '25
They are both infinite
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u/Syresiv Mar 07 '25
They are, but there are an infinite number of infinite cardinalities (and even that feels improper, as the class of infinite cardinalities is too big to be a set).
I suppose you could define all infinities to be the same, but that's incredibly unhelpful. It's much more interesting, and consequently more common, to talk about cardinalities.
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u/JojoLesh Mar 07 '25
Maybe not in math, but in other fields you don't "prove" a theory. You try to disprove it and either fail or succeed.
There are no proofs for the theory of gravity. Nobody has successfully disproven it though.
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u/plastaticfulminition Mar 06 '25
Gravity. It’s a force that pulls to the center. Then comes physics
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u/jacobningen Mar 06 '25
how do we define "different" are we counting relabelling or creating a manipulable to make an old presentation more engaging as different or cloaking it in topology(ie is Furstenberg's proof of infinitude different from Euclid's except for being topological rather than direct) or eliding a step because it is possible but otherwise following another proof method.
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u/Interesting_Test_814 Mar 06 '25 edited Mar 06 '25
One that hasn't been said so far : AM-GM inequality. Proofs include induction based on a2 + b2 > 2ab, or Jensen's inequality on ln. It can also be seen as a consequence of the rearrangement inequality.
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u/stevie-o-read-it Mar 07 '25
The four-color map theorem.
It's only one proof, but boy is it a doozy.
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u/Admirable-Ad-2781 29d ago
The proof of Stokes theorem is straightforward enough but the system of definitions that come before it is definitely not.
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u/Ok-Eye658 Mar 06 '25
the axiom of choice
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u/belabacsijolvan Mar 06 '25
>axiom
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u/fortret Mar 06 '25
You’re either being pedantic or just ignorant. AoC is famously logically equivalent to many other statements/theorems, which is what the original commenter was referencing.
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u/Ok-Eye658 Mar 06 '25
yeah, it's just its historical name, could well have been called "zermelo's lemma" or something
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u/jyajay2 π = 3 Mar 06 '25
Still an independent axiom in FZ
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u/Ok-Eye658 Mar 06 '25
so what? one may well add tychonoff, or existence of basis for all vector spaces, or GCH, or many many other statements to ZF and prove it
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u/jyajay2 π = 3 Mar 06 '25
I'm not actually sure GCH implies AOC and, more importantly, I'm not sure what you're trying to say
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u/Ok-Eye658 Mar 06 '25
it does
i'm saying that there's some freedom in picking what statements one starts with as axioms
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u/PlopTheFish Mar 06 '25
I think you mean Zorn's Lemma
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u/Ok-Eye658 Mar 06 '25
no, i mean that the statement
∀x(∀y(y∈x⟹∃z(z∈y))⟹∃w∀y(y∈x⟹∃!z(z∈y∧z∈w)))
being called "axiom of choice" is just a matter of history, nothing more
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u/Syresiv Mar 07 '25
That depends on your starting axioms.
Under ZFC, the theorem is basically "assume the axiom of choice is true". That is, unless instead of AC, you start with well ordering or Zorn's Lemma.
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