Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content for viewers that are eager to attain a greater understanding of the world around them. Math is pervasive - a robust yet precise language - and with each episode you’ll begin to see the math that underpins everything in this puzzling, yet fascinating, universe.
Previous host Kelsey Houston-Edwards is currently working on her Ph.D. in mathematics at Cornell University.
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Decades of research she says...after how long have our brains been here?!?
stacking higher dimensional oranges, a problem the world was anxious about
is it just me or is there really a spadeload unspoke assumptions? I can't even see the 1st graph is biunivocally coherent with the 1st cube according to the just stated rules. Reminds me of high schoolers in the US getting unnecessarily positively lost when simple equations systems are written so "informally" (even by teachers), say without brackets, modding equations in place, adding or removing bits, one soon can't even read which equations are part of the system, which are not, which step one is reading or writing, if a bit has been parsed or not, etc. to the point one really needs 30 extra IQ points just to be able to wade through the totally avoidable ambiguity. I def don't have that much spare IQ.
I mean, it’s not REALLY a paradox if the paradox stems from an axiom and that axiom is required to save the ramblings of a madman (Cantor) 😅
Math (teachers) never looked so good.
"Are prime numbers just a figment of our collective imagination?" The human race does not have a collective imagination, so, no.
but how do we find the most efficient strategy to catch the robber on a cop win graph
Again, I miss this show.
Oh I miss this one…
the plank length disallows an infinitely small piece of any real object and therefore disallows making two objects out of one so maybe get your logic in line with reality before declaring a paradox
The axiom delivers a reset within the hierarchal parameters of some other train of reasoning. It avoids mathematical rigor without account.
Fun Fact removing the Axiom of Choice and Replacing it with Hahn Banach will give non measurable sets not Vitali tho
I don't understand 7:15 , the transition from 1/2 1/2 = to zero because the area is zero?... I don't understand
Part of the 'problem' [esp for communicating with the lay person] is that 'infinity' itself isn't that well understood in the first place. The set 'goes on and on' aspect, and the separate 'counting' aspect are distinct concepts that get confounded when the set is the 'integers' that appear to match the countings. The bijection between the positive integers and the evens is between _different_ sets (and their particular orderings). Both sets 'go on and on' in a definite countable order so are of the same 'countable size'. For the rationals, the ordering isn't (for the purpose here) by linear value, rather by one of the diagonalization orders. It is that ordering which makes the set 'countable'. Having decided that one _can_ count the rationals, there is a flip to an order that doesn't appear to have the countable property (but is the same set) that is then used to show that the reals are definitely larger even though we get into the 'alternating' vs 'between' problem of reals and rationals (i.e. reals having smaller infinitesimals that the rationals ;-) If you want to further confuse the issue you get into the 1.000000... being preceded by 0.999999... for some arbitrarily small infinitessimal ! Monty-Hall had it easy.
Well we know that 1.0 is EXACTLY equal to 0.99999..., they are the same number
@@farkler4785 Who is the "we" that you talk of; why does it (0.99..) keep coming up? Both are valid representations (i.e. all the base-1 repeating digit representations) that can be in used in any of the diagonalization arguments. There is a blind spot as to how 3/3 = 1 but (1/3)*3=0.99..... It's all about communicating the key steps that either suspend disbelief or 'jump the shark', or flip philosophy to explain just how certain apparent impossibilities happen (counting to infinity squared, etc.) One solution is to invoke the distinction between 'arbitrarily large' and then 'countable infinity' and how they differ as to the _conventions_ they invoke.
@@philipoakley5498 I’m not following how any of this relates to cantors diagonalization theory. And the “we” is mathematicians as a whole
Still looks out of phase!
👍. Great explanation.
Interesting. As a beginner chess player, it seems it would be possible to implement infinite chess online as a game in the same way finite chess is implemented. I understand there is often a rule that if a piece is not captured or there is no check within N moves, where N is specified before the game starts, the game is a draw. So maybe infinite chess with finite N could be implemented or maybe the most up to date chess bot or algorithm would be upgraded to play infinite chess and set to play itself to find interesting infinite chess games and strategies. Or maybe infinite chess puzzles could be implemented?
I wish they did category theory before ending 💢
I'm so glad that people in my country can be that smart! I hope there will be more people like Maryna! (She's from Ukraine)
Way better than the woke mind virus info people deal with
I ca make a tuple theoretic model for the finite surreal numbers. That would be the Dyadics, which are the numbers you find on an old fashioned ruler with imperial divisions (fractions of inches).
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The Silver Spin... its power could surpass even the Golden Spin, and the Bronze Spin could surpass the Silver Spin. But how can one go beyond infinity?
This only seems paradoxical on the surface with these artificial examples that seek every mathamatically possible corner case regardless of its probability. In practice, the system is attempting to solve the particular real problems of a specific election involving a population of real people who lack complete information. How often is an anomaly expected for the given parameters, how probable and impactful is a type of strategic manipulation and tactical voting in actual practice? Something that may be plausible for a group of 12 close aquaintences, may be nearly imposible to acheive in a population of 100k strangers, half of which may choose not to vote at the last moment. The normal concern is not so much getting the 1st verses 2nd choice correct but rather avoiding an accidental worst case, or preventing some long term degredation like system capture by one or two groups.
I think the set S at 5:02 has measure 0. It's all the rationals between -1 and 1. Since they are the rationals, they can be ordered, and they are countably infinite. Put a little open interval of length a around the first one, a/2 around the second, a/4 around the third, etc. Add up all the lengths, and you get a/(1 - a). You can make a as small as you want, so the measure of the set S is zero.
That's not the set in question. Let's try it again: consider an interval from 0 to 1. That's an uncountable set. Split it into equivalence classes using the relation: x~y, if x-y is an irrational number. Each equivalence class is a countably infinite set (it's a shifted copy of a subset of rational numbers); and so there are uncountably many classes. Finally, take a set containing a single element from each equivalence class; that's an uncountable set. (That such a set exists is a consequence of the axiom of choice.) For example, one equivalence class is the set of all rational numbers from that interval, the other is the set of numbers which differ from pi by a rational, and so on; the non-measurable set (or Vitali set) contains a single element from each such class. Now we want to prove that this set is non-measurable; and that is proven by constructing countably many shifted copies of that set by all rational numbers from -1 to 1. The union of such copies is a superset of an interval from 0 to 1, but a subset of an interval from -1 to 2. So: if the Vitali set is measurable, it has either a measure of 0, or it has a non-zero measure. But neither is compatible with the countable additivity of the Lebesgue measure (that is also a consequence of the axiom of choice). If the Vitali set is measurable, then so are all its copies, and they all have the same measure. So if the set has measure 0, then the union of countably many copies of the set also has measure 0 (measure of a union of countably many sets is equal to the infinite sum; and infinite sum 0+0+0+... is equal to 0, by the definition of an infinite sum as a limit of the sequence of partial sums). And if it has some non-zero measure m, then the union has measure m+m+m+..., and that infinite sum diverges to infinity. But as I have said, the union is a subset of an interval of length 3, and a superset of an interval of length 1; and neither 0 nor infinity is between 1 and 3. It follows that Vitali set doesn't have a measure.
I wish this series back 😢
"What are numbers made of? [...] Nothing, provided at least that you stipulate nothing exists." Incredible statement.
What a well produced and great series. Such a shame that they stopped.
"We can simulate the second round". If only it was so simple. In reality, other candidates give support to the remaining top two candidates, depending who won the first round. Furthermore, voters tend to not turn up in round two if their preference did not win.
This is so good. Set theory is such a weird and cool space of research. Is the non measurable aspect of these sets used to understand quantum mechanics? Is the immeasurability of S similar at all the the immeasurability of quantum variables? And could the space of all functions (with the axiom of choice) that you can apply to S generate a set of all possible combinations of all elements? Would that set also be non measurable?
Absolutely a great video
If an infinite series which gets smaller and an infinite series which gets larger meets, does that mean the whole universe exists in each atom of my body? If so, am i god?
The core problem is: is mathematics a natural phenomena that we are discovering or have we made it up and the structure of our observational reality is reflective of it???
she is very pretty 😶
If we compare this to wavelengths of light, a sphere would be the colour yellow.
I have seen this video YEARS ago as a teenagers and I am only understanding it fully now that I am taking Computer Organization in college and I suddenly remembered about this video mid-lecture. Thank you PBS Infinite Series for implanting this in my brain until now 😅
Finally I found an explanation of quantum computing that kinda makes sense! Thank you
It seems weird to call S sizeless. Where R, Q, and Z are defined as the number of real, rational, and integer numbers respectively, something with size 1 consists of R/Z total points. We divide this size 1 into R/Q sets of Q/Z points (length Q/R). We pick 1 point (length Z/R) from each of the R/Q sets (Z/R * R/Q) to form S with length Z/Q, which might be evaluated 0 because there are infinite rational numbers for every integer. We then take 2Q/Z copies of S for a total length of 2.
What on Earth is "Z/R" or "R/Q"? By 'size' here is meant measure - the extension of the notion of length of an interval (or in a higher dimension: area of a shape, volume of a solid body, and so on); and measure of a set is either a non-negative real number, or infinity. The usual definition of measure on real numbers is the Lebesgue measure. So a one-element set definitely has a measure, and that measure is 0. Conversely, the Vitali set doesn't have a measure; the measure can't be 0, and it can't be non-zero. Union of countably many sets of measure 0 has measure 0 (assuming axiom of choice, from which it follows that Lebesgue measure is countably additive). But that says nothing about a union of uncountably many sets; such a set can have measure 0, or a finite positive measure, or an infinite measure, or no measure at all.
4:48 Why cannot I extend the (0,1) semicircle into a 3/4th circle or a complete circle excluding two points to argue that all the real numbers have a one to one correspondence with just the semicircle and I still have infinitely many points left over in the interval (0,1) to show that (0,1) is bigger than the set of all real numbers?
because first you can just apply a transform function to the semicircle and get it back to this form
There is an unfortunate inaccuracy around 4:37. The statement is based on the false premise that a game of chess can only be won or lost rather than drawn. But by considering a few closely related games where draws are redefined as wins for one side, we can reach the correct conclusion that either (1) there is a strategy for white to win or (2) there is a strategy for black to win or (3) both sides have a strategy that guarantees at least a draw. Most believe (3) is very likely true, but it has not been proven and doing so appears computationally impractical.
Ever since finding this concept in like, first year math, I found it fascinating!!
Cute but incomprehensible 🤔
According to an article published in May 2024, any number can be divided by zero with a simple elementary school-level operation. According to the article, the operation 50/0 is performed as follows: 50 / 0 = 0 A( 50 ) (A is the exponent above the parenthesis, but unfortunately I couldn't write it here). In this operation, the zero after the equal sign is the value of the quotient and A(50) is the remainder. The proof of this operation is as follows: 50 / 0 = 0 A( 50 ) 50 = 0 x 0 A( 50 ) 50 = ( 0 x 0 ) + 50 50 = 0 + 50 50 = 50 This article examines this issue in detail. It also tries to refute one by one the explanations of why a number cannot be divided by zero. For those interested, information about the articles is below: 1.Article Information Article Title: The problems created by zero in the division operation, their reasons and an attempt at a solution Article Address: www.academia.edu/118447190/The_problems_created_by_zero_in_the_division_operation_their_reasons_and_an_attempt_at_a_solution_S%C4%B1f%C4%B1r%C4%B1n_b%C3%B6lme_i%C5%9Fleminde_olu%C5%9Fturdu%C4%9Fu_sorunlar_nedenleri_ve_%C3%A7%C3%B6z%C3%BCme_y%C3%B6nelik_bir_deneme_%C3%A7al%C4%B1%C5%9Fmas%C4%B1 2.Article Information Article Title: A study to prove that the denominator can be zero in fractional numbers Article Address: www.academia.edu/118448116/A_study_to_prove_that_the_denominator_can_be_zero_in_fractional_numbers_Kesirli_say%C4%B1larda_paydan%C4%B1n_s%C4%B1f%C4%B1r_olabilece%C4%9Fini_kan%C4%B1tlamaya_y%C3%B6nelik_bir_%C3%A7al%C4%B1%C5%9Fma&nav_from=a54f476f-5cf7-4264-ab57-7383abc3d91f&rw_pos=0 (This article has been translated into English with a translation program.)
My idea is that S is not Lebesgue measurable but is measurable by another type of measure. For me, the paradox comes from the discretisation of the unit interval. The passage from continuity to the discrete world is the cause of trouble.
Sure, the set is measurable under the following measure: m(X)=1 if 0.5∈X, and m(X)=0 otherwise. That's a function on the system of sets of real numbers which satisfies the definition of a measure, except for that it's not the case that measure of an interval is equal to its length.
Please define Pi? - write down Dedekind cut for Pi?
Is it too early for infinitesimals yet?
Ok trying to work this out, the number of equivalence classes is c (order of the continuum) divided by ℵ0 (countable infinity), so that is the number of elements in S. c is 2 to the power of ℵ0, so it's 2^ℵ0/ℵ0 which is infinite, but what order of infinity is it? Note that the Lebesque measure of c differs depending on how much space it represents and the Lebesque measure of ℵ0 is zero, different from the number of elements. Anyway it's obvious that to give S a size, you have to extend your number system to include infinitesimals.
It can be proven: the set of equivalence classes has at least continuum-many elements (by constructing a specific set of continuum-many real numbers, none of which differ from each other by a rational). But can you inject the set of equivalence classes into real numbers? If you say: "yes, by picking a single element from each equivalence class", then you are using axiom of choice. Without choice, it's consistent that equivalence classes can't be injected into real numbers; in other words, it's consistent that a set with the cardinality of the continuum can be split into more partitions than it has elements. I'll give you a simpler example: consider the set of all infinite sequences of digits 0 and 1. Obviously, this set has cardinality of the continuum. Now consider the equivalence relation: A~B, if sequences A and B differ at finitely many positions. How many equivalence classes are there? We're going to split natural numbers into countably many countable sets: S1 is the set of all numbers not divisible by 2 ({1, 3, 5, ...}), S2 is the set of all numbers divisible by 2, but not by 4 ({2, 6, 10, ...}), S3 is the set of all numbers divisible by 4, but not by 8 ({4, 12, 20, ...}), S4 is the set of all numbers divisible by 8, but not by 16, and so on. Now take a sequence of digits 0 and 1: ABCDEF... . For any such sequence we're create a sequence where at positions of the n-th previously defined set there's the n-th number in the sequence; that gives the sequence ABACABADABACABAEABACABADABACABAF... . So this yields continuum-many sequences, none of which differ from each other at finitely many positions; it follows that there are at least continuum-many equivalence classes. But if you want to prove the other relation - that there are at most continuum-many equivalence classes - you need the axiom of choice (by picking a single element from each equivalence class).
My favorite consequence of the axiom is that anything I chose is legit choice function. Just pick your favorite number in the set.
Wait so what is even the purpose or usefulness of this concept? Why would we want sizeless sets??
And then, a K-pop band started playing Ska, and we all fell into a black hole.