paradoxes

[JaBlue]

When I went to take a crap today I took Paradoxes by R M Sainsbury (http://www.amazon.com/Paradoxes-R-M-Sainsbury/dp/0521483476) with me. It has some cool stuff in it.

My plan is to relay the paradoxes in the book and have the forum talk about and respond to them. If you are familiar with the paradox in question, please don't respond. Let others think about it for themselves. Similarly, don't look up information about how to respond to the paradox. Just don't be an ass. Lets see what we can come up with.

We'll do one at a time and discuss them until we are at a reasonable stopping point. If nobody can produce anything on their own that is valuable, which may very well be the case for a lot of these, I'll introduce a response from the book.

Note that the wording of the paradox is going to be directly plagiarized from the book so I don't muddle it up for everyone.
So here goes:

The first:

Barber Paradox
In a certain remote Sicilian village, approached by a long ascent up a precipitous mountain road, the barber shaves all and only those villagers who do not shave themselves. Who shaves the barber? If he himself does, the he does not (since he shaves only those who do not shave themselves); if he does not, then he indeed does (since he shaves all those who do not shave themselves).

[Howard Treesong]

After reading this, I am no longer sure that I know precisely what a paradox is. I thought it was two plausible statements that appear to be consistent but are not. Am I close on the definition?

[Golden_Rhino]

Are we supposed to solve it like a riddle, or just discuss the paradox? I could be wrong, but isn't the whole point of a paradox that there isn't a solution?

[Bork]

Goat Paradox
On top of a mountain there lives a goat who is fatter than himself.

[milliondollaz]

Is responding to a Paradox "solving" it?

Are you going to post true paradoxes (like women, wave/particle) riddle paradoxes (like the town chief only tells the truth to people he shaves, you can ask him one question....) or previously baffling, but answered paradoxes (like the Twin Paradox about effects of time dilation in the theory of relativity)? Which one is the Barber Paradox? I guessed it was a riddle.

[econophile]

JaBlue,

I too am a little confused by the OP. Maybe you should describe what you had in mind a little more.

[Golden_Rhino]

This thread reminds me of the Star Trek episode where Kirk made the android's head blow up by telling him:

Everything I tell you from this point forward is a lie; I am lying.

[.Alex.]

The barber does not live in the village?

[Potvaliant]

Maybe I'm misreading it - kind of awkward sentence structure in the first sentence - but why does anyone have to shave the barber?

[gamblore99]

You have a whole book of paradoxes and THIS was your choice?

[FoxwoodsFiend]

[ QUOTE ]
This thread reminds me of the Star Trek episode where Kirk made the android's head blow up by telling him:

Everything I tell you from this point forward is a lie; I am lying.

[/ QUOTE ]

philosophers refer to this as "the liar's paradox." weird huh?

[GuyOnTilt]

I like the idea behind this thread, but I don't think it will work well with paradoxes like this. To solve these you need to be pretty well versed in set theory, and if you are you've already learned all about Russell's Paradox.

[b0000000000m]

[ QUOTE ]
I like the idea behind this thread, but I don't think it will work well with paradoxes like this. To solve these you need to be pretty well versed in set theory, and if you are you've already learned all about Russell's Paradox.

[/ QUOTE ]

Now, Russell's is a quality paradox. I remember musing on that one for a week straight in Intro to Metaphysics & Epistemology.

[milliondollaz]

[ QUOTE ]
Now, Russell's is a quality paradox

[/ QUOTE ]

out with it!

just guide the discussion better. it's really hard not to wikipedia this stuff...i'm holding out though

[GuyOnTilt]

[ QUOTE ]
[ QUOTE ]
I like the idea behind this thread, but I don't think it will work well with paradoxes like this. To solve these you need to be pretty well versed in set theory, and if you are you've already learned all about Russell's Paradox.

[/ QUOTE ]

Now, Russell's is a quality paradox. I remember musing on that one for a week straight in Intro to Metaphysics & Epistemology.

[/ QUOTE ]
OP's is pretty much an imperfectly worded version of Russell's Paradox. ZFCftw.

Note: I have no training in and know very little about set theory or ZFC, but like to read about such things anyway. /images/graemlins/smile.gif

[b0000000000m]

[ QUOTE ]
[ QUOTE ]
Now, Russell's is a quality paradox

[/ QUOTE ]

out with it!



[/ QUOTE ]

Is the "set of all sets that do not contain themselves" a member of itself, or not?

[GuyOnTilt]

[ QUOTE ]
[ QUOTE ]
Now, Russell's is a quality paradox

[/ QUOTE ]

out with it!

just guide the discussion better. it's really hard not to wikipedia this stuff...i'm holding out though

[/ QUOTE ]

Russell's Paradox is:

Does the set of "all sets that do not contain themselves as members" contain or not contain itself as a member?

The equation would be:

{x | x not in x}

The Barber's Paradox is an analogy of this.

[El Diablo]

Ja,

I'm drunk right now and not in the mood to focus, but yeah, this is a fantastic thread idea. Nice. If there are still open questions by tomorrow I'm on it.

[Zeno]

[ QUOTE ]
The equation would be:

{x | x not in x}


[/ QUOTE ]

I'm bad on symbolic nomenclature...but wouldn't the equation (symbolism) be:

{X | X not in X | }

-Zeno

[alphatmw]

i love paradoxes. here's one of my favorites.

i give you and your friend 2 envelopes, each with a mystery amount of money. one envelope, however, has twice the other one. before you open yours, i offer you to switch with your friend. should you?

if yours had the bigger amount, you end up with x/2. if he had the bigger one, you end up with 2x. so your EV is 5x/4 which is greater than x. therefore, you should switch.

now you have the other envelope. should you switch back? once again the answer is yes. is it logically sound that you are increasing your EV everytime you switch envelopes?

[b0000000000m]

[ QUOTE ]

so your EV is 5x/4 which is greater than x.

[/ QUOTE ]

/images/graemlins/confused.gif /images/graemlins/confused.gif /images/graemlins/confused.gif


I don't think this is a paradox so much as faulty reasoning.

If the amount in the smaller envelope = x/2, and the amount in the larger envlope = x, then:

If you have x/2 and swap, you win x/2
If you have x and swap, you lose x/2

Seems EV neutral.

[JaBlue]

The paradoxes in the book are famous and very tough. The barber paradox was comparatively stupid and actually given in the introduction of the book. According to the author, if one could rank paradoxes on a scale of 1-10 of deepness the barber would be 1 and the paradoxes in the book would be 6+. So I had the barber paradox in mind as a warm-up.

Sainsbury says he thinks of a paradox as "an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable. So generally either the conclusion is not really unacceptable or the starting point or reasoning has some obvious flaw."

The task in this thread will be to try to figure out if the conclusion of the paradox is not really unacceptable or there is something wrong with the premises or reasoning.

Nobody really went for the barber paradox but since it seems less interesting to the forum than others, I will answer it: the barber cannot exist. Because the conclusion is unacceptable, it is impossible for such a barber to exist. All details except those concerning the barber are superfluous and distract you from the situation at hand.


So on to attempt #2.

Someone hinted at the liar's paradox, which goes like this:

One version of the liar's paradox
Consider a man who says "what I am now saying is false." Is what he says true or false? The problem is that if he speaks truly, he is truly saying that what he says is false, so he is speaking falsely; but if he is speaking falsely, then, since this is just what he says he is doing, he must be speaking truly. So if what he says is false, it is true; and if it is true, it is false.

And please don't introduce other paradoxes in the middle of discussing one. I don't mind if you have one you'd like to talk about, but save it for when we're at a suitable stopping point for our current one.

[josh_x]

I geuss a conclusion might be that it is logically impossible for someone to always tell the truth or to always lie?

[daryn]

[ QUOTE ]
I geuss a conclusion might be that it is logically impossible for someone to always tell the truth or to always lie?

[/ QUOTE ]

no.

[xorbie]

JaBlue,

Given your "solution" to the first paradox, I'll just say that his statement is incomprehensible. Is the statement "All round squares are red" true or false? There's no truth value to the statement whatsoever.

[JaBlue]

Xorbie,

I don't see the answer as being analogous to the first, for it is perfectly possible for any person to say "what I am now saying is false," while it is not possible for a barber to have the property of shaving all and only persons who do not shave themselves for the very reason that is the conclusion of the first paradox, namely "who shaves the barber?"

Maybe your objection is really that the question "is what he says true or false?" is incomprehensible in this situation.? In that case I think you have to say what truth and falsity are and why they apply to some things and not others for you to be able to say what they do not apply to.

[xorbie]

JaBlue,

I'm not really sure "this statement is false" is a statement of any sort that holds meaning. There's just no content whatsoever.

[JTrout]

[ QUOTE ]
[ QUOTE ]
I geuss a conclusion might be that it is logically impossible for someone to always tell the truth or to always lie?

[/ QUOTE ]

no.

[/ QUOTE ]


How about,
It is impossible for someone to always tell the truth or to always lie, and make the statement "I am lying."


Concerning the barber paradox (which I realize has been answered), I like the answers:
1. the barber is a woman (barber-ette?)
2. the barber is 10 yrs. old.

[TheIrishThug]

Hold it Santa! Consider this: you are programmed to destroy the naughty, but many of those you destroy are in fact nice. I submit to you, that you are in fact naughty, and that, logically, you must destroy yourself.

http://img72.imageshack.us/img72/632/sf3acv031thws2.jpg

[maddog2030]

I agree with xorbie that the statement is meaningless. Much like a statement along the lines "X exists but doesn't exist".

[Borodog]

A straightforward geometric consequence of the invariance of the speed of light is the relativity of time, i.e. "time dilation." A clock at rest in a frame that moves relative to you moves more slowly than it would when at rest relative to you.

A straightforward implication of this is the relativity of length, i.e. "length contraction." An object at rest in a frame moving relative to you is shorter along the direction of motion than it would be if it were at rest relative to you.

A farmhand runs headlong towards a barn that is 10 feet deep while holding a ladder 18 feet long parallel to the ground. The barn doors are open, and the man runs fast enough that the ladder is length contracted to only 9 feet. After he and the entire ladder have run into the barn, a farmer closes the barn door behind him.

But in the man's frame, the barn is contracted to only 5 feet, and there is no way the ladder can fit.

How can this be?

[econophile]

OK. So I see how the logic implies that no barber exists. But when you use the phrase "the barber shaves," common understanding of language implies a barber does exist. So the set up of this paradox seems slightly misleading, mainly because people understand "the baber shaves" to imply "the barber exists."

[econophile]

I think the liars paradox may be base on a false assumption that all statements are true or false.

[tabako]

[ QUOTE ]
i love paradoxes. here's one of my favorites.

i give you and your friend 2 envelopes, each with a mystery amount of money. one envelope, however, has twice the other one. before you open yours, i offer you to switch with your friend. should you?

if yours had the bigger amount, you end up with x/2. if he had the bigger one, you end up with 2x. so your EV is 5x/4 which is greater than x. therefore, you should switch.

now you have the other envelope. should you switch back? once again the answer is yes. is it logically sound that you are increasing your EV everytime you switch envelopes?

[/ QUOTE ]

I have talked to my friends extensively about this, and I was able to convince them that it is not a paradox (which is most people's original reaction). Like another poster above me said, its faulty probabilistic analysis.

[milliondollaz]

in the running mans' frame, the barn is approaching him at a speed making the barn only 5 feet deep, so the ladder doesn' fit?

or something like, by the time you close the door behind the ladder, from the perspective of the barn, the front of the ladder is still in, so we think it fit in. but from the perspective of the dude w/ the ladder, it's out already. so it depends on which frame you are in as to what answer you give.

boro, explain after the liars paradox is over...

[danzasmack]

My favorite paradox of all time - Schrodinger's Cat (http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat)

I read an article about it YEARS back about quantum computing and was fascinated.

"A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

[felson]

[ QUOTE ]
I think the liars paradox may be base on a false assumption that all statements are true or false.

[/ QUOTE ]

Okay, so maybe some statements cannot be determined to be either true or false. Let's say that "this statement is false" is neither true nor false, but actually "indeterminate."

Now, is the following statement true, false, or indeterminate?

"This statement is either false or indeterminate."

[felson]

[ QUOTE ]
Is the statement "All round squares are red" true or false?

[/ QUOTE ]

It's true. The convention in logic is that "All X are Y" is equivalent to "if p is X, then p is Y." If there are no counterexamples to the claim, then the statement is true. In this case X is the empty set, so there are no counterexamples.

The next two statements are also true:
All round squares are blue.
All round squares are not red.

The next statement is false:
It is not the case that all round squares are red.

[felson]

[ QUOTE ]
JaBlue,

I'm not really sure "this statement is false" is a statement of any sort that holds meaning. There's just no content whatsoever.

[/ QUOTE ]

Does the statement "The next statement is true" have any meaning? I think you would have to agree that it does. So, how about this?

The next statement is true.
The previous statement is false.

One attempt to resolve the liar's paradox is to disallow self-referential phrases like "This statement is X." As you can see, this doesn't solve the problem.

[GuyOnTilt]

[ QUOTE ]

Nobody really went for the barber paradox but since it seems less interesting to the forum than others, I will answer it: the barber cannot exist. Because the conclusion is unacceptable, it is impossible for such a barber to exist. All details except those concerning the barber are superfluous and distract you from the situation at hand.

[/ QUOTE ]
That's not really the "solution" to this problem at all. While it is true that such a barber cannot exist, saying a true statement isn't the same as solving a problem. The real conclusion to be drawn this paradox you told is that the axioms the story was based on were incorrect, incomplete, or otherwise flawed. The problem is with the system itself, not the barber.

As has already been mentioned, the Barber Paradox is simply an attempt to put Russell's Paradox into english: {x : x not in x} , which can be put into english as "the set of all sets that do not contain themselves." The problem arises when you pose the question, "does that set contain itself?" If some of these phrases seem confusing, think of the term sets as lists. We can make lists of lots of things. Of anything we want to, in fact. For instance:

A = Board games currently in my closet
B = Positive odd integers
C = Positive odd integers divisible by 2

You'll notice that list B (or set B) is infinite. So it's easily possible to form an infinite set. You'll also notice that list C contains 0 members (or elements). So it's also possible to have an empty set. But what about sets that contain other sets? That's pretty easy too if we again think of lists. I can easily make a list that contain other lists:

1. Board games currently in my closet
2. Positive odd integers
3. Positive odd integers divisible by 2

If I call the above list, list D, then I now have a list of lists, or set of sets. You get the idea.

It's also possible for a set to contain itself. It's easier to first think of sets that do not contain themselves though, such as "the set of all automobiles." That would not contain itself as an element, since the only things including in that set are automobiles, and the set itself is...well, a set, and not an automobile. But others sets can contain themselves, such as "the set of all non-automobiles." Since a set is not an automobile, all sets would be elements of this set, including itself.

So back to the paradox: The set of all sets that do not contain themselves.

As we've shown, it's easy to come up with sets that do not contain themselves; there are lots of them. But what if we took all of those sets and labeled that a set. Is that okay to do? Sure it is. Again, that's just a set of sets, or list of lists, which are perfectly acceptable. We can notate this as {x : x not in x}. Let's call this set "D".

Now, as we've seen, some sets can be including as elements of themselves (such as non-automobiles). Is D such a set?

Since D contains all sets that do not contain themselves, then if D does contain itself, then it does not. And if it does not contain itself, then it does. And so on and so on (just as in the Barber Paradox).

So we've got ourselves a paradox. Great. But where did things break down? Did we make a mistake constructing set D? Are we not allowed to have such a set? Well, sure we are. We've already shown that. Sets most definitely do exist that do not contain themselves, and there's nothing wrong with grouping them all together and calling them "Set D".

So the problem isn't in the set we created, it's in the system of logic we're using, the axioms we've subscribed to. Russell realized that this was the problem, that they currently had a flawed system of formal logic, and went on to develop a new kind of set theory called Theory of Types that got around the problems that his paradox exposed in the current theory, which was naive set theory. Another form of set theory was developed that also stood up to this paradox, called Zermelo-Fraenkel Set Theory w/Choice, or ZFC.

So just as in Russell's paradox, the problem wasn't with the set we created and tested, but with the rules of logic we were using, the problem in the Barber Paradox isn't with the barber, but rather the story we were given. Saying that the solution is "the barber doesn't exist" doesn't address the problem.

The barber can exist; it's the rules that we were told he follows that cannot exist.

Note: I am not well-versed in this stuff and the little I know is only based on casual reading over the past couple years. That coupled with me trying to further water down my already watered down understanding of this stuff into plain english means there are probably some errors that people can point out. I don't know enough about ZFC to be able to show work on how it solves Russell's paradox, I only know that it does. Also, 2+2 should support LaTeX /images/graemlins/mad.gif

[Borodog]

[ QUOTE ]
in the running mans' frame, the barn is approaching him at a speed making the barn only 5 feet deep, so the ladder doesn' fit?

[/ QUOTE ]

That's the paradox. In the barn's frame, the farmer closes the barn door behind the ladder; the ladder is entirely contained within the barn. But in the ladder's frame, there is no possible way that the 18 foot ladder can be contained within the 5 foot barn and the door closed behind him.

How can this be?

[ QUOTE ]
or something like, by the time you close the door behind the ladder, from the perspective of the barn, the front of the ladder is still in, so we think it fit in. but from the perspective of the dude w/ the ladder, it's out already. so it depends on which frame you are in as to what answer you give.

[/ QUOTE ]

No quite sure what you mean here, but you definitely cannot have different events in different frames. For example I can't switch a light on in one frame but not switch in on in another. You either flip on the light or you don't.

[econophile]

is this thread going to lead up to goedel's incompleteness theorem?

[econophile]

Borodog,

That farmer must be moving very quickly!

[Borodog]

[ QUOTE ]
Borodog,

That farmer must be moving very quickly!

[/ QUOTE ]

He's the slow one. Why do think he got the other guy to carry the ladder?

[GuyOnTilt]

[ QUOTE ]

One version of the liar's paradox
Consider a man who says "what I am now saying is false." Is what he says true or false? The problem is that if he speaks truly, he is truly saying that what he says is false, so he is speaking falsely; but if he is speaking falsely, then, since this is just what he says he is doing, he must be speaking truly. So if what he says is false, it is true; and if it is true, it is false.

[/ QUOTE ]

JaBlue,

If I may, I'd suggest not spelling out the reason for the paradox to us. I realize you're just quoting straight from a book, but I think it'd be better if you left off the end part. To be honest, I think it'd be more fun if you left off the question that reveals paradox altogether. Like in the Barber Paradox, if you had simply told the story and then we had to figure out why it was paradoxical (i.e. come up with the question, "who shaves the barber?") before tackling the seeming paradox itself. I feel like that would be something more people would have a real chance at truly solving which would lead to this thread being more fun for more people.

[Magic_Man]

[ QUOTE ]
A straightforward geometric consequence of the invariance of the speed of light is the relativity of time, i.e. "time dilation." A clock at rest in a frame that moves relative to you moves more slowly than it would when at rest relative to you.

A straightforward implication of this is the relativity of length, i.e. "length contraction." An object at rest in a frame moving relative to you is shorter along the direction of motion than it would be if it were at rest relative to you.

A farmhand runs headlong towards a barn that is 10 feet deep while holding a ladder 18 feet long parallel to the ground. The barn doors are open, and the man runs fast enough that the ladder is length contracted to only 9 feet. After he and the entire ladder have run into the barn, a farmer closes the barn door behind him.

But in the man's frame, the barn is contracted to only 5 feet, and there is no way the ladder can fit.

How can this be?

[/ QUOTE ]

I'm pretty sure that in the pole's frame of reference, the front edge of the pole will crash into the wall of the barn. Then the pole will compress from its momentum, and the door shuts behind it, trapping it in a compressed state (unless it just busts through the walls), right?

[MatthewRyan]

What do u guys think about this:

For any consistent formal theory that proves basic arithmetical truths, an arithmetical statement that is true1 but not provable in the theory can be constructed. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete.

and also:

For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

? Ill share my thoughts later

[DrunkHamster]

What Russell's paradox showed is that naive set theory, i.e. where sets could be created simply based on a defining characteristic, was as the name suggests, too naive a view. What the ZF axioms do is formalize the notion of a set by inductively defining what sets are allowed to exist (and of course, there are restrictions on self referencing sets which cause all the trouble).

[Pokerlogist]

Yes the two sentence version is supposed to get around the slef-contradiction solution. IMHO the self-contradiction solution still holds since by paradox we mean an "idea" is impossible. Either the two sentences are meant to be linked together as an "idea "or they are not. If they are not, then no paradox. If they are meant to be linked then they are essentially the same as single statement that has been artifically split in two parts.

IMHO the solution to all these paradoxes is that "such a barber can't exist" ,ie, self-contradictory statements are not allowed.

[Ra_]

[ QUOTE ]
i love paradoxes. here's one of my favorites.

i give you and your friend 2 envelopes, each with a mystery amount of money. one envelope, however, has twice the other one. before you open yours, i offer you to switch with your friend. should you?

if yours had the bigger amount, you end up with x/2. if he had the bigger one, you end up with 2x. so your EV is 5x/4 which is greater than x. therefore, you should switch.

now you have the other envelope. should you switch back? once again the answer is yes. is it logically sound that you are increasing your EV everytime you switch envelopes?

[/ QUOTE ]

It would be +ev if for any value of either envelope there was an equall probablity that the other envelope was double as it was half. if this truely the case then the average value of either envelope would approach infinity, so really it doesn't matter.

[gumpzilla]

Here are a couple. The first one isn't exactly a paradox, but it's sort of in the spirit of this thread and 2+2 in general.

Suppose I offer you the opportunity to play a game, a wondrous game indeed. I will flip a fair coin repeatedly until it lands heads. If it lands heads on flip 1, you get $1. If it lands heads on flip 2, you get $2. If it lands heads on flip 3, you get $4, and so on, doubling at each step. Calculating the EV of this game, you find that there's a 1/2 chance you get $1, a 1/4 chance you get $2, a 1/8 chance you get $4, etc., so that the EV is 1/2 + 1/2 + 1/2 . . ., which doesn't converge. So you should be willing to pay ANY amount of money to play this game. If you would be interested in paying me $2 million for the opportunity to play this game, please PM me. (I forget what this one is called, but I think it might be the St. Petersburg Paradox.)

Another one which is kinda cute is Newcomb's paradox. (http://en.wikipedia.org/wiki/Newcomb%27s_paradox) . In this one, you are offered the opportunity to play a game. Your opponent purports to have the ability to see the future, and he has laid out a bizarre scenario. He puts in front of you two boxes, A and B. In box A, for sure, is $1000. In box B there are two possibilities. If the predictor has decided that you will take both A and B, then box B is empty. If the predictor has decided you will take just B, then box B contains $1 million. Some tests conducted before this game lead you to believe that his predictive ability is, at least, substantially better than chance alone would predict, and thus there is some merit to his claim. Once you start the game, the boxes aren't touched. So whether or not there's $1 million in box B, you're always going to do better by taking both, it would appear.

[gumpzilla]

[ QUOTE ]
What do u guys think about this:

For any consistent formal theory that proves basic arithmetical truths, an arithmetical statement that is true1 but not provable in the theory can be constructed. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete.

[/ QUOTE ]

Godel's theorem doesn't really seem like much of a paradox.

[Borodog]

[ QUOTE ]
[ QUOTE ]
A straightforward geometric consequence of the invariance of the speed of light is the relativity of time, i.e. "time dilation." A clock at rest in a frame that moves relative to you moves more slowly than it would when at rest relative to you.

A straightforward implication of this is the relativity of length, i.e. "length contraction." An object at rest in a frame moving relative to you is shorter along the direction of motion than it would be if it were at rest relative to you.

A farmhand runs headlong towards a barn that is 10 feet deep while holding a ladder 18 feet long parallel to the ground. The barn doors are open, and the man runs fast enough that the ladder is length contracted to only 9 feet. After he and the entire ladder have run into the barn, a farmer closes the barn door behind him.

But in the man's frame, the barn is contracted to only 5 feet, and there is no way the ladder can fit.

How can this be?

[/ QUOTE ]

I'm pretty sure that in the pole's frame of reference, the front edge of the pole will crash into the wall of the barn. Then the pole will compress from its momentum, and the door shuts behind it, trapping it in a compressed state (unless it just busts through the walls), right?

[/ QUOTE ]

Essentially yes; but for me this still leaves the crux of the paradox. How can the ladder be whole and in the barn and not crashed into the wall in one frame, but crashed into the barn while the door is still open in the other?

There's an important concept ("hidden" in my original post on it) that resolves the paradox.

[Boris]

Why is it paradoxes and not paradoxen?

[ahnuld]

[ QUOTE ]
Why is it paradoxes and not paradoxen?

[/ QUOTE ]

I knew paradoxes sounded wrong but couldnt think of the correction. Nice call


edit: nm, just checked dictionary.com and it is paradoxes.

[Yads]

[ QUOTE ]
A straightforward geometric consequence of the invariance of the speed of light is the relativity of time, i.e. "time dilation." A clock at rest in a frame that moves relative to you moves more slowly than it would when at rest relative to you.

A straightforward implication of this is the relativity of length, i.e. "length contraction." An object at rest in a frame moving relative to you is shorter along the direction of motion than it would be if it were at rest relative to you.

A farmhand runs headlong towards a barn that is 10 feet deep while holding a ladder 18 feet long parallel to the ground. The barn doors are open, and the man runs fast enough that the ladder is length contracted to only 9 feet. After he and the entire ladder have run into the barn, a farmer closes the barn door behind him.

But in the man's frame, the barn is contracted to only 5 feet, and there is no way the ladder can fit.

How can this be?

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Time is not absolute, from the farmer's perspective the man has run inside the barn, but from the man's perspective he has already crashed the ladder into the wall.