I really appreciate how you want to share this knowledge in a digestible way. I have only gotten up to DifEq so far, so I understand pretty much nothing you say, but I find these videos enjoyable nonetheless. I'm strongly looking forward to your measure theory series!
Cantor's diagonal argument (the version that shows the uncountability of the power set of N) was one of the first proofs that blew me away, I still think it's magical.
Emile Borel is someone worth reading about if you are an aspiring analyst. His output was insane and he was an authority figure on the subject for a long time. He was the analyst for some time.
Mathematics is one of my majors (I’m a double major) and the reason I developed a passion for math is because for me it’s always been a challenging subject. I’ve always struggled with math, which motivated me to actually learn and study and get good at it. I didn’t really start getting good at math until I reached high school. Basic arithmetic and mental math was always my weak point because I was behind in elementary school and never developed strong skills early on. Don’t get me wrong, I’m good at math. I just got good at it later. So when it came time to take undergrad real variables, I struggled a lot because I didn’t understand decimals. I ended up dropping my first semester of real variables because when we started getting into the Cantor set and devil’s staircase I just got totally lost because of my lack of understanding and skill with decimals. I can do a calculus problem no sweat but when it comes to basic arithmetic and decimals I have no clue what I’m doing. I just never picked up on it very strongly during those critical periods of development during childhood. I plan to retake real variables, but this time I want to be more prepared going into it. I might have to review some stuff. My real variables professor was Dr. Steen Pedersen, I think you had him as well. He just retired but when I took Real Variables with him he was on his last year of teaching so he was kind of winding things down. He didn’t do lectures and had worksheets for each class that we did in groups, he was no longer using his textbook. Steen is a really smart guy and a great professor, I just wasn’t prepared for that class.
I often think of Maryam Mirzakhani, winner of the Fields Medal in 2014. She died of cancer in 2017. She worked at Princeton and Stanford universities. She is not so much my favorite as a mathematician, for much of her work is beyond my comprehension. But as a person, she is unforgettable.
To me its Euler. There are 2 lectures on Euler on youtube by a historian. It blew actually my mind what he actually found out, even being blind. I can only recommend it.
Georg is not pronounced George although it's a common anglicization. Cantor did unravel infinity and he's a deserving pick. My favorites are Euler and Noether. Euler for doing everything and Noether for solving physics.
Just read about Cantor quite recently in Abbott's book. What a guy. These amazing mathematicians are so inspiring. They all struggled just like we have, but they didn't stop, and we shouldn't either. Best of luck on your PhD endeavor! I hope to join you when the time is right!
I'm love with Galois, his youth struggle, he is the perfect romantic poet, expect he was a matematician :D btw would be really enjoyable a video about Galois's theory, can't wait to study it at university!
17:54 There’s a piece of music I really like called “L’escalier du diable” (The devil’s staircase) written by composer György Ligeti written for piano and it basically sounds like this Cantor-Lenesque function. You’ll have to listen to it to understand what I mean by this because the piece itself feels like it’s structured off of this function. I have no idea if the composer was trying to depict this in the music but I wouldn’t be surprised if he did. Here’s a link to my favorite performance of it: ua-cam.com/video/bpOubpwv0CQ/v-deo.htmlsi=dFiw3y9w7CKcT-lR
The eternal struggle for me in mathematics is admiration for a proof like Cantor's diagonalization one and disappointment that I can't independently rediscover such a proof on my own. So I have this ongoing ambivalence towards maths- a love-hate relationship really. Can you offer any advise on reconciling these two feelings: my thought process usually goes: why couldn't I think of that; well, don't feel too bad, it took all the brilliant mathematicians that ever lived until the early 1900s to see this; then I try to get to the heart of why I couldn't see it and why I do now- a kind of reverse engineering and overanalysis e.g. the general feeling for why the diagonalization works here is because he's creating a truly independent number from the others - it feels like he's creating another dimension out of a basis of numbers; it feels linear algebra-ish with infinite degrees of freedom. But I never feel fully satisfied in drawing every insight I can from a problem to easily rediscover other proofs on my own. I guess it's only when you can do that you have mathematical maturity. I often wonder how many top mathematicians today could rediscover fundamental stuff like this independently and I suspect probably not. I'd be interested to hear your thoughts on all the above. Also: I'd go for Euler as one of the genuine greats.
Euler is my favourite mathematician purely cos of how much he did and continued when he was blind. Re continuity at approx 19 minutes, I initially agreed and after thinking about it, I disagree cos let's say we go from 0.2 to 0.21, there is an infinite amount of numbers in between, as well, so it isn't continuous nor is it discrete. It is neither so I think we need to create a new set with a new name. This is the first time ever after thinking about this again that I realised there is something that is neither continuous nor discrete. Maybe this is a "hole" in mathematics that can be researched if it hasn't been done so already.
As someone who doesn't differentiate Algebra and Geometry, I love Grothendieck. His view of mathematics is extremely beautiful either from sets to categories. His study of spaces is a profound gift to all of us. I love every aspect of his life, as I myself plan to be outside of humanity, one who truly loves mathematics is one who tries to discover new forms as he said in one of his quotes. A lot of people see this now as obvious, our lives are synthetically sustainable.
@@PhDVlog777 actually i am already an electronics undergrad student...but i am definitely gonna pick maths courses for my open electives whenever i am gonna get the chance
I have many historical & modern favorites, but I will have to add u, good sir to the lot! It's not everyday that an aspiring analyst can get FREE lessons from a budding PhD Mathematician. Your content is equally on par with the likes of Michael Penn, Math Sorceror, Numberphile & Sybermath to name a few Channel faves....LOL
My bachelor degree is in Environmental Science. So your background I don't think matters too much, as long as it is STEM related. The biggest piece of advice I can give is to get some good books, and read through them. Try some problems, the more you do the better off you will be. And don't be afraid to look stuff up on the internet. For me, as long as I understood what was going on the proof, that was enough. Eventually, the math will come naturally. I hope this helps!
Galois is my favourite mathematician. I learned some basic modern algebra in undergrad (but didn’t learn it well). Hopefully I can learn Galois theory some day to appreciate his work more deeply
Great video!! Cannot wait for your measure theory series. I'm trying with great difficulty to teach myself the subject. It's very challenging. On the Cantor diagonal argument--I think the easiest demonstration is to consider the numbers in the set (0,1) in binary expansion, so all digits are 0 or 1. Using your notation for x_i=(d_i1)(d_i2)(d_i3)..., we see that not all the d_ii = 0 and not all the d_ii = 1, since 0.111...=1 which is not in (0,1), and likewise, 0.000...=0 which is not in (0,1). Then we define x=(d_11)'(d_22)'(d_33)'..., where (d_ii)'=1 if (d_ii)=0, and (d_ii)'=0 if (d_ii)=1. From above, not all the (d_ii)' = 0 and not all the (d_ii)' = 1 either. Then your argument shows that x is not any of the x_i, and thus x is not in (0,1), a contradiction since x is clearly in (0,1) as defined, thus proving that (0,1) is not countable.
As someone who has been thinking of majoring in mathematics in the near future (hopefully by the next year i will get into a university once my current course is over), I have to say that your channel is absolute awesome!
I’m approaching my Associate degree in Computer Science then possibly pursuing my Bachelor’s. I’ve been toying with the idea of Minoring in Mathematics while majoring in CS. I really enjoyed Calculus and am currently taking Discrete Math. Would you recommend double majoring or may that be too much?
Oh my god, I understood that. Not just the gist, or maybe a sense of what was being communicated but I actually understood it. Shocking. Do you take requests? Linear algebra please. Matrices and vectors and all that stuff. Something like you did here. Something delightful.
@@ILoveMaths07 Oh, matrices and vectors are the greatest thing ever. I don't know if you programme but if you ever build a graphics engine from the ground up then it's all vectors. Neural networks and just glorified matrix transforms, they're fun too. Did he explicitly say he didn't like them? Or are you projecting your own point of view?
@@davidmurphy563 He says it a lot in his videos. I dislike applied maths... and I absolutely hate programming/coding. I hate anything to do with computers and technology.
A couple questions from a non-mathematician: 1) Does the concept of different infinities have any practical utility, beyond creating proofs like the Cantor diagonalization proof shown here? Does it ever help in other proofs, e.g. 2) How do we know that this diagonalization proof is not just another infinity paradox? Why would we assume this implies multiple infinities? Is there some connection between the concept of a paradox and having multiple infinities? (Sorry, that might have been more than 2 questions - but I did say that I’m not a mathematician).
Hi! To respond to your question - yes, although "practical utility" might be too strong of a phrase depending on what you mean. For instance, functions can be integrated even if they have a countably infinite number of discontinuous points.
For your second question, we know that Cantor's diagonal argument is not a paradox because it is a verifiably-correct proof, i.e. every step follows from the previous ones by rules of logic and the axioms. Any result in mathematics can never be "outright true"; a result can only be true in a given axiom system, i.e. a consequence of the axioms. This is the case even for basic arithmetic with results like 2+2 = 4; here the standard axiom system is the Peano axioms (which, if you are curious, are [1] x + 0 = x, [2] x + (y+1) = (x+y) + 1, [3] x × 0 = 0, [4] x × (y+1) = x×y + x, [5] x + 1 = y + 1 ⇒ x = y, [6] 0 ≠ x+1 for all natural numbers x, [7] mathematical induction: if a statement P(n) is true for P(0), and P(k) ⇒ P(k+1) for all natural numbers k, then P(n) is true for all natural numbers n). In the context of Cantor's diagonal argument, the axiom system is an axiomatic set theory, the standard one being Zermelo−Fraenkel-Choice set theory, ZFC (although Cantor's argument does not make use of all the axioms). ----------------------------------------------- A technical note which is likely alarming to a non-mathematician, so proceed with caution: we say an axiom system is *consistent* if it does not lead to any contradictions (e.g. something like 0 ≠ 0). We do not know that ZFC is consistent - to be technically correct, ZFC cannot prove that ZFC is consistent. Gödel showed more generally in 1931 with his *incompleteness theorem* that any "sufficiently strong" axiom system _A_ cannot prove that _A_ is consistent. ZFC is "sufficiently strong", as are almost all set theories. It turns out that Euclidean geometry (high school geometry) is not "sufficiently strong" and therefore is not susceptible to Gödel's incompleteness theorem - that is, there is an axiom system _E_ for Euclidean geometry which is capable of proving its own consistency, and therefore the geometrical mathematics of the Ancient Greeks is consistent. The alarming part: Since Euclidean geometry is consistent - that is, does not lead to any contradictions - one would certainly expect that basic arithmetic is consistent. However it turns out that the Peano axioms *are* "sufficiently strong", and so basic arithmetic cannot prove that basic arithmetic is consistent. This has the alarming consequence that absolutely elementary statements such as 2+2 = 4 cannot be proven - within Peano arithmetic itself - to not be nonsense. If basic arithmetic *were* inconsistent (no mathematician believes that it is, but by the incompleteness theorem it is impossible to rule out the possibility) then the logical system of Peano arithmetic would - as logicians say - explode, meaning that *all statements would be true* : 2+2 = 5, 2+2 = 4, 2+2 ≠ 4, 0=0, 1=0, 2=0, 3=0, ..., 0 ≠ 0. Although Peano arithmetic PA cannot prove its own consistency, we _can_ prove that PA is consistent using a stronger axiom system - for example ZFC can prove that PA is consistent. But PA is "sufficiently strong" and ZFC is stronger than PA, so Gödel's incompleteness theorem also applies to ZFC, so the problem of being able to internally answer the question "is PA consistent?" has just been relocated, to "ZFC ⇒ PA is consistent, but is ZFC consistent?". To answer the latter question one again has to go to a stronger axiom system, one such system being e.g. Morse−Kelley set theory, MK, so then one has "MK ⇒ ZFC is consistent ⇒ PA is consistent, but is MK consistent?", and this hierarchy can go on forever: one has "A_n is consistent ⇒ ... ⇒ A_3 is consistent ⇒ A_2 is consistent ⇒ A_1 is consistent ⇒ ZFC is consistent ⇒ PA is consistent" for stronger and stronger axiom systems A_1, A_2, A_3, ..., A_n, but Gödel's theorem will always apply to the last system A_n, so we can never prove "absolutely" that PA is consistent, or that any "sufficiently strong" axiom system is consistent - for instance ZFC, the axiom system used by all mathematicians except some mathematical-logicians. So, ZFC _is_ (the current consensus for) "mathematics", and mathematics cannot be mathematically proved to not be nonsense.
Regarding how the diagonal argument implies the existence of different infinities, that has to do with how size is usually defined in maths. Two sets are said to be the same size if there exists a one-to-one mapping between them, which is proven to be impossible for the natural numbers and real numbers (from 0 to 1) in the video. This means both sets have different sizes, so there must exist at least two different infinities (because both sets are infinite). Cantor didn't stop there, the diagonal argument can be generalised to prove that for any set A, the set of its subsets has strictly greater cardinality (i.e. size), so given any infinite set you can construct a set of greater infinite size. What hasn't been proven is that the real number infinity is the smallest infinity greater than the natural numbers', in fact I believe that proof has been shown to be impossible.
"Gay-Ork" is the correct pronunciation
😄
I really appreciate how you want to share this knowledge in a digestible way. I have only gotten up to DifEq so far, so I understand pretty much nothing you say, but I find these videos enjoyable nonetheless. I'm strongly looking forward to your measure theory series!
Cantor's diagonal argument (the version that shows the uncountability of the power set of N) was one of the first proofs that blew me away, I still think it's magical.
You're My favourite mathematician.
I am not worthy but thank you 🙏
Emile Borel is someone worth reading about if you are an aspiring analyst. His output was insane and he was an authority figure on the subject for a long time. He was the analyst for some time.
Mathematics is one of my majors (I’m a double major) and the reason I developed a passion for math is because for me it’s always been a challenging subject. I’ve always struggled with math, which motivated me to actually learn and study and get good at it. I didn’t really start getting good at math until I reached high school. Basic arithmetic and mental math was always my weak point because I was behind in elementary school and never developed strong skills early on. Don’t get me wrong, I’m good at math. I just got good at it later. So when it came time to take undergrad real variables, I struggled a lot because I didn’t understand decimals. I ended up dropping my first semester of real variables because when we started getting into the Cantor set and devil’s staircase I just got totally lost because of my lack of understanding and skill with decimals. I can do a calculus problem no sweat but when it comes to basic arithmetic and decimals I have no clue what I’m doing. I just never picked up on it very strongly during those critical periods of development during childhood. I plan to retake real variables, but this time I want to be more prepared going into it. I might have to review some stuff. My real variables professor was Dr. Steen Pedersen, I think you had him as well. He just retired but when I took Real Variables with him he was on his last year of teaching so he was kind of winding things down. He didn’t do lectures and had worksheets for each class that we did in groups, he was no longer using his textbook. Steen is a really smart guy and a great professor, I just wasn’t prepared for that class.
What is your other major?
I had Steen as a professor a few years ago.
I often think of Maryam Mirzakhani, winner of the Fields Medal in 2014. She died of cancer in 2017. She worked at Princeton and Stanford universities. She is not so much my favorite as a mathematician, for much of her work is beyond my comprehension. But as a person, she is unforgettable.
To me its Euler. There are 2 lectures on Euler on youtube by a historian. It blew actually my mind what he actually found out, even being blind. I can only recommend it.
Georg is not pronounced George although it's a common anglicization. Cantor did unravel infinity and he's a deserving pick. My favorites are Euler and Noether. Euler for doing everything and Noether for solving physics.
Just read about Cantor quite recently in Abbott's book. What a guy. These amazing mathematicians are so inspiring. They all struggled just like we have, but they didn't stop, and we shouldn't either. Best of luck on your PhD endeavor! I hope to join you when the time is right!
Euclid is my favourite mathematician. Geometry is so underrepresented in mathematics.
Your videos always fill me with determination to pursue maths. And give me a sneak peek of what is to come, thank you.
Can you please make a video about Grigori Perelman and the The Poincaré Conjecture?
You should definitely change the title of this video to “Our favorite mathematician”
I'm love with Galois, his youth struggle, he is the perfect romantic poet, expect he was a matematician :D
btw would be really enjoyable a video about Galois's theory, can't wait to study it at university!
I really want him to have lived longer
Galois is my favourite mathematician 😊
Why wait?
17:54 There’s a piece of music I really like called “L’escalier du diable” (The devil’s staircase) written by composer György Ligeti written for piano and it basically sounds like this Cantor-Lenesque function. You’ll have to listen to it to understand what I mean by this because the piece itself feels like it’s structured off of this function. I have no idea if the composer was trying to depict this in the music but I wouldn’t be surprised if he did.
Here’s a link to my favorite performance of it:
ua-cam.com/video/bpOubpwv0CQ/v-deo.htmlsi=dFiw3y9w7CKcT-lR
The eternal struggle for me in mathematics is admiration for a proof like Cantor's diagonalization one and disappointment that I can't independently rediscover such a proof on my own. So I have this ongoing ambivalence towards maths- a love-hate relationship really. Can you offer any advise on reconciling these two feelings: my thought process usually goes: why couldn't I think of that; well, don't feel too bad, it took all the brilliant mathematicians that ever lived until the early 1900s to see this; then I try to get to the heart of why I couldn't see it and why I do now- a kind of reverse engineering and overanalysis e.g. the general feeling for why the diagonalization works here is because he's creating a truly independent number from the others - it feels like he's creating another dimension out of a basis of numbers; it feels linear algebra-ish with infinite degrees of freedom. But I never feel fully satisfied in drawing every insight I can from a problem to easily rediscover other proofs on my own. I guess it's only when you can do that you have mathematical maturity. I often wonder how many top mathematicians today could rediscover fundamental stuff like this independently and I suspect probably not. I'd be interested to hear your thoughts on all the above. Also: I'd go for Euler as one of the genuine greats.
Euler is my favourite mathematician purely cos of how much he did and continued when he was blind.
Re continuity at approx 19 minutes, I initially agreed and after thinking about it, I disagree cos let's say we go from 0.2 to 0.21, there is an infinite amount of numbers in between, as well, so it isn't continuous nor is it discrete. It is neither so I think we need to create a new set with a new name. This is the first time ever after thinking about this again that I realised there is something that is neither continuous nor discrete. Maybe this is a "hole" in mathematics that can be researched if it hasn't been done so already.
How come there is no mention of grothendieck in the comments? Grothendieck was just a gigachad all around
As someone who doesn't differentiate Algebra and Geometry, I love Grothendieck. His view of mathematics is extremely beautiful either from sets to categories. His study of spaces is a profound gift to all of us. I love every aspect of his life, as I myself plan to be outside of humanity, one who truly loves mathematics is one who tries to discover new forms as he said in one of his quotes. A lot of people see this now as obvious, our lives are synthetically sustainable.
My favorite mathematician is Grigori Perelman.
mine is uncle ted
Can't ever give up on Ramanujan from my heart and soul, for what he did, how he did, and the fact he never gave up.
Yeah :D. So unfortunate of a genius. He could have done better.
Ramanujan has a lot of fan boys/girls for some reason. Probably because he was self-taught and died tragically. I'm surprised galois isn't as popular.
Everyone loves cantor, Euler gussa
André Weil, of course
Euler? Gauss? Godel? Newton? Taylor? Do these guys just get forgotten at the graduate level?
Not Godel, others maybe so.
my is Ramanujan and his Goddess
watching your videos makes me wanna pursue mathematics for higher studies lol
Do it
@@PhDVlog777 actually i am already an electronics undergrad student...but i am definitely gonna pick maths courses for my open electives whenever i am gonna get the chance
@@ayushgupta4725 u can start with book of proofs
I have many historical & modern favorites, but I will have to add u, good sir to the lot!
It's not everyday that an aspiring analyst can get FREE lessons from a budding PhD Mathematician.
Your content is equally on par with the likes of Michael Penn, Math Sorceror, Numberphile & Sybermath to name a few Channel faves....LOL
That is a very high honor, and I hope I live up to it lol. Thank you :)
any advice for someone who wants to get a math PhD but has a stats bachelors?
My bachelor degree is in Environmental Science. So your background I don't think matters too much, as long as it is STEM related. The biggest piece of advice I can give is to get some good books, and read through them. Try some problems, the more you do the better off you will be. And don't be afraid to look stuff up on the internet. For me, as long as I understood what was going on the proof, that was enough. Eventually, the math will come naturally. I hope this helps!
Fun fact that cantor set is homeomorphic to {0,1}^\N with respect the product topology.
Love the videos!
Gauss
How would you explain math subjects to non mathsy people? I find analysis particularly hard to explain.
Galois is my favourite mathematician. I learned some basic modern algebra in undergrad (but didn’t learn it well). Hopefully I can learn Galois theory some day to appreciate his work more deeply
Great video!! Cannot wait for your measure theory series. I'm trying with great difficulty to teach myself the subject. It's very challenging.
On the Cantor diagonal argument--I think the easiest demonstration is to consider the numbers in the set (0,1) in binary expansion, so all digits are 0 or 1. Using your notation for x_i=(d_i1)(d_i2)(d_i3)..., we see that not all the d_ii = 0 and not all the d_ii = 1, since 0.111...=1 which is not in (0,1), and likewise, 0.000...=0 which is not in (0,1). Then we define x=(d_11)'(d_22)'(d_33)'..., where (d_ii)'=1 if (d_ii)=0, and (d_ii)'=0 if (d_ii)=1. From above, not all the (d_ii)' = 0 and not all the (d_ii)' = 1 either. Then your argument shows that x is not any of the x_i, and thus x is not in (0,1), a contradiction since x is clearly in (0,1) as defined, thus proving that (0,1) is not countable.
Georg Cantor and Emmy Noether are my two favorite mathematicians. Cantor is really great and a fairly interesting person to learn about
I watch your videos as ASMR, really soothing
As someone who has been thinking of majoring in mathematics in the near future (hopefully by the next year i will get into a university once my current course is over), I have to say that your channel is absolute awesome!
I'd love more vids about Cantor sets. Would love to hear more about the research you've done regarding them
I love listening to people talk about math.
Haskell Curry but I'm a CS major
Nice video! Cantor is definately one of the greats!
I love grothendieck
What's pen do you use?
Hilbert is my favorite mathematical superstar but Cedric Villani and his popular books motivated me along my journey.
If you love matrix theory and linear algebra. Hilbert is your man.
I’m approaching my Associate degree in Computer Science then possibly pursuing my Bachelor’s. I’ve been toying with the idea of Minoring in Mathematics while majoring in CS. I really enjoyed Calculus and am currently taking Discrete Math. Would you recommend double majoring or may that be too much?
I’m also thinking of doing a minor in math but I’m a first year haha
Very good choice.
Oh my god, I understood that. Not just the gist, or maybe a sense of what was being communicated but I actually understood it. Shocking.
Do you take requests? Linear algebra please. Matrices and vectors and all that stuff. Something like you did here. Something delightful.
He hates matrices.
I don't like them, too!
@@ILoveMaths07 Oh, matrices and vectors are the greatest thing ever. I don't know if you programme but if you ever build a graphics engine from the ground up then it's all vectors. Neural networks and just glorified matrix transforms, they're fun too.
Did he explicitly say he didn't like them? Or are you projecting your own point of view?
@@davidmurphy563 He says it a lot in his videos.
I dislike applied maths... and I absolutely hate programming/coding. I hate anything to do with computers and technology.
@@ILoveMaths07 You're so funny. Have a nice weekend. Unless you hate those too! Ha
A couple questions from a non-mathematician: 1) Does the concept of different infinities have any practical utility, beyond creating proofs like the Cantor diagonalization proof shown here? Does it ever help in other proofs, e.g. 2) How do we know that this diagonalization proof is not just another infinity paradox? Why would we assume this implies multiple infinities? Is there some connection between the concept of a paradox and having multiple infinities? (Sorry, that might have been more than 2 questions - but I did say that I’m not a mathematician).
Hi! To respond to your question - yes, although "practical utility" might be too strong of a phrase depending on what you mean. For instance, functions can be integrated even if they have a countably infinite number of discontinuous points.
For your second question, we know that Cantor's diagonal argument is not a paradox because it is a verifiably-correct proof, i.e. every step follows from the previous ones by rules of logic and the axioms. Any result in mathematics can never be "outright true"; a result can only be true in a given axiom system, i.e. a consequence of the axioms.
This is the case even for basic arithmetic with results like 2+2 = 4; here the standard axiom system is the Peano axioms (which, if you are curious, are [1] x + 0 = x, [2] x + (y+1) = (x+y) + 1, [3] x × 0 = 0, [4] x × (y+1) = x×y + x, [5] x + 1 = y + 1 ⇒ x = y, [6] 0 ≠ x+1 for all natural numbers x, [7] mathematical induction: if a statement P(n) is true for P(0), and P(k) ⇒ P(k+1) for all natural numbers k, then P(n) is true for all natural numbers n).
In the context of Cantor's diagonal argument, the axiom system is an axiomatic set theory, the standard one being Zermelo−Fraenkel-Choice set theory, ZFC (although Cantor's argument does not make use of all the axioms).
-----------------------------------------------
A technical note which is likely alarming to a non-mathematician, so proceed with caution: we say an axiom system is *consistent* if it does not lead to any contradictions (e.g. something like 0 ≠ 0). We do not know that ZFC is consistent - to be technically correct, ZFC cannot prove that ZFC is consistent. Gödel showed more generally in 1931 with his *incompleteness theorem* that any "sufficiently strong" axiom system _A_ cannot prove that _A_ is consistent. ZFC is "sufficiently strong", as are almost all set theories. It turns out that Euclidean geometry (high school geometry) is not "sufficiently strong" and therefore is not susceptible to Gödel's incompleteness theorem - that is, there is an axiom system _E_ for Euclidean geometry which is capable of proving its own consistency, and therefore the geometrical mathematics of the Ancient Greeks is consistent.
The alarming part: Since Euclidean geometry is consistent - that is, does not lead to any contradictions - one would certainly expect that basic arithmetic is consistent. However it turns out that the Peano axioms *are* "sufficiently strong", and so basic arithmetic cannot prove that basic arithmetic is consistent. This has the alarming consequence that absolutely elementary statements such as 2+2 = 4 cannot be proven - within Peano arithmetic itself - to not be nonsense. If basic arithmetic *were* inconsistent (no mathematician believes that it is, but by the incompleteness theorem it is impossible to rule out the possibility) then the logical system of Peano arithmetic would - as logicians say - explode, meaning that *all statements would be true* : 2+2 = 5, 2+2 = 4, 2+2 ≠ 4, 0=0, 1=0, 2=0, 3=0, ..., 0 ≠ 0.
Although Peano arithmetic PA cannot prove its own consistency, we _can_ prove that PA is consistent using a stronger axiom system - for example ZFC can prove that PA is consistent. But PA is "sufficiently strong" and ZFC is stronger than PA, so Gödel's incompleteness theorem also applies to ZFC, so the problem of being able to internally answer the question "is PA consistent?" has just been relocated, to "ZFC ⇒ PA is consistent, but is ZFC consistent?". To answer the latter question one again has to go to a stronger axiom system, one such system being e.g. Morse−Kelley set theory, MK, so then one has "MK ⇒ ZFC is consistent ⇒ PA is consistent, but is MK consistent?", and this hierarchy can go on forever: one has "A_n is consistent ⇒ ... ⇒ A_3 is consistent ⇒ A_2 is consistent ⇒ A_1 is consistent ⇒ ZFC is consistent ⇒ PA is consistent" for stronger and stronger axiom systems A_1, A_2, A_3, ..., A_n, but Gödel's theorem will always apply to the last system A_n, so we can never prove "absolutely" that PA is consistent, or that any "sufficiently strong" axiom system is consistent - for instance ZFC, the axiom system used by all mathematicians except some mathematical-logicians.
So, ZFC _is_ (the current consensus for) "mathematics", and mathematics cannot be mathematically proved to not be nonsense.
@@schweinmachtbree1013. Surprisingly, almost all of that made sense to me. Perhaps I missed my true calling in life 😉. Thanks!
Regarding how the diagonal argument implies the existence of different infinities, that has to do with how size is usually defined in maths. Two sets are said to be the same size if there exists a one-to-one mapping between them, which is proven to be impossible for the natural numbers and real numbers (from 0 to 1) in the video. This means both sets have different sizes, so there must exist at least two different infinities (because both sets are infinite).
Cantor didn't stop there, the diagonal argument can be generalised to prove that for any set A, the set of its subsets has strictly greater cardinality (i.e. size), so given any infinite set you can construct a set of greater infinite size.
What hasn't been proven is that the real number infinity is the smallest infinity greater than the natural numbers', in fact I believe that proof has been shown to be impossible.
Schrodinger....is he mathematician or physicist?
Physicist. I don't know of any mathematical results proven by him.
He’s a physicist but he can hang.
@@PhDVlog777 He's also a pedophile so maybe not. Like he kept detailed notes on it and everything. Really sick stuff.
@@abebuckingham8198 I did not know this…
@@PhDVlog777 It's only recently come to light.
en.wikipedia.org/wiki/List_of_things_named_after_Georg_Cantor