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0:00
our today we are going to have what I consider to be my really fun watcher about Finny we're going to talk about infinity some pretty fun ways I think I'm not really actually going to follow the book too great on you I'm gonna be less formal with infinity then we are in this real analysis class to the textbook we're gonna make a bunch of pictures and we are going to okay and you want to address some homework problems I turn it to anybody else
0:38
if you want to look at those because there's important points to bring out but but let me first wet your appetite for infinity by talking about something called a ping pong ball conundrum okay I did not make this up this credit goes to Edward Berger and Michael starburns from their books of harder mathematics which we use in creative problem-solving here a bubble it's a really cool of okay let's run this color here and see what happens okay so there we
1:28
have the infinity girl and we've got some numbered circles there one of those number of circles those with ping-pong balls I could be a track for the ping-pong balls to slide down I didn't bother the ping pong balls are going to go in the barrel and in fact I should like before we run this I should you know that these ping-pong balls go off to infinity there's one ping-pong ball from every natural number okay so we see the first three here one two three as they start moving the
1:59
slider as I start letting time go by we see these ping-pong balls going into the infinity barrel you see the four the air 5 6 etcetera they're coming in coming in and so far about 10 seconds of gun by and we have a Bahji - almost three balls in the barrel and then as we approach 30 seconds there's seven eight nine ten you're approached thirty seconds over about 18 seconds here's when I did it I could have done it differently we take ping-pong ball number one out of the barrel okay
2:36
so now we're getting really close to three sevens we've seen the first 10 when in the barrel but we're taking one out so ultimately after the first 30 seconds we've got ping-pong balls to droop 10 in the barrel but then when we get to 30 seconds now we bring in more ping pong balls 11 through 20 in fact they continue going in the barrel and they're actually coming in faster now you can't
3:03
really see it so well since I'm just letting in my hand they're going in faster than they were before and when we get close to 45 seconds elapsed well we'll have numbers 19 20 shot as well but we're going to take ping-pong ball there comes to out of the barrel and when we get to 45 seconds we are we definitely have ping-pong balls 3 through 20 in the barrel okay now what you know about 45 seconds compared to 30 seconds in a minute well it's you know 30 seconds went by in the first
3:43
time in their bowl 15 half of 30 went by in the second able to get ping-pong balls 3 through 20 in the barrel and one and two are out of the barrel when we fix 45 seconds we continue with ping pong balls 21 30 30 now they're going in even faster than before and in fact what's half of 15 at seven and a half after seven and a half more seconds when we get to 50 2.5 seconds we will have ping-pong balls well for through 30 in the bharatas threes going to come out here now right
4:23
now there comes threes coming out of the girl so by the time we get to 50 2.5 seconds we're going to have ping-pong balls forward to the three in the barrel we did put one two and three in there but we took them out can you guess the pattern seven and a half seconds cut that in half to three point seven five seconds starting at time 52 point 5 we're going to start playing 31 through 40 in
4:48
the barrel and when we get to let's see fifty two point five plus three point seven five but with MP 56 point two five we do that right we're gonna have well there comes four out of there now we're gonna have five through the 40 in the barrel and then we're going to cut time in out again let's half of three point seven five one point eight seven tell you they're right I'm afraid I did it
5:23
wrong but oh well we're gonna take we're putting 41 through 15 in the barrel but we're taking the five out we keep cutting the time in half we keep now we're gonna put then we're gonna put up 51 through 60 and take six hours and then 61 through semi and take seven out and at the end we get to 60 seconds one minute has gone by and well we've done an infinite number of things ignore the word no Laverne essentially that was my little computer I'm crashing program about to go out
5:55
that part you continue doing them process speeding out twice as fast over each half of the whole time here's the question at the end of a minute what balls are in the barrel
6:17
infinity Indiana so that we have a vote for infinity and beyond what balls are in the barrel after the first 30 seconds it was 2 through 10 the next 15 seconds it was what three through 20 the next seven and a half seconds it was fourth you there good another constantly changing it is constantly changing and I keep taking half intervals and going twice as fast and evidently this process of going faster and faster is happening until you get to 60 when it's gotta stop
6:48
guess what does anybody feel confident that they know the answer well we're pretending we do we do have to go intently fast which means we're going faster than the speed of light which of course is impossible but in real life but this is a mental experiment again there are no ping pong balls and barrel that's the right answer what first are you telling them 3 to 20 it kept getting bigger than either there are no ping pong balls in the barrel oh
7:28
that's right and I have a regular time let it run for a minute here slower slower slower this is almost real-time here we started over back
7:52
this is almost real-time here we're close to it watch the balls go in the barrel let it right here zooming the thingy - you see they're all number can you see that with the camera there comes one end up for 30 seconds that's pretty close to real time you come faster than before I took one out let them sell for sure
8:20
- comes out for sure it's not in the barrel faster and faster three comes out for sure here comes right now there goes three four comes out five comes out six seven eight nine okay I of course I can't continue that prospect over forever hey this a finite amount of code so I had to stop it somewhere because you can say the moment in time when every number natural was pulled out that means the barrel 71 came in at 30 seconds - came out about 45 seconds by 45
9:04
seconds 3 came out by 52 point 5 seconds 4 came out I was in fifty six point two five seconds you keep cutting that interval in half you take out every ball before you get to 60 seconds
9:22
it's a conundrum it seems like paradox the ping-pong ball coming through infinity he's got lots of strange things that happen have you heard of Zeno's paradox how can I ever reach that computer because to get there I have to cover half the distance and then how my every other region Xena was a Greek philosopher back who argued then that any motion is impossible because for any small motion you'd have to cover half the distance and a half that isn't have this and so
10:09
how you ever going to get there from here to there of course we know that we do move so it's a pair of paradoxes can be resolved maybe the resolve Zeno's paradox yeah I don't know maybe you have to get into quantum physics maybe it's time just time neither discrete nature in reality instead of a continuous nature I don't know for sure I think some businesses think so ok before we come back
10:35
to infinity let's do some actual real analysis where you re are thinking about things carefully let's take a look the problems that you did for today I hope with enough struggle that you got them and then you got them with pretty short elegant approaches you can do these problems with pretty short and elegant proofs let's take a look at them you don't think you want to look at your book again and first let's take a look at problem number 12 from section 152 page 17
11:07
suppose you're given some real numbers suppose X is less than n plus epsilon for all positive numbers epsilon proved in fact that X is less in reconvey probably the most direct proof here is by contradiction that's kind of a the most direct proof contradiction is also called indirect group that's why okay assume the contrary
12:04
that X is greater than a let me not finish the sentence here right right there then we have so extra money to say is there going through I'm going for the most elegant proof that I throw okay this is going to be short and sweet trumpets we in Delhi there's cinema cephalic elegant basically
12:34
then hey auntie showed me her prefers like this it was what do you do on do a plus X movie is a well okay on the add something in here then X is less than a plus X minus a because X long as a is UPS as other than it can be treated as an Epsilon I don't have to say loaded equal epsilon I could well that means X done that's a contradiction X can't be less than itself
13:17
it is more clear to you you could say over here what epsilon equal X minus a so X is less than X plus Epsilon oops but less than a plus Epsilon you could do it that way that's more clear to you the X minus a is acting like the epsilon for your hypothesis but again that's contrary putting a month three arrow is putting each other that's not a star is two arrows pointing at each other contradiction meaning your original
14:06
assumption of the contrary must be false therefore X is less than or equal to a might not be a bad idea to put that as the last sentence but I'm not going to write it
14:18
technically speaking that's good enough to emphasize the contradiction but for clarity state you might also want to add therefore assumption that the contrary must be false and therefore X must be less than or equal a short get to the right conclusion I tell it's all right let's going into the next number eight in one point three crew without an empty set that is valued above has only once I bring them
15:10
it is assumed without staying here that this says the sub some of the remembers all right not the complex numbers or something they say over the numbers we could well unless stated otherwise you can always assume your sets are going to be subsets of the real numbers that is found in the following my radiation for my limit is Vee has only one supreme perhaps especial to do that by contradiction well
16:04
suppose you got a subset of the real numbers that is bounded above let this be bounded above and let beta-1 and beta-2 be two distinct soups of X okay soup is an abbreviation for supremum soos is going to be our abbreviation for supreme omens or supremo might be the best little hole just a serves two distinct soups of s what do you think I should write in next there's an abbreviation that would be good to write
16:58
a four letter abbreviation long without loss of generality beta 1 is less than beta 2
17:16
you're basically done and almost Cruz itself okay they're both serve some ass they're both supreme of ass therefore you could say BAE the truth the least upper bound on the paint paint was less than a to do so I can't be an upper bound because that's what it means to be able you stop without contradiction okay minute right now you can also say beta one's the least upper bound the native who's bigger than it so it can't be the least upper
17:50
bound you can do it either way I'll say those in you can either do it by saying hey diffuse the soup it's the least upper bound beta 1 is less than it so it can't be number balance so it also can't be the soup or you could say beta one's the least upper bound and beta 2 is bigger than it so it can't be policed remember this it's always good to have the freeze all best
18:23
okay should write in another sentence but I'm not bothering to and these proofs are a little strange and a tricky once you've seen the trick they're actually pretty shortly number 12 let house be non-empty set of real numbers as far as I will prove you want us to like let's take what we know and then say prove that start there - I'll show you where our proof is starting or you don't have to restate the problem if that's the first question you don't have to
19:03
restate the problem you can if it if it's helpful for you you can say I'm assuming this and I want to show that at various spots in the group if that's helpful for you I'm just it's not markets everybody you're looking for I mean I will create the proof as a full credit if it is technically correct logically correct I this elegance thing you should just strive for yeah it's better for you in terms of just learning how to write hellion proofs and
19:34
it's also easier for me because then it makes your purse easier to read and easier to grade but I will struggle through long unpleasant on elegant proofs if I have to get over a sore okay
19:51
but it doesn't need to be technically correct okay house is a subset of the real Honduras assets out above number 12 here I just found proof for all the positive epsilon there exists an X in s such that X is bigger than beta minus s whole big word of warning do not assume s as an interval here it's an arbitrary subset of the real numbers it's bounded above not necessarily an interval so
20:36
therefore they mean the thing that people often say is well just pick X to be some number between beta minus epsilon and beta that's in that interval but you don't know it's a Nance that's the problem s is not necessarily an interval but the proof like that part once you see what to do
21:04
and you should be when I send in the note that I sent you with the other day you should as a first sentence say let epsilon greater than 0 be given you want to prove this is true for all epsilon 0 so the standard way to do that is to say give me an arbitrary positive epsilon it here's the main point in trying to mimic give me an arbitrary positive epsilon if I can show the inclusion of holds for that arbitrary positive epsilon then
21:37
this offense statement essentially this implication will be true for all epsilon so I want you to start with this epsilon greater than zero to give them in fact I want you to pretend it's a stamp you got some ink pad and you just oh there Upsilon greater those are will be given at the start of this kind of proof where you are trying to prove something follows so for all x1 that kind of proves now that I'm gonna prove that kind of
22:15
since beta the super best suit we know exists since s is not nothing common above I should say they say that yeah this is not empty FP mt7 doubt equal to BM consent since baby the soup which is beliefs upper bound beta minus epsilon is not a number of minutes and we're basically done because what does it mean to not be an upper bound of us it means there's something in S that's bigger than theta minus epsilon done this is unpacking that definition therefore there
22:59
exists an ass X and a subset X is bigger than 3d PCL everyone
23:16
is that short and that doesn't that you gotta the key issues Epsilon's arbitrary positive number there is an axle here's my beta my cell phone because basically at least a propellant made of minus epsilon not an upper bound of us there must be something in s that's bigger than myself that's it but that doesn't mean s was an Internet it could be a finite set of points it could be a sequence of points in sequence okay there are lots of things that s
23:50
QP alright on to the fun stuff about infinity here well okay I do have this section of Polish polish group and score to exist I don't want to read it to you here but I will put this on Moodle and I would suggest that you look it through it's a nice logical proof based on the scratch work we did the other day last Friday it's a nice logical proof that screw to exists
24:27
okay so I would encourage you to read that I'm gonna skip the crazy function the parts here there's anything between infinity cover I do want to ensure a couple crazy functions maybe on Friday I should also mention before we get to infinity that you should definitely pay attention to the Archimedean property which which you cannot prove just with the ordered field not Radice even though the rationals for example are in order to feel and do actually satisfy the Archimedean property
24:56
because there are some of the reels you in this framework you need the completeness axiom to prove it so you should make sure you read that TF s 18 is the following statements are equivalent these are all the Archimedean property all four of those statements here mean the same thing essentially and you can use them very deeply I don't want to spend time talking about the book but I will show
25:27
you I mean some code that illustrates it here give a name to positive numbers there is some whole number multiple the smaller one that eventually gives you higher than the bigger one is deep
25:49
all right let's get on through the affinity I am going to start a little bit formally here but ultimately we're gonna have lots of fun with affinity and draw a bunch of pictures on the board okay I think it is good to be a little bit formal and the beginning to talk about what it means for two sets to have the same cardinality the book alludes to this idea of cardinality but they kind of take a slightly different approach after all there's a definition this definition applies
26:30
to any sense not just sense of real numbers but a and B be any to sense and I do mean any to sense Y could be a set of numbers one could be a set of colors you get a set of suits of cards kind of sets of people okay any sense whatsoever we say a and B have the same cardinality and a strange word same cardinality yeah who taught algebraic structures therapists never use the word ejection
27:28
we talking about advice of ice of or physics our by ejection is is anybody know what by ejection means by dejection yep if there exists a but by ejection kind of fancy carries in that word a one-line I do functions or maybe or for me to a doesn't matter this is a one-to-one and onto option
28:20
it's a parody so to speak every element of a has paired up with a unique element of V and every element of B is coming up with the unique element of it one to one means distinct inputs two distinct outputs no two different points getting back to the same point on two means every point over the target set is an output or something okay that I'm just saying verbally isomorphisms group isomorphism use to bring my sources and filter by junctions
29:00
actually when it's a one-to-one you can also call the injection and when it's odd you can call the search action okay you don't farm who calls it a one-to-one correspondence that's what it says four to seven senior gallant you let the absolute value of a set no just looks like out some value it's kind of analogous down to value equal to the cardinality of a rich I'm not being very precise rich is equal to the number of elements in a it is fine
30:00
in a if a is finite although I haven't said what fun it is yet in Twitter's we go I should have defined fine interest and something else if a is infant okay I should define you find that an infinite first and you know you intuitively know what those are let me define finite and infinite for here I should have done them about to do first a set is finite a set a
30:56
that's the word a that's a set a the set a is finite if it has the same cardinality as the set 1 2 3 through n for some positive integer n I use as the enemy for the natural numbers the book uses E Plus positive integer that's what it means for a set I guess I didn't need the letter A in there set a to be fine it has the same cardinality as this set in other words there is a bijection no wonder one correspondence from a to that set a function that map's every element of a to a
31:57
unique element of B and no two elements of a get back to the same thing and everybody every element over here is not by the function something negative it you can pair things up okay you can lay away in such a situation a 1 a 2 3 and that's a week late way when you know it's fine
32:25
doesn't either an order and a doesn't even have to be a set of real numbers so no ordering necessarily a set a is infinite now that there's a bijection to say all the natural numbers but just if it's not finer anything means not finance there's no ejection to a set of this one
33:22
a set a is said to be countably infinite if it's a certain kind of infinite sight if there exists well if any has the same cardinality have the cardinality of the natural numbers in magnification if there exists a bijection from a to the set of natural numbers
34:06
you might also want to write that too if there exists of our objection from a to the seven natural runners today is uncountable if a is infinite and there is no like action from a to the set of natural numbers you can say cardinality is different but I would encourage you to also write there is no bisection from a now clearly and itself this is a countably infinite because you can match elements
34:58
of an upward themselves it's not clear that there are even any uncomfortable sense what if there are no infinite sense that cannot be paired up with elements of a man wonder Wharf
35:19
what is gonna get fun with me first fact back one we're going to be building up some interesting facts here the set of all even natural numbers this heavily I do that to Z
35:55
so for the set of about me even injured only gonna be even a solution to in and it's not a group sees a group disease our ranking but this is a citation from the set of all positive natural numbers that are even this is one definition that I skipped by the way it is what it means for something countable palpable means either finite or countably infinite an uncommon was purchase back one was the proof
36:53
I've got to come up with a my attention from true intends to show they have the same cardinality a once when I not to function define F from two ends and actually with the other way as well commanded to in by this is okay have you seen this one connotation before to turn pro season but the functioning in the colon domain codomain arrows indicating it's a function mapping define a function with this domain in this photo meaning
37:43
with this formula not going to finish the proof to do so here but I want you to believe it's pretty easy to finish definitely this function is well defined give me any even positive integer even natural number and we're true is going to be still a natural number you know that it's not something you have to prove isn't one to Lebanon to yeah gimme two distinct even numbers when you divide them by two they're still distinct one to one is it odd to give me an arbitrary natural
38:25
number whether it even they're odd if I multiply it by two that was the number that gets matric for example if the number I'm thinking about a seven fourteen it's not to seven if I'm thinking about eight sixteen is not eight for every injury if you multiplied by two that's the number in the domain that got Mac two kids now so one day one an onto function on this to this this is Hadley infinity
38:56
it's not the same cardinality as the natural numbers even though it's got fewer elements should I use that word in fewer there's no odds in here back to I continued all in this vein and continued considerating length sum of all multiples of 3 7 all multiples of 4 and show those are all count the infinite or I can start out in fun and show that let's just write this back as an equation Z is heavily in the Sun of all integers meaning includes 0 and negative integers as well it's
39:55
a little trickier finding a one-to-one and onto function from luzina and Orpha magazine it's extra tricky though not super hard to find a formula for such a function we are going to be satisfied with the picture and actually in some of the cases of the book is satisfied with a picture too
40:16
making a little chart here the natural numbers are up here the integer is going to be down there I am going to indicate our wonderful an analogy function by a pattern rather than a formula I need to say one each natural number it's matched up within the integers and I need to guarantee though it's a one-to-one and onto function that's pairing these things up that can be done pretty easily with the pattern in the following rate for example
40:50
let that pattern continue so the idea here is the function this time I'm letting env2 domain max 1 2 0 2 2 1 3 2 negative 1 4 2 2 5 2 negative 2 6 2 3 7 2 negative 3 8 2 4 9 to the negative 4 etc 5 negative 5 or 6 negative 6 7 negative 7
41:27
it's one-to-one through their two distinct numbers and the natural are going to match it to think distinct integers that's clear from the pattern and it's on through giving any integer like a hundred what gets mapped to it I guess by the pattern a would be 200 what about a hundred and one excuse me what about negative hundred I guess my pattern with each 100 more 200 is not 200 to a 1
42:14
that's true fact 3 I never saw any fifth surprising the several rational numbers is counted anthem has the same cardinality as the 7th of natural numbers that's definitely less obvious I mean think about a number line you know that between any two real numbers no matter how close they are there is a rational in there in fact you could say there's infinitely many rationals in any little tiny interval you know somehow the
42:54
rationals are going to the same cardinality as the natural numbers they are infinite
43:08
how in the world would you prove such a thing again we're not going to do it rigorously because we're not going to come with a formula but I'm going to draw a diagram that helps you believe with this facts
43:22
a few different ways to draw this diagram your so I'm going to do it in the diagram
43:52
and by the pattern you see here think about it every rational number is somewhere in this diagram it one for everything indicated pattern in fact every rational number except zero is in this diagram infinitely often 1 1 1 1 half 1 half 3 6 would be 1/2 right every number is not only in this diagram every rational everybody's in there infinitely often so I'm going to come up the by action from and the cube with the SATA gram I'm going to starting
44:51
then with zero and by circling that and starting their own indicating that one gets mapped to zero naturally it's not to zero then I'm going to go all right on the diagram in a spiraling path by circling that next that means 2 gets mapped to 1 circling this way 3 is not 2 negative 1 continually inspired won't for is any good map to 1/2 should I go there or there it doesn't matter
45:26
um skip that one because that's one we already go to goods map-21 so we skip them all over here and where am I at one gets out there two three four five it's not - six - snap - negative 1/2 skip this one's already that does 7 gets mapped to negative 1 sit that one - no no don't scare selling it's not too negative - it gets up to 1/3 9 to 2/3 10 to 3 halves 11 to 3 it's ceteris funnel around every integer is going to get back
46:17
to something unique because of the skipping and every rational number gets hit eventually so this diagram indicates helps you believe without being a Reuters proof this is true
46:47
any with me here okay it gets more fun okay it's even more fun actually I saw professor here performing the fits once I I wrote down somewhere I can I write down babies maybe in my own workshop somewhere or my filing cabinets paper not electronic stuff I saw professor give a formula for that fact for the set of real numbers does not have the same cardinality is the set of natural numbers and in fact people even write the set of
47:33
real numbers of carnality is strictly greater than strictly greater than the cardinality of their homes thus every lowers is uncomfortable it is a bigger size of infinity
47:54
it's you can you have me say this to the cardinality of the natural numbers is less than the cardinality of the reals because definitely there's an injective function 1 to 1 from the naturals in the reals but there's no no such function is an odd function
48:20
the book gives a nice scrupulous it's a very nice group actually based on the completeness axiom you should read and study that proof I'd probably go over it someone Friday maybe just with a picture I try to encourage you to draw a picture as you read the books proof that they even drop to 40 but you read right it's a pretty nice proof that actually once you study it and understand it you you probably could believe you couldn't come
48:46
up with it instead I'm going to do what's called Cantor's dying validation argument
49:11
it's even funnier than this actually any tiny little interval of real numbers no matter how small and you can be super close together no matter how small okay again I'm not gonna write out the arguments as a literal argument with words and sentences I'm just gonna draw a diagram again and then this diagram the numbers I put on the Left are going to be elements of the natural numbers and the other side on the right are going to be numbers in the open interval from 0 to 1 now what
50:05
kind of argument could you possibly do to prove to prove this is true contradiction assume the contrary that these are equal these part Nelly's that are Zn have the same size infinity assume the contrary that already is countably infinite then there must be some one-to-one correspondence some pairing here every natural number being paired up with a unique number in the interval from 0 another reason this is not going to be rigorous is because we're
50:45
going to use decimal representations every number between 0 and 1 can be represented by a decimal and just for the sake of clarity assume we're not using the alternative representation of some decimals with the infinite number of nights
51:06
now you don't know what this correspondence is but since you're assuming in the contrary that this is not true there must be some correspondence so there must be some desk in here that can snap match type of one those azar digits 0 through 9 there must be some decimal representation of some number between 0 & 1 that get snatched up good to go 3 it's sort of it's like you have an infinite matrix here on the right
52:04
and every number between zero and one every real number is I'll be in here somewhere I haven't lost now create get a contradiction by creating a number that can't possibly be over
52:22
define X to the zero point B 1 B 2 B 3 etc we're beating one visit digit not equal to a 1 what does anyone I don't know anyone I should say what does anyone one say it 7 pick something other than something for me the second digit is not equal to K 2 2 second digit after the decimal in the second number in the room what does a 2 - I don't know if it were 3 picks I think you're further than 3 these 3 is different
53:11
from 83 3 etc that pattern continues you are making sure this number you're creating does not match these numbers along the diagonal of this diagram that's why it's called Cantor's diagonalization argument and therefore you have created a number that isn't that easily
53:34
right it can't be personal because it doesn't match the first day and the second is it doesn't National second it can be the third does it doesn't mention the 37
53:50
what are these commonalities the cardinality of the natural numbers has a special symbol it's called elephant not looks like this
54:09
is actually a Hebrew letter Elif I'm not sure very pronounced the right what about the cardinality of a real what you'd call that well some people write it like this 2 to the L of not what does it mean to raise to 2 in the infinite Cardinal other times people just abbreviated C for continuum really big kind of like the speed of light is really big and see where everyone's the speed of light in physics ok this is not the speed of
54:53
light see here stands for killing anyone okay maybe for a tram business we make an old face
55:05
is there any Cardinal number between these two cardinal numbers with me every few moments is there any number between these two cardinal numbers tantaran hypothesized that there wasn't but there's no infant set that is not countable yet is also less than the cardinality variants that he struggled for years to try to prove it or disprove and the story is you went crazy because I don't know if that was a reason
55:44
it's called the continuum hypothesis that there is no car DNA between these two and the answer to the continuum hypothesis was completely unexpected the continuum hypothesis can be proved to neither be true nor false it can be proven to neither be true nor false it's not true it's not false it's undecidable and what does that mean it really means that should be an axiom that's what it means it's a statement independent of our usual accent hang with me a
56:31
little more I'm gonna point you to some videos online they're gonna help you to explore this a little bit further but let me just say to conclude today then this fun with infinity can keep going there are not only two sides of infinity there are infinitely many sizes of them to me and ultimately you can get to the biggest I call it the ultimate paradox the ultimate contradiction
57:01
and I can freeze it in one question does the set of everything contain yourself what is the set of everything I mean the set of everything the collection of all things all numbers all letters all colors all people all atoms all my ears does the senator everything doesn't contain itself in loss because it contains everything but I'll give you second teen itself how do you resolve that that does seem like a paradox doesn't I didn't resolve it we'll talk about that refund right
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