I found a non trivial zero off the critical line, but there is not enough room in these comments to write it out, and they keep deleting my comments...
@@dentonyoung4314 No, it's not, as there is no such thing as a universal, context free Gödel-undecidable sentence. A sentence may be Gödel-undecidable only in the context given by a formal system (strictly, an axiomatic formal deductive system) such as e.g. ZF, and moreover Gödels procedure gives a proof that the sentence is true (in context). Thus, if you could show that the Riemann hypothesis is Gödel-undecidable in ZF, you have a proof that given ZF, then Riemann is true.
Please do more videos like this! It's always fascinating to hear from the author about their thought process in their own words, which is often a big part that's missing in the final paper.
Provable for finite sets - ok so that means the counterexample isn't finite - save the link to the paper for when I have an unbounded amount of time (after Christmas?).
ha posting this in exam season might have been cruel:D But yes, if it is a finite set then for sure it has a fishbone and the counterexample actually has to be crazy large, not just infinite but infinite with specific weird structures.
I am chuffed I can follow this video. Have no background in maths to speak of, but in distributed computing partially ordered sets describe the events that occur on a network of computers. If I follow your terminology correctly - if two events (elements in a set) are an anti-chain, that means they happened concurrently, or completely independently of each other. One event could not have influenced the other. You might see this a computer/phone user when something that syncs your data tells you you have a "conflict" - there's no total order of events, so you have to decide which bit of the anti-chain is valid. EDIT: Ok so when you resolve a conflict when syncing data, what you're doing I think is defining the "spine". There's some algorithms that can define spines for you, but for a lot of data it's best to just ask the user what they wanted (this file edited on this machine, this file on another machine, which version do you want?).
An example of a poset that I've constructed in my work is the org chart of a company where some people report to managers (and there are no cycles--generally there isn't--I suppose it would still be a poset if you have situations where someone has "dotted lines" to multiple managers, though my code didn't handle that). A chain in that poset would be any subset of a literal chain of command, and an antichain would be some set of people who are not in each other's management chain. But, of course, this is a finite graph. Posets involving the time-ordering of events even occur in physics, in the theory of relativity. If one event is in the future or past light-cone of another, then there's an invariant time ordering, but outside of the light-cone, when the interval separating them is a faster-than-light path, there isn't--which order they happened in depends on one's frame of reference. In quantum field theory, it's very important for the preservation of any notion of causality for certain quantum-mechanical operators to *commute* for events outside each other's light-cone; this is why non-local quantum correlations can't actually be used to send faster-than-light messages. And all this is assuming that the space has no closed timelike loops, in which case God knows what happens (but no one has ever found any...) But *that* poset is actually an uncountably infinite one, assuming a continuous classical spacetime, which is another big if. (A fishbone on it would be pretty simple to construct, though, at least for most of the spaces physicists actually think about: if you can put any timelike coordinate on it, you can define a chain, and then the "space" slices are the antichains.)
@@MattMcIrvin My understanding is the partial order of events used in computing is very much derived from physics (Lamport '78 mentions this explicitly). Much like there's no one global time in relativity, there's no one global time with different machines, so deciding the causality between events is key.
The problem with "larger infinities" is that these types of combinatorial questions are not well posed for uncountable structures, the answer always depends on what model of set-theory you feel like using today. That's because those larger infinities are not "absolute" in a technical sense, any uncountable set can be shoehorned into a countable one by forcing it to collapse onto the set of whole numbers So this guy is selling himself short, he probably has said the most that can be said about this conjecture for any cardinality..
@@creativenametxt2960 I didn't read his paper yet, I don't even know what he did exactly. I just know that whatever he did, there's no way "going uncountable" is going to add anything new, because it never does.
Hi professor! I would like to ask you why we can't compare all nodes in our ordering? timestamp: 01:19. For example you said that we cannot compare C and G, but I can imagine in a sort of 3blue1brown grabbing the branch where G belongs ( branch e and g ) and kind of stretching it so it were to be inline with the road of B and C. In this way, for it to be a straight line it would be G -> E -> B -> C. Here we have an ordering, no?
You could do that, it's just that you could also NOT do it:D That is, there are many ways to define an ordering. One could be "more to the right" or one could be "more to the right AND on the same road" etc. So some the orderings that people define are partial and some are not.
That was another conjecture I'd never heard of. So it's provably true in most case, but there's this one weird condition where if if something really bizarre happens it can be false.
You could have made the Cartesian example more concrete by comparing apples and oranges. Two apples and three oranges are less than three apples and four oranges, but can't be compared to three apples and two oranges.
Sounds like this pertains to the Collatz conjecture! The patterns in that problem vacillate between trending upward infinitely and trending downward infinitely.. perhaps looking at those Collatz sequences as fishbones would be fruitful. I imagine this has been done, I wonder where it leads.
Have you seen the video on the moving sofa problem by the channel "Wrath of Math"? (It does not go into Baek's proof in much detail, but it gives an overview of the problem and an outline of the basic proof idea)
@@DrTrefor And yet also expected, in the sense that the shape that turned out to be optimal was proposed in 1992. (Coincidentally, this is the same year that the fishbone conjecture was posed.)
9:05 This isn't really a valid justification, is it? You could easily construct a finite partition of antichains of a poset (infinite) for which a chain is found connecting them (each at one point). e.g. from the given example, also remove the points (x, y) for which y
It just so happens that such a partition is not possible given the triangular structure of the lattice shown. But that is not a necessary fact from the choice of a finite connecting chain (leaving infinitely many points left to form antichains).
So to have a fishbone, we'd need to be able to connect that point to the spine via an antichains. That we could do, but the requirement of a fishbone is all the anti-chains are DISJOINT and so it would overlap with one of the existing antichains.
Hi professor! I hope you reply to my comment I want to pursue theoretical physics and am a Physics major(3rd semester) currently we're studying Bessel's functions and frobenius method, laplace transforms etc in ODE. Since these problems get pretty Lengthy, how much would you suggest i practice and how much do you suggest I focus on the proofs(there aren't many in this course) and also what areas of Mathematics do i need to look into for studying General Relativity?
I do think practice makes perfect, but the big goal is to UNDERSTAND why you are doing what you are doing. That is don't focus necessarily on all the technical details in a long computation, do you understand the big picture of what say a laplace transform and inverse laplace transform is doing to solve an ODE?
@DrTrefor Coincidentally that is the exact question i was going to ask my professor but he'd left his office early today due to an emergency. I was going to ask him how does someone come up with something that looks so random as if it was a revelation only they knew of and then they were somehow mysteriously guided that this reduced ODEs to algebraic problems! Thank you very much prof
I'm at the start of the interview, and if the counterexample turns out to be dependent of the Axiom of Choice I'll be quite annoyed. Let's unpause. EDIT: well, I still have that question and now I have another: is the counterexample countable?
My issue with your first example is that it can easily be changed to show infinite antichains + fishbone. The written conjecture is fine (no infinite antichain => fishbone), but I don't believe your "exclusive or" presentation makes sense.
The NxN example? That doesn't have infinite antichains. Given a point, anything else on an antichain is either down and to the right OR left and up. But since there is only a finite amount of down and a finite amount of left possible in NxN it has to be finite.
Sorry, I meant the one at 6:37. This example would work even if the y-axis were extended infinitely down (into negatives), no? In which case those antichains become infinite (not just arbitrarily large).
My gut tells me, Collatz is wrong. Just some weird loop with crazy big numbers that have an interesting property. It feels like the 4-2-1 loop is just the trivial unit case. But we'll see I guess 😅
If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....)...
Huge thank you to Lawrence Hollom for talking to us about his cool new paper!. Check out the interview starting at 10:41 in the video.
can't wait for Hollom to disprove the Reimann Hypothesis, cuz so many of my friends believe it has to be true intuitively.
I found a non trivial zero off the critical line, but there is not enough room in these comments to write it out, and they keep deleting my comments...
If its disproven, physics would be in a pretty tough place...
The Riemann hypothesis is probably Godelian-undecidable -- meaning it's true but we can never prove that fact.
Fermat2.0@@thomashoglund5671
@@dentonyoung4314 No, it's not, as there is no such thing as a universal, context free Gödel-undecidable sentence. A sentence may be Gödel-undecidable only in the context given by a formal system (strictly, an axiomatic formal deductive system) such as e.g. ZF, and moreover Gödels procedure gives a proof that the sentence is true (in context). Thus, if you could show that the Riemann hypothesis is Gödel-undecidable in ZF, you have a proof that given ZF, then Riemann is true.
Please do more videos like this! It's always fascinating to hear from the author about their thought process in their own words, which is often a big part that's missing in the final paper.
Glad to hear that! I definitely loved doing the interview bit and if people enjoy it I'll definitely do more of that style.
apparently, the moving sofa problem was just solved. "The Optimality of Gerver's Sofa" by Jineon Baek.
Amazing
"Lawrence Fishbone" really just makes me think of Morpheus from The Matrix.
Provable for finite sets - ok so that means the counterexample isn't finite - save the link to the paper for when I have an unbounded amount of time (after Christmas?).
ha posting this in exam season might have been cruel:D But yes, if it is a finite set then for sure it has a fishbone and the counterexample actually has to be crazy large, not just infinite but infinite with specific weird structures.
I loved the interview
10:10 note that we have an... anti-fishbone?
We could have the (0,0), (1,0), (2,0), ..... anti-chain, and all vertical disjoin chains.
I am chuffed I can follow this video. Have no background in maths to speak of, but in distributed computing partially ordered sets describe the events that occur on a network of computers.
If I follow your terminology correctly - if two events (elements in a set) are an anti-chain, that means they happened concurrently, or completely independently of each other. One event could not have influenced the other. You might see this a computer/phone user when something that syncs your data tells you you have a "conflict" - there's no total order of events, so you have to decide which bit of the anti-chain is valid.
EDIT: Ok so when you resolve a conflict when syncing data, what you're doing I think is defining the "spine". There's some algorithms that can define spines for you, but for a lot of data it's best to just ask the user what they wanted (this file edited on this machine, this file on another machine, which version do you want?).
I actually really love this, building connections to your own area of expertise is awesome
An example of a poset that I've constructed in my work is the org chart of a company where some people report to managers (and there are no cycles--generally there isn't--I suppose it would still be a poset if you have situations where someone has "dotted lines" to multiple managers, though my code didn't handle that). A chain in that poset would be any subset of a literal chain of command, and an antichain would be some set of people who are not in each other's management chain. But, of course, this is a finite graph.
Posets involving the time-ordering of events even occur in physics, in the theory of relativity. If one event is in the future or past light-cone of another, then there's an invariant time ordering, but outside of the light-cone, when the interval separating them is a faster-than-light path, there isn't--which order they happened in depends on one's frame of reference. In quantum field theory, it's very important for the preservation of any notion of causality for certain quantum-mechanical operators to *commute* for events outside each other's light-cone; this is why non-local quantum correlations can't actually be used to send faster-than-light messages. And all this is assuming that the space has no closed timelike loops, in which case God knows what happens (but no one has ever found any...) But *that* poset is actually an uncountably infinite one, assuming a continuous classical spacetime, which is another big if. (A fishbone on it would be pretty simple to construct, though, at least for most of the spaces physicists actually think about: if you can put any timelike coordinate on it, you can define a chain, and then the "space" slices are the antichains.)
@@MattMcIrvin My understanding is the partial order of events used in computing is very much derived from physics (Lamport '78 mentions this explicitly). Much like there's no one global time in relativity, there's no one global time with different machines, so deciding the causality between events is key.
The problem with "larger infinities" is that these types of combinatorial questions are not well posed for uncountable structures, the answer always depends on what model of set-theory you feel like using today. That's because those larger infinities are not "absolute" in a technical sense, any uncountable set can be shoehorned into a countable one by forcing it to collapse onto the set of whole numbers So this guy is selling himself short, he probably has said the most that can be said about this conjecture for any cardinality..
could you elaborate how this specific problem is not well-posed and how it is different in different models?
@@creativenametxt2960 I didn't read his paper yet, I don't even know what he did exactly. I just know that whatever he did, there's no way "going uncountable" is going to add anything new, because it never does.
Wonderful video, very nice!
That is correct for discrete math.
Hi professor! I would like to ask you why we can't compare all nodes in our ordering? timestamp: 01:19. For example you said that we cannot compare C and G, but I can imagine in a sort of 3blue1brown grabbing the branch where G belongs ( branch e and g ) and kind of stretching it so it were to be inline with the road of B and C. In this way, for it to be a straight line it would be G -> E -> B -> C. Here we have an ordering, no?
You could do that, it's just that you could also NOT do it:D That is, there are many ways to define an ordering. One could be "more to the right" or one could be "more to the right AND on the same road" etc. So some the orderings that people define are partial and some are not.
That was another conjecture I'd never heard of. So it's provably true in most case, but there's this one weird condition where if if something really bizarre happens it can be false.
You could have made the Cartesian example more concrete by comparing apples and oranges.
Two apples and three oranges are less than three apples and four oranges, but can't be compared to three apples and two oranges.
Sounds like this pertains to the Collatz conjecture! The patterns in that problem vacillate between trending upward infinitely and trending downward infinitely.. perhaps looking at those Collatz sequences as fishbones would be fruitful. I imagine this has been done, I wonder where it leads.
How cool would it be to study with Dr. Bazett??
Wow so cool
Does the existence of the counterexample rely on the axiom of choice? It sounds like that from the interview.
I don't think so. It is an explicitly constructed ordering on NxNxN.
Have you seen the video on the moving sofa problem by the channel "Wrath of Math"? (It does not go into Baek's proof in much detail, but it gives an overview of the problem and an outline of the basic proof idea)
Ya that was a cool one for sure, such an u expected shape
@@DrTrefor And yet also expected, in the sense that the shape that turned out to be optimal was proposed in 1992. (Coincidentally, this is the same year that the fishbone conjecture was posed.)
9:05
This isn't really a valid justification, is it?
You could easily construct a finite partition of antichains of a poset (infinite) for which a chain is found connecting them (each at one point).
e.g. from the given example, also remove the points (x, y) for which y
It just so happens that such a partition is not possible given the triangular structure of the lattice shown. But that is not a necessary fact from the choice of a finite connecting chain (leaving infinitely many points left to form antichains).
Finally discrete math comes in useful
lol "finally" what you on about:D
People use discrete math every time they do arithmetic on integers (or integer multiples of some unit).
Discrete math is quite important in computer science!
I don't understand the problem with the (4, 4) point at 9:35. What are the intersections that must be avoided?
So to have a fishbone, we'd need to be able to connect that point to the spine via an antichains. That we could do, but the requirement of a fishbone is all the anti-chains are DISJOINT and so it would overlap with one of the existing antichains.
This is quite poggers
Man this would have been nice for the exam I took 4 hours ago 😭
Ya you would def have gotten an A+ if you just wrote "no fishbone lol"
dr. bazett, please try to prove or disprove the Collatz Conjecture.
Brb solving;)
So b and c are same depth in the network, so equivalence is in the eye of the beholder?
Hi professor!
I hope you reply to my comment
I want to pursue theoretical physics and am a Physics major(3rd semester) currently we're studying Bessel's functions and frobenius method, laplace transforms etc in ODE.
Since these problems get pretty Lengthy, how much would you suggest i practice and how much do you suggest I focus on the proofs(there aren't many in this course) and also what areas of Mathematics do i need to look into for studying General Relativity?
I do think practice makes perfect, but the big goal is to UNDERSTAND why you are doing what you are doing. That is don't focus necessarily on all the technical details in a long computation, do you understand the big picture of what say a laplace transform and inverse laplace transform is doing to solve an ODE?
@DrTrefor Coincidentally that is the exact question i was going to ask my professor but he'd left his office early today due to an emergency.
I was going to ask him how does someone come up with something that looks so random as if it was a revelation only they knew of and then they were somehow mysteriously guided that this reduced ODEs to algebraic problems!
Thank you very much prof
this got anime power scalers mad as hell now
I'm at the start of the interview, and if the counterexample turns out to be dependent of the Axiom of Choice I'll be quite annoyed. Let's unpause. EDIT: well, I still have that question and now I have another: is the counterexample countable?
The counter example is countable, and um don’t quote me but I think it doesn’t depend on AoC, it’s entirely constructed
My issue with your first example is that it can easily be changed to show infinite antichains + fishbone. The written conjecture is fine (no infinite antichain => fishbone), but I don't believe your "exclusive or" presentation makes sense.
The NxN example? That doesn't have infinite antichains. Given a point, anything else on an antichain is either down and to the right OR left and up. But since there is only a finite amount of down and a finite amount of left possible in NxN it has to be finite.
Sorry, I meant the one at 6:37. This example would work even if the y-axis were extended infinitely down (into negatives), no? In which case those antichains become infinite (not just arbitrarily large).
7:53 Christmas?
Damn. i couldnt even pass calculus 100 three times in a row..·´¯`(>▂
SHOW US A PICTURE OF HIS COUNTEREXAMPLE.
The counterexample if quite complicated, but you can check it out in the paper if you like
At this rate Collatz Conjecture is gonna be debunked too ahah
In before Lawrence debunks imo
My gut tells me, Collatz is wrong. Just some weird loop with crazy big numbers that have an interesting property.
It feels like the 4-2-1 loop is just the trivial unit case.
But we'll see I guess 😅
Inst this the hood will hunting thing
well that problem from the movie was in graph theory, graph theory is a big subject and it wasn't this one:D
❤❤❤
Translation to Arabic 😢
If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....)...