# Origional solution to Achilies and tortoise paradox

## Recommended Posts

I solved this paradox in high school when the math teacher presented it in limit theory.

Since its still listed in the unsolved problems, I guess it didnt take back then. Let me see if I can make it more transpareent.

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.[9]

Solution:
The falacy of the paradox is that the distances applied for Achilles to cover is provided in like some kind of rapid decay series governed by the pace of the tortoise, that looks like a 1/10th decay.

d = 100m + 10m + 1m...

The serries wil continue on forever, so I assume the paradoxical effect is that Achilies can never seem to be able to pass the tortoise?

However they didnt provide the constant rate of speed or a series for time, so technically its an invalid data set, but to solve the series lets assume Achilies constent velocity is 10m/s.

The logical falacy is that we are worried about an infinite series of distance but are forgetting that it’s a race of speed and both have a constant speed.

That means an infinite  series of time also must coexist with the series for distance:
d = 100m + 10m + 1m... =  111.111111m
t = 10s + 1s + 0.1s... = 11.1111111s
Va =10m/s,
Vt = 1m/s

So the infinite series just disappears when including both terms and we end up with two linear slopes that cross where Achilies passes the tortoise.

To clarify, the two serries each resolves down to infinitely repeating 1’s. where distance has one extra decimal place. to get the velocity we take
v =d/t = 10 m/s
Therefore the two infinite serries simply cancel out.

Edited by TakenItSeriously

##### Share on other sites

So you have "solved" the paradox by just ignoring it. Well done.

• 1

##### Share on other sites
7 minutes ago, Strange said:

So you have "solved" the paradox by just ignoring it. Well done.

or you could say, I annihilated it

##### Share on other sites
6 minutes ago, Strange said:

So you have "solved" the paradox by just ignoring it. Well done.

I think this is a tad unfair as TIS has the germ of the resolution, but has just not stated it clearly.

However in view of the reception I got in the last thread about these ancient paradoxes, I am disinclined to help further.

##### Share on other sites
9 minutes ago, studiot said:

I think this is a tad unfair as TIS has the germ of the resolution, but has just not stated it clearly.

However in view of the reception I got in the last thread about these ancient paradoxes, I am disinclined to help further.

Why? did I say something that was offensive. I certainly didn't mean to but if I did offend, I’m sorry.

##### Share on other sites
45 minutes ago, TakenItSeriously said:
I solved this paradox in high school when the math teacher presented it in limit theory.

Since its still listed in the unsolved problems, I guess it didnt take back then. Let me see if I can make it more transpareent.

Listed where?

##### Share on other sites
15 minutes ago, studiot said:

I think this is a tad unfair as TIS has the germ of the resolution, but has just not stated it clearly.

Yes, looking at it again, I think you may be right. It looks like a rather garbled approximation to the usual resolution (in terms of limits).

47 minutes ago, TakenItSeriously said:

Since its still listed in the unsolved problems

Whatever that list is, it seems to be very out of date.

##### Share on other sites
11 minutes ago, swansont said:

Listed where?

No listing, he just showed the riddle, and I found a solution. I thought it had already had been solved back then.

In fact today was the first time I saw that it didnt have a solution.

Oh I see what you meant, I googled unsolved logic problems and found it listed under Zihnos paradoxes. is said there were a series with 9 that still survived which I thought meant 9 had remained unsolved. then I saw that the millet problem had a solution given

Edited by TakenItSeriously

##### Share on other sites
1 minute ago, TakenItSeriously said:

In fact today was the first time I saw that it didnt have a solution.

You apparently saw that in a "list" ("its still listed in the unsolved problems"). What list was that?

##### Share on other sites
1 minute ago, TakenItSeriously said:

No listing, he just showed the riddle, and I found a solution. I thought it had already had been solved back then.

In fact today was the first time I saw that it didnt have a solution.

Where did you see that it didn't have a solution? (Please don't make me ask a third time)

##### Share on other sites
34 minutes ago, swansont said:

Where did you see that it didn't have a solution? (Please don't make me ask a third time)

I guess we cross posted. see three posts up.

I’ll try and find the link

Something is wrong with my browser I got to that page and everything froze

##### Share on other sites

Well, I wouldn't hire Zeno as a running coach! "See where the competitor in front is? That's your finish line... no, where he is now! Dammit, he keeps moving! Where he is now, Now! Now, now, no-, n-, n- ... Oh, you've passed him! And I thought that wasn't possible!"

I never thought it was much of a paradox - more an example of a mis-stated problem leading to a mis-taken conclusion.

##### Share on other sites
1 hour ago, TakenItSeriously said:

Something is wrong with my browser I got to that page and everything froze

It was obviously trying to catch up with a slower computer.

1 hour ago, TakenItSeriously said:

I guess we cross posted. see three posts up.

I’ll try and find the link

"Some of Zeno's nine surviving paradoxes..."

Which is (not very clearly) referring to the fact that he wrote a book with 40 or so paradoxes in. Of those 40, only 9 are now known about (that is the sense in which "survived" was used).

• 1

##### Share on other sites
3 hours ago, Strange said:

It was obviously trying to catch up with a slower computer.

How do you mean? As in someone redirecting my google results?

Edited by TakenItSeriously

##### Share on other sites
3 hours ago, Strange said:

"Some of Zeno's nine surviving paradoxes..."

Which is (not very clearly) referring to the fact that he wrote a book with 40 or so paradoxes in. Of those 40, only 9 are now known about (that is the sense in which "survived" was used).

I see, that makes more sense. Thanks.

##### Share on other sites
3 hours ago, TakenItSeriously said:

How do you mean? As in someone redirecting my google results?

No it was a joke and quite funny I thought.

##### Share on other sites

It sounds like something I saw before... Archers Paradox? or Arrow paradox? - something like like....  It is the same...  The arrow can 'never' hit the target as it always HAS to travel halfway to the target before it reaches it...  but of course when it has got half way it then has to go half way of the remaining distance and so on. As it has to go through infinite 'half ways' it can never reach the target. This is of course a misunderstanding and ignores the fact that time flow steadily in relation to the arrow flying and if you plot distance vs time you will see the point on the graph at the precise time the arrow impacts.   The failing of the paradox can be shown by standing in front of an arrow after it has been fired and observing it actually hitting your body and passing straight through it if fired from a decent enough longbow.

Also, if you want to take bets, I'd happily race a tortoise over 400 metres to prove the point. I'll give it a 370 metre head start.  Normal tortoise mind you, not one of those big ones!

• 1

##### Share on other sites
46 minutes ago, DrP said:

........ if you want to take bets, I'd happily race a tortoise over 400 metres to prove the point. I'll give it a 370 metre head start.  Normal tortoise mind you, not one of those big ones!

Are you sure DrP? Some of those tortoises can move at 5mph! And if it's Russian, forget it !

• 1

##### Share on other sites
9 minutes ago, Tub said:

And if it's Russian, forget it !

Haven't they just been banned from taking part?

• 1

##### Share on other sites
5 minutes ago, Strange said:

Haven't they just been banned from taking part?

Ha,ha..... touche! So go for it, DrP - but make it just 300 metres to be on the safe side.

##### Share on other sites
23 minutes ago, Tub said:

Ha,ha..... touche! So go for it, DrP - but make it just 300 metres to be on the safe side.

lol - It would have been a close race in my prime from 370 meters...  Nowadays though I wouldn't stand a chance if it goes at 5mph in a straight line. I am banking on the tortoise not knowing wtf is going on though so it won't be going at the full 5mph and maybe not in the right direction....  hopefully it will stop to eat the grass too. I will out smart the thing.

• 1

##### Share on other sites
22 hours ago, DrP said:

...... I will out smart the thing.

Again, DrP, are you sure? Tortoises may not be as thick as they look - after all they've been around for millions of years and, if going by your picture, you're a bass-player, it might be a very, very close call. No offence....honestly.

• 1

##### Share on other sites
12 minutes ago, Tub said:

Again, DrP, are you sure? Tortoises may not be as thick as they look - after all they've been around for millions of years and, if going by your picture, you're a bass-player, it might be a very, very close call. No offence....honestly.

Ha ha ha ha! Very funny!   Maybe I'll race you instead then tubby! ;-) lol

• 1