Zeno's Paradox-Series on Infinity Part 1

[JayGatsbyOdysseus]

The universe is expanding .8 C (speed of light) beyond the outter most reaches. Where and when does it stop? I don't think it does. This proves our consciousness moves beyond death. Death is like a hiccup, a cosmic speed bump, where uniform motion meets some turbulance and there is light dilation and length contraction. He should show the calculus for this hiccup to show how we get beyond 1. This is the speed bump into the next realm, a different dimension, the door.

[HIONH]

The paradox and all the infinity paradoxes of this kind are actually rejected by the atomic theory of Democritus. Mankind didnt had to wait for calculus, sorry!

[knowitall23]

Very interesting. Hope I can find the solution video

[soslothful]

I thought it was a fine looking turtle.

[MisterrLi]

By adding 1/2, 1/4, 1/8... you will get 1 in the end ONLY if you disregard the

infinitesimal difference between the sum and 1, as the standard real numbers

are defined to do. They are Dedekind complete, which means that infinitesimal

differences are neglected from the real numbers. If you use hyperreal numbers

that include infinitesimals, the sum of 1/2, 1/4, 1/8... will not reach 1. Just wanted

to make this clear, as geometry is not necessary about standard real numbers.

[lTHRICEl]

I know what you mean when you say it is cool that these thinkers were discussing these ideas at the time. I felt as if these philosophers were scratching the surface of quantum physics in a way, but of course did not know it. lol

[mechkapechka]

1 thing i don't get please someone explain to me how he gets from 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... and so on how did he gets 1 ?

[SMWTas]

So, does this mean I shouldn't ever buffer? :P

[zadeh79]

Actually, what your looking at is an example of einstein's relativity. Pretty damn interesting.

[RUL1S88]

Nothing is true. Everything is forbidden.

[bedwarri0r333]

if a limit exists on infinity, and a limit is defined as "A point or level beyond which something does not or may not extend or pass" (Google.com), since infinity doesnt end, the limit needs to end (hence it being called a limit.) But since that limit has a start and finish isnt that a line? which is infinity? if you dont understand then you dont know math and therefore i dont care for your input. i need a math perspective or else it wont make sense. if it doesnt, i already accepted that.

[bedwarri0r333]

he reaches the turtle because he is real. because he occupies space. if you see the two as a point having no magnitude of any sort then yeah, one point can not overtake another point. but the points themselves are nothing since they have no magnitude. how can that situation exist if the things in that situation don't?

[bedwarri0r333]

@VitalSigns1 but time is infinite and we are inside time. are we not real then? theoretically there is nothing after infinity. so a beyond infinity isnt real. however infinity is a real thing. we might not be able to perceive infinity as a whole, because if we could then there would be nothing. so, if time is infinity and we are inside time the same way a limit is inside infinity, then we wouldn't exist without infinity because a limit cant exist without infinity. it has to be real or we arent.

[bedwarri0r333]

@VitalSigns1 it does if you arent dumb and include a limit. if you say achilles has to end at the tortoise why not have the tortoise just not move at all? without a limit on his speed he should overtake the tortoise. so by having a limit already you are saying that he cant overtake the turtle. without a limit he would keep going being infinity. so a limit says infinity exists the same way a limit exists because of infinity.

[mgcgyugitradeco]

finding sums of infinite series is in algebra 2 I did it in that class and all you do is take the first term divided by 1-r where r is the common ratio and it has to converge

[zadeh79]

@VitalSigns1 Exactly. The theory of resolving an infinite sums, does not help us understand this paradox. To me it suggests that time and/or space are discrete, and that is how Achilles, ultimately, surpasses the turtle.

[bedwarri0r333]

it proves your time is discrete. not time in general. you would not be here if time ended with your parents...

[zadeh79]

I always used to think this paradox proves space and/or time are discrete.

[kumiwanakilya00]

thats a fast turtle

[MagnaCarta0]

@VitalSigns1

Please go take a calculus course.

[LogosNigrum]

@VitalSigns1 Any real number can be expressed as a sum of an infinite number of components. E.G. The infinite series for pi.

[Toca91]

I heave a problem understanding this paradox... If the runner doesn't reduce speed after passing half of the way, then he can pass the turtle!

Maybe this is the problem:

-He is set a target and he is running for a target.

-The turtle is his target.

-Target moves...

-So, he slows down he could reach the target same way as if it didn't move...

-But the target moves, so he slows down constantly = Paradox.

Or

-He set target in front of turtle.

-He ran at constant speed until he passed the target.

[UltimaWeapon9339]

@Toca91 The paradox is not that there is a goal that both of them are running toward but simply that achilles is just trying to catch up to the turtle. When achilles reaches the turtle, the turtle has already moved further than it was. I think the paradox is easier to understand this way: as you reach toward an object, the distance between your hand and the object decreases, right? But does the distance ever reach zero or does it just keep dividing into infinitum? That's the paradox.

[nickcorn93]

Heisenberg's uncertainty principle doesn't play into this argument because this is a quantummechanical effect. Achilles and the Turtle are not particles. They don't even have to have mass and therefore momentum. We know that both start the race in the same location and just by measuring the speed of each we can determine their precise location and when they will pass eachother.

[nickcorn93]

I think the easiest way to explain this to people who are easily fooled by magic tricks is that not only the distance in each stage gets shorter, but the time each stage takes is shortened by the same factor (assuming both racers race at the same speed). We can divide a time interval into another infinite amount of intervals, but that doesn't mean it will take forever for this time interval to pass.

[DojiSan]

@nickcorn93 Does Q E's Uncertainty Principle play into this argument? You cannot know both the position and momentum of a particle. You can solve for the time, but once known, you would not know where the particle is (ie Achilles and the Turtle).

[StitchedMusik]

at some point achilles passes the turtle, the stages cant be shorter and shorter

[danielcarmi305]

@MathDoobler prove it, considering your so brillian. what was wrong with my theory?

[MathDoobler]

@danielcarmi305

What theory?

Anyways, the claim is that 1/2 + 1/4 + 1/8 + 1/16 ... = 1. Seems counterintuitave. But it's true. Are you familiar with proof by contradiction? It goes as follows: Suppose otherwise, and that 1/2 + 1/4 + 1/8 + 1/16 ... != 1 (where != is "does not equal"). Then 1 - (1/2 + 1/4 + 18/8 + 1/16...) is a positive nonzero real number. Call it Q.

[MathDoobler]

@danielcarmi305

Ok, the claim is that 1/2 + 1/4 + 1/8 + 1/16 ... = 1. Seems counterintuitave. But it's true. Are you familiar with proof by contradiction? It goes as follows: Suppose otherwise, and that 1/2 + 1/4 + 1/8 + 1/16 ... != 1 (where != is "does not equal"). Then 1 - (1/2 + 1/4 + 18/8 + 1/16...) is a positive nonzero real number. Call it Q....

[MathDoobler]

@MathDoobler

There is always an whole number power of 1/2 that is smaller than Q. This seems intuitively obvious, but you can prove it, since you can just compute the integer ceiling function of logbase0.5 of Q (and a little review of logarithms will show you that 1/2 to that power is indeed smaller than Q). Call this integer N. You can see that the Nth partial sum of the series is 1 - (1/2 + 1/4 + 1/8 .... + (1/2)^N) = (1/2)^N (this step can be proven by induction, its easy). But then

[MathDoobler]

@MathDoobler

... but then the partial sum is strictly smaller than Q. But the complete sum 1 - (1/2 + 1/4 + 1/8 ...) is even smaller! Hence the complete sum is strictly smaller than Q! This is a contradiction, because we said the complete sum equaled Q, but then we proved it is not equal to Q (and was in fact smaller). Hence no such positive real number Q can ever exist.

Once you convince yourself that 1 - (1/2 + 1/4 + 1/8...) is not negative either (using the same method), the only remaining

[MathDoobler]

@MathDoobler

... possibility is that 1 - (1/2 + 1/4 ...) = 0, since we showed it can't be positive nonzero or negative nonzero.

Hence 1/2 + 1/4 +.... = 1.

There are other proofs I'm sure, but I thought you might enjoy a proof by contradiction.

[PacRimJim]

That's why Newton and Leibnitz invented the calculus. Infinity, addition, infinite series, etc., are concepts, not concrete objects, and thus are not subject to physical limitations of time, etc. The calculus obviates the need to add discrete quantities infinite times. It's a fiat mathematics. Fiat sum (or whatever the Latin inflection is).

[Knukleheada]

my friends don't believe me when i mathematically prove them that 0.9999... = 1

infinity is difficult.

[mateo3470]

@Davidson1956

Set up a system of equations and the feeble-minded 'real' analysis disappears. These set of hacks are nothing to build on top of. It is also increasingly useless for physical understanding. Thinking that you are adding an infinite amount of numbers reveals a serious misunderstanding. Even theory needs to be non-contradictory. Even the most elementary acquaintance with foundational issues in mathematics concerning completed infinity would reveal it to be entirely untenable.

[MathDoobler]

@mateo3470

Um, how specifically is it untenable?

[mateo3470]

Search "PlanetMath: Zeno's Paradox" then consider the following:

A simple system of equations makes the problem easy. D[A] = 10T D[T] = T + 10. Substitute D and solve for T and you get T = 10/9 (rational!).

A bad modeling of the problem says nothing about accomplishing some feat of non-enumerable complexity.

Michael Livshits ideas on how to simplify Calculus are interesting.

[mateo3470]

See: "Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and Parmenides' prohibition" by A Papa-Grimaldi

[istabuinface]

.999... does equal 1! Have .999=1 thus make variable x=.999... then multiply both sides to make the problem (10)x=(10).999... then simplify to 10x=9.99... then subtract the x value from both sides giving you 10x-(x)=9.99-(.999...) thus simplifying down to 9x=9 then divide both sides by 9 which gives you x=1 Now since x=.999... like we stated at the beginning .999...=1

[minikakapowpow]

Thanks for explaining it to me.:)

[dineshssairam]

At each 'stage', the distance that the turtle moves away from Achilles keeps decreasing. At some point, Achilles will overtake.. or probably step on the turtle and squash it. Zeno was a retard.

[SSCCMath142]

It's not rounding. If 0.999... does not equal 1, then please exhibit a number between 0.999... and 1.

[netzweltler]

@mpattym

It is true to say 9/10 + 9/100 + 9/1000 + ... at no step of computation reaches 1. But infinity means "after all steps".

[maribakumon]

eventually you will have to put the fractions in to decimals and to the ancients it wont be one but now once you reach 0.9999999... + the next smallest step = 1

[netzweltler]

@maribakumon

0.99999... equals 1. No next step needed. Because 3 times 0.33333... = 0.99999... = 1 .

[maribakumon]

one problem with this is also the greeks number line ended at 10,000 therefore nobody belived in infinity

[Paddis92]

I hate this. I really do hate these kinds of things because I no matter how much I think about it I just can't seem to wrap my head around it. It's a great way to waste time if you've got nothing better to do though.

[clemzzz]

This is only a paradox because an element ( TIME ) is missing.

and thus all this adding infinite numbers nonsense is a waste of effort.

the mathematical solution is just that ..... "mathematical", and has no basis in reality

[sonnetxi]

@clemzzz youre telling me an apple and a banana dont make 2 fruits?

[netzweltler]

@clemzzz It was Zeno´s suggestion to use mathematical points with no dimension to describe reality. Isn´t this already "mathematical"? So the mathematical solution is the right answer.

[netzweltler]

Let´s assume there is a system, where Zeno´s argument is valid, you as a part of this system can never prove a paradox. If you are watching the clock during the race, the hand of the clock will never reach the next second, because it first has to reach half a second, then another quarter and so on. So the clock and even you and your consciousness as a part of this system are subject to the same rules, and time and space is still linear within this system, and Achilles eventually wins the race.

[netzweltler]

Doesn´t it end up in the question: Does 1 divided by infinite yield 0? If it yields 0, there is no Zeno´s paradox. If it yields not exactly 0 (just close to 0), then we have got a paradox. Am I right?

[DavidsonSSCC]

Infinity is not a number in any conventional sense. All we can say is that dividing 1 by increasingly large numbers will give a result that is increasingly close to zero. This is the essence of the rigorous definition of a limit as n approaches infinity.

Ultimately, the paradox is dispelled by seeing that infinitely many steps may be taken in a finite amount of time in the context of Zeno's situation.

(But you can count on some stubborn folks disagreeing no matter what the math shows!)

[NastyNassim]

lim (n-1/n) =1 n-> infinity  =)

[netzweltler]

@NastyNassim

If you say:

lim((n-1)/n) = 1 n-> infinity

you have proven that we have got a paradox.

If you say:

(n-1)/n = 1 n-> infinity

you have proven that there is no Zeno´s paradox.

[albpeter]

This by no means solves or refutes Zeno's Paradox.

[bennyb4]

Hmmmm momentum and inertia and multi dimensionally existing characters..... why is one in linear motion to the mathematical trail or history, equationatically, wait is that a word, of the second character in this forum. Sounds like many rules have not been stated or were remitted due to the large task which would ensue of defining the entirety of the characters distinctions and the environments. This might be why we should all stop trying to figure out who we are....... too much data to unfold

[lanmancz]

I see a major problem with this Achiles and turtle example of the math experiment. There are two separate actions and they happen simultaneously. There's no step one : move Achiles, step two : move turtle. So don't see no paradox. The math part makes more sens tho.

[Just42day88]

This was helpful thanks

[philosophywpaul]

Thanks for posting. Good stuff. :)

The problem I see is that mathematically we can make sense of convergence/limit and this math works in the world. However, conceptually (using our exp & conceptions of space/time), the problem remains. If real space is not really inf divisible, what is the most basic unit? A quantum? But doesn't that exist IN space (conceptually)? I'm still confused...

In short, I think we have solved the problem... but not on the conceptual level.

What do you think?

[DavidsonSSCC]

Great questions!

Mathematicians around the 1850's created the rigorous, logical language that gives a sound foundation to calculus. On that basis, we are able to handle difficult problems involving infinity and infinitesimal.

I'm not a physicist, but if real space is not infinitely divisible, the smallest length may be the Planck length. If true, I'd think that ol' Achilles would still best that turtle!

[paulio2293]

i agree, geometric serieses are not very intuitive on the mind, it's only when you see the algebra that u can undertand why the series converges, otherwise you would just think its infinity

[myohme]

----decay----

Physics

[downcard11]

At what point does it become one? What fraction brings it to make it a whole one?

[downcard11]

First time I heard this in college it flipped my mind. Thanks for sharing.

[MorbidAngel4444]

Zeno's paradox and the concept of infinity makes perfect sense mathematically, but I just don't see it happening in real life. Suppose there are two people. The first one runs very slowly the second one is a speedy runner. Even if the first runner starts ahead of the second, wouldn't the second person eventually catch up and run past the first one at some point, because he/she is faster?

[x1101011x]

Achilles doesn't need an infinite amount of time to reach the turtle. Just like the distance, the amount of time that passes in each iteration is getting smaller very fast and converges to the time required for Achilles to catch up with the turtle.

Achilles's progress doesn't depend on your adding up those numbers.

Infinity, like all of Mathematics, is a representation of real-world concepts. Of course it works in the real world; that's where it came from.

[ANDREWSABAT]

you are awesome man!

[mistershithead]

Zeno's paradox: if you want to travel a distance, you've to 1st travel an infinite number of small distances. (1/2, 1/4, 1/8, 1/infinity etc)

but the loophole here is that if you can travel 1/2 of the distance, then you'll be able to travel the rest.

zeno hinted at it by saying that it was like incorporating zero into maths thus introducing a flaw by allowing nothing to be something.

he was playing with logic.

he threw truth in your face, and then made it disappear. walla, illusion incarnate!

[mistershithead]

"If you wish to make an apple pie from scratch, you must first invent the universe."

=D

[ReikaiDemon]

I hate it when people say viola, but spell it walla...

[x1101011x]

I hate it when people intend to write voilà but spell out a stringed instrument instead.

[doomsdaymonster7]

@x1101011x voila

[BritneyAwesomeSpears]

@mistershithead : The problem with your solution is, to travel 1/2 of the distance, you have to travel 1/2 of 1/2 of the distance, and so on.

Zeno intended to say that motion is impossible.

(Sorry if I'm wrong I'm 13 years old)

[kingkilla312]

I'm only 15 and i'm very intrested in this. And strangely I can't wait to learn about this soon :]

[danalan666]

this seems to be mathematics answer to god - either that or the math itself is simply wrong - so what you're telling me is that .999 to infinity IS1? It takes a leap of faith and fills the gaps and says 'oh well - its just a value it isn't" How can this be? its time thats an illusion - the greeks were right for a change

[hirofan]

I thought about this stuff as a kid long before I learned that the Greeks pondered it. Math can never resolve this paradox. The universe clearly doesn't exist. Through meditation (altered state of consciousness) one may experience this moment which is all there is.

[slikyle]

ah, thus the infinite geometric progression is born

[Davidson1956]

Merry Christmas, my friend! You amuse me. Sophistry can be fun, but I wouldn't get too wrapped up in it. If Achilles was not faster than the turtle then we wouldn't have Zeno's Paradox to kick around, would we?

[StunnerJohnKim]

well said.

[richardhewitt124]

Please post a video on how to work out pi.

[Davidson1956]

If by working out pi you mean calculating a decimal version of pi, there are numerous ways to do it. Understanding why any of them work requires at least some knowledge of calculus, I think. My favorite is adding the squares of the reciprocals of positive integers to get pi square divided by 6. I can prove why it works, but it's probably too deep for YouTube. You would also need to know about Fourier Analysis.

[WiAmp]

there is a -relative- easy way to calculate pi if

you are able to pull roots by hand.

Actually you really need to be able to do that.

Otherwise you need a calculator

to pull the roots so everything is in vain.

2-3 years ago i calculated pi on enrolled squares.

I could make quite good narration, i think to a few

1/1000 th ..

[WiAmp]

+ , unfortunately with every narration my

method .. which i "invended" from scratch in my free time leads to recusive roots. so for each narration

wich decreases the with of the intervall of pi

i get a "root in a root" , "root in a root in a root" and so forth .. pulling it by hand means proportionally

more work for proportionally more accurracy.

Im sure if i wouldnt be such a looser in anylsis i could solve that problem. ;-)

[FreakyFeline88]

thanks it was very helpful! I recentley started learning philosphy and well Zeno's paradox is pretty tough to understand what's going on, but well now I understand it a little better ^^

[Davidson1956]

You're quite welcome! It's been a pleasure to know that some people have gotten something out of this.

[d34dly4rr0w]

Infinitesimal? Pfft. Saying that you can't see it but it does exist is a self contradicting statement. If for example, there was an infinitesimal object and it does exist, that must mean it has some kind of height, width and length.

[deyaa1989]

Very interesting... Thank you

[nothingstopskings]

Thank you Sir. I knew in my gut that Xeno's argument was flawed but my philosophy teacher thought otherwise. I will forward him this video.

You are Awesome, and I will subscribe. I really really love math. Thank you so much!

[Davidson1956]

Thank you for the very kind comments! It is a real pleasure to share these great math ideas with people who appreciate them. I hope to add more videos on a variety of topics very soon.

I think there are a few holdouts out there on Xeno's Paradox. Mathematicians had to cheat a little to debunk Xeno by defining ways to deal with infinity, and that may be the philosophers' complaint.

Soon I will offer a video that illustrates one way of formally and rigorously settling this paradox.

[didi0014]

200 views! nice

[sammygee17]

nice vid :)