Paradoxes!

[sprinks]

Does anyone else here love paradoxes too? I like how they really make you think. Right now I'm reading a book FULL of them :D and I wondered if anyone else had a paradox for people to think about or something. (I'm sure there's not already a thread exactly like this)

I prefer the ones that don't have answers and just go in loops, but here is one where the answer is really quite simple, but for some reason it took me a while to think of it:

If a hotel with an infinite amount of rooms is fully occupied, can it still accommodate a new guest?

Hopefully you will all realise the answer faster than I did because it really is quite simple! Have fun thinking and I'll probably post some more paradoxes soon, but feel free to post your own! :)

[Sir Bartholomew]

yes, because it has an infinite amount of rooms.

[sprinks]

yes, because it has an infinite amount of rooms.

That is the response of most people, but keep in mind EVERY room is ALREADY full in this paradox, even though there is an infinite amount... So basically you're half right but there is more to it.

[white camellia]

It can when one guest checks out.

[sprinks]

It can when one guest checks out.

Very clever :lol: but in this paradox no one ever checks out... For some reason they just constantly live in the room. No idea why you would do that though... Anyhow so maybe to make it clearer it is:

If a hotel with an infinite amount of rooms is fully occupied, can it still accommodate a new guest if no one ever checks out?

[Sir Bartholomew]

what do you mean by fully occupied rooms?

[sprinks]

Each room has someone in it, none are vacant

Basically there is a hotel, it has an infinite amount of rooms, but every room is already taken by someone. How can this hotel take more guests if each room is already full?

P.S. *hint* Note that I said no one moves OUT, but nothing about changing rooms! So perhaps you were closer than I implied, white camellia! :)

[Sir Bartholomew]

i give up :p

[sprinks]

Should I tell you the answer or should we see if someone else can figure it out? :idea:

[Sir Bartholomew]

let them answer

[white camellia]

let them answer
:nod: ;)

[Lioness_Heart]

If there are an infinite number of rooms, could an existing person move to another room, thus creating a new room and a space for a new person?

And what is this book called? Paradoxes are fun: they're why I want to study physics!

[Seabird111]

Very clever :lol: but in this paradox no one ever checks out... For some reason they just constantly live in the room. No idea why you would do that though... Anyhow so maybe to make it clearer it is:

If a hotel with an infinite amount of rooms is fully occupied, can it still accommodate a new guest if no one ever checks out?

:idea:

Well, when you think about it, if every room is fully booked in a hotel with an infinite amount of rooms, then another guest could be accomodated if they go into an existing room.

[Dori]

I like Zeno's Arrow.


"The flight of an arrow, said Zeno, is an apparent example of motion. But at any given moment of its flight, the arrow is either where it is or where it is not. If it moves where it is, it must be standing still, and if it moves where it is not, then it can't be there; thus it can't move." (An Incomplete Education, William Wilson and Judy Jones)

[Seabird111]

I like Zeno's Arrow.


"The flight of an arrow, said Zeno, is an apparent example of motion. But at any given moment of its flight, the arrow is either where it is or where it is not. If it moves where it is, it must be standing still, and if it moves where it is not, then it can't be there; thus it can't move." (An Incomplete Education, William Wilson and Judy Jones)

Good one!

[Seabird111]

:idea:

Well, when you think about it, if every room is fully booked in a hotel with an infinite amount of rooms, then another guest could be accomodated if they go into an existing room.

Did I get it right?

[evt1]

I believe that an infinite number of additional guests can be accomodated? Isn't that the whole idea of infinity--either uncountably or countably infinite?? I guess Ibetter re-read Russell and Whitehead. I believe they got quite involved with that whole issue.

It's late, and I'm not very reliable even in the best of times, but I do enjoy paradoxes, riddles, logic etc. so keep on posting!! Thanks!

P.S. This is my very first post in any site--except, of course when I send nasty emails to microsoft.

[Lily Adams]

I adore paradoxes. The world is one big paradox. I've said that earlier.

Like, being different is really great, but it's also what causes wars in this world.

I think about that one a lot.

[sprinks]

If there are an infinite number of rooms, could an existing person move to another room, thus creating a new room and a space for a new person?

And what is this book called? Paradoxes are fun: they're why I want to study physics!

Yes you got it right! :D Each person moves up a room, creating a vacancy for another person!

The book I'm reading is Paradoxes from A to Z by Michael Clark... I randomly found it when I was at the library and I'm hoping they have more! :D



Well, when you think about it, if every room is fully booked in a hotel with an infinite amount of rooms, then another guest could be accomodated if they go into an existing room.
If you meant what Lioness_Heart said about guests moving rooms to empty another than that is exactly right, otherwise I think the idea is that the new guests go into a vacant room, they don't move in with other guests already there.


I believe that an infinite number of additional guests can be accomodated? Isn't that the whole idea of infinity--either uncountably or countably infinite?? I guess Ibetter re-read Russell and Whitehead. I believe they got quite involved with that whole issue.

It's late, and I'm not very reliable even in the best of times, but I do enjoy paradoxes, riddles, logic etc. so keep on posting!! Thanks!

P.S. This is my very first post in any site--except, of course when I send nasty emails to microsoft.
You are right, the only thing is that it's so easy to say "well yes because it's an infinite amount", but there's an explaination for why. I hope you keep posting, especially in this thread! :D

[sprinks]

I like Zeno's Arrow.


"The flight of an arrow, said Zeno, is an apparent example of motion. But at any given moment of its flight, the arrow is either where it is or where it is not. If it moves where it is, it must be standing still, and if it moves where it is not, then it can't be there; thus it can't move." (An Incomplete Education, William Wilson and Judy Jones)

I like this one too, although its worded slightly different in the book I'm reading...

An arrow cannot move in the place in which it is not. Nor can it move in the place in which it is. But a flying arrow is always at the place at which it is. Therefore, it is always at rest.

I think the version you stated is clearer though. :)

Shall we all discuss Zeno's arrow? :D Or does anyone else have other paradoxes they'd like to put out out there for people to think about?

[sprinks]

I adore paradoxes. The world is one big paradox. I've said that earlier.

Like, being different is really great, but it's also what causes wars in this world.

I think about that one a lot.

Sounds like you'll be a valuable member of discussion in our paradox thread then! :D (of course everyone who posts here is a valuable member of discussion :))

I agree, the world really is one big paradox!

[amanda_isabel]

about the hotel room thing.. is it possible that there still is someone with no room, seeing as there are an infinite number of rooms?

am i way off base here? lol...

[johann cruyff]

Well,other than the Zeno's arrow paradox,there are two more that are quite famous,the dichotomy paradox and the Achilles and the tortoise paradox.Actually,these are called aporia in philosophy,and there are many more.

Also,very interesting are the sophisms.They're means of intentionally deceiving people,whilst being logical at a first glance.

[sprinks]

Well,other than the Zeno's arrow paradox,there are two more that are quite famous,the dichotomy paradox and the Achilles and the tortoise paradox.

Achilles and the tortoise - thats the one where Achilles gives the tortoise a head start so how can he catch up to the tortoise? (of course the real thing makes more sense then that :lol:)


Also,very interesting are the sophisms.They're means of intentionally deceiving people,whilst being logical at a first glance.

Would you give me an example of a sophism? please? :D They sound interesting...

[johann cruyff]

Achilles and the tortoise - thats the one where Achilles gives the tortoise a head start so how can he catch up to the tortoise? (of course the real thing makes more sense then that :lol:)

Yes,something like that:).

The dichotomy paradox is basically this: to get from point A to point B,you'd first have to cross a half of the distance.But to cross that,you first have to cross a half of that half,or one quarter of the way,and so on ad infinitum.


Would you give me an example of a sophism? please? :D They sound interesting...

We can both agree that a drop of petrol would cost you nothing,right?And since a full tank contains nothing but a big number of drops,by that logic,a full tank of petrol shouldn't cost you anything either.

Obviously,this isn't a real sophism(dating from Ancient Greece,that is),but it should give you an idea of how they work.If you're interested in logical fallacies such as this,try searching for the problem of Protagoras and Euathlus,or the number of hairs you'd have to pluck to make a man bald,they're nice examples of sophisms.

[Remarkable]

A sophism would be this:A man says:"I support poligamy because it has to do with some insuppresible insticts and it's neccesary for the well being of some persons".An other person replies:"Well then,murder is justifiable too,since it has to do with insticts and the spiritual or mental well being of the murderer".I've seen people use this in both sides...

[kitten]

there wouldn't be anyone else to check in.

[sprinks]

We can both agree that a drop of petrol would cost you nothing,right?And since a full tank contains nothing but a big number of drops,by that logic,a full tank of petrol shouldn't cost you anything either.

Obviously,this isn't a real sophism(dating from Ancient Greece,that is),but it should give you an idea of how they work.If you're interested in logical fallacies such as this,try searching for the problem of Protagoras and Euathlus,or the number of hairs you'd have to pluck to make a man bald,they're nice examples of sophisms.

Thanks :D That makes me think of The Heap paradox... 10 000 grains suitably arranged make a heap. But, at no point can you convert a collection of grains that is a heap into one that is not, simply by removing a single grain. So it follows that a single grain makes a heap. For if we keep removing grains over and over again, say 9 999 times, at no point does it cease to be a heap. Yet we obviously know that a single grain is not a heap.

[Lioness_Heart]

Yes you got it right! :D Each person moves up a room, creating a vacancy for another person!


*feels smug*

I'm not sure that I quite get the Zeno's arrow one though... perhaps I need to read it again...

But that Heap paradox and the oil one are so cool!! Although the oil one is based on the premise that a drop costs nothing, whereas it surely does?!? Or am i missing the point entirely?

I love the way that paradoxes make your (I'm assuming it's the same for everyone) head feel.. kind of confused and satisfied at the same time.

[Dori]

*feels smug*

I'm not sure that I quite get the Zeno's arrow one though... perhaps I need to read it again...

But that Heap paradox and the oil one are so cool!! Although the oil one is based on the premise that a drop costs nothing, whereas it surely does?!? Or am i missing the point entirely?

I love the way that paradoxes make your (I'm assuming it's the same for everyone) head feel.. kind of confused and satisfied at the same time.

Yeah, for now on I'm calling every grain a heap. :p Some people might argue that every drop counts while others think it is worthless. My dad would argue the former. :rolleyes:

I thought the Zeno's arrow one was easy to understand. Maybe I could explain it better...okay if you take an arrow at any point in its flight it is either where it is or where it isn't, right? If it is where it is, it is stationary, and if it's where it isn't it doesn't exist. Obviously the arrow cannot be where it isn't because it does indeed exist, so the the arrow is where it is and thus is stationary. I think I got it right. :)

[Lily Adams]

Sounds like you'll be a valuable member of discussion in our paradox thread then! :D (of course everyone who posts here is a valuable member of discussion :))

I agree, the world really is one big paradox!

Thank you.

[sprinks]

*feels smug*

I'm not sure that I quite get the Zeno's arrow one though... perhaps I need to read it again...

But that Heap paradox and the oil one are so cool!! Although the oil one is based on the premise that a drop costs nothing, whereas it surely does?!? Or am i missing the point entirely?

I love the way that paradoxes make your (I'm assuming it's the same for everyone) head feel.. kind of confused and satisfied at the same time.

I had that same thought on the oil one, surely even though it would cost an incredibly small amout it would still cost something? Oh well :lol:

I'm the same with how paradoxes make your head feel! :D That's why I wanted to start a thread where we can all discuss them :)

[sprinks]

I found one partially related to literature :D (Well it talks about films but it could apply to novels :p)

The Paradox of Fiction:

We can be afraid of something that does not in fact exist, but it seems we must at least believe it exists. Again, we cannot hate or love anything unless we believe it exists. But we also have emotional responses to fiction: we can be afraid of a fictional criminal in a film or feel vengeful when fictional injustice is perpetrated, and in these cases we know the objects of our emotions do not exist.


or simply,

How can we have emotional responses to fiction, when we know that the objects of our emotions do not exist?

[johann cruyff]

I had that same thought on the oil one, surely even though it would cost an incredibly small amout it would still cost something? Oh well :lol:


Well that's why I said it wasn't really a sophism at all,but just something to give you the general idea of how they're meant to deceive.Again,I recommend you search for The Court Paradox(the one with Protagoras and Euathlus),now that's a mind bender.

[Roivas]

Move each guest to the next room, so that Room #1 is empty and can accommodate a guest.

[Lioness_Heart]

I thought the Zeno's arrow one was easy to understand. Maybe I could explain it better...okay if you take an arrow at any point in its flight it is either where it is or where it isn't, right? If it is where it is, it is stationary, and if it's where it isn't it doesn't exist. Obviously the arrow cannot be where it isn't because it does indeed exist, so the the arrow is where it is and thus is stationary. I think I got it right. :)

Aaah, I see. Thank-you!! :D






How can we have emotional responses to fiction, when we know that the objects of our emotions do not exist?

Well, I suppose even though we know that it's not real, for a time (whilst reading) our disbelief is suspended... for a time, they do exist, if only in our minds...

[byquist]

A major paradox is the query: "Is life a paradox?" And to what percentage, 10% or 95%? And is that good and beneficial, or detrimental to even consider?

[sprinks]

This one confuses me:

Tibbles the cat is sitting on a mat. Tibbles minus any one of 1000 hairs is also a cat. If any one of those hairs had been missing we should still have had a cat there on the mat. So it seems we have 1001 different cats on the mat, for Tibbles has precisely the hairs he does at the time, and so cannot be identical with any of the cats with only 999 hairs.

I don't entirely get it.

[sprinks]

Heres some more paradoxes! :)

Self-deception:

I can be a victim of another's deception but I can also be a victim of self-deception. You cannot succeed in decieving me if I know what you are up to. So how can I decieve myself? Won't I know what I am up to, and won't thing necessarily undermine the self-deception?

The Spaceship:

A spaceship travels in a straight line. It doubles its speed after half a minute, doubles it again after another quater of a minute, and continues successively to double it after half of the last interval. Where is it at one minute? It is neither infinitely far away, nor at any finite distance either.
(Reminds me a little of Zeno's arrow :p)

The Toxin Paradox:

You are offered a million dollars to form the intention of drinking a vile potion which, though not lethal, will make you unpleasantly ill. Once you have formed the intention the money is handed over, and you are free to change your mind. The trouble is that you know this, and it will prevent you from forming the intention, since you cannot intend to do what you know you will not do.

Any thoughts one this one?

[PeterL]

Heres some more paradoxes! :)

Self-deception:

I can be a victim of another's deception but I can also be a victim of self-deception. You cannot succeed in deceiving me if I know what you are up to. So how can I deceive myself? Won't I know what I am up to, and won't thing necessarily undermine the self-deception?

Self-deception is simply the unwillingness of an individual to see himself or herself objectively. Since others see us through the lenses of their prejudices, the situation isn't much different between self-deception and being deceived by another.



The Spaceship:

A spaceship travels in a straight line. It doubles its speed after half a minute, doubles it again after another quater of a minute, and continues successively to double it after half of the last interval. Where is it at one minute? It is neither infinitely far away, nor at any finite distance either.
(Reminds me a little of Zeno's arrow :p)
[B]

This is not a paradox. It is simply an incompletely stated problem. If the initial velocity or the velocity at any time interval were known, then it would be easy to determine the spaceship's location and velocity.



The Toxin Paradox:

You are offered a million dollars to form the intention of drinking a vile potion which, though not lethal, will make you unpleasantly ill. Once you have formed the intention the money is handed over, and you are free to change your mind. The trouble is that you know this, and it will prevent you from forming the intention, since you cannot intend to do what you know you will not do.
Any thoughts on this one?

This is an example of where self-deception comes in handy. Tell yourself that the money will be paid when one actually drinks the stuff, and in addition to the money one will receive a few other nice things. That way, you would believe that the payment would come with the action.

[PeterL]

How can we have emotional responses to fiction, when we know that the objects of our emotions do not exist?

Emotions are not part of the rational mind; they are physical reactions to inputs, both sensory and imaginative. There might be an emotional reaction to reading about an airplane crash, but the reaction is independdent of whether the crash happen or was an product of fiction.

[Pensive]

The concept of American Dream confuses me.

[Taliesin]

Heres some more paradoxes! :)


The Spaceship:

A spaceship travels in a straight line. It doubles its speed after half a minute, doubles it again after another quarter of a minute, and continues successively to double it after half of the last interval. Where is it at one minute? It is neither infinitely far away, nor at any finite distance either.
(Reminds me a little of Zeno's arrow :p)



Any thoughts one this one?

Well, concerning the general relativity, it can't be more farther than light can cover in a minute, which is a finite distance.:p
Let it cover a distance of p in the first half-minute. Then, in the coming quarter, there is half the time and double the speed - and again it will cover a distance of p. It is easy to prove by induction that it will cover a distance of p in all those decreasing parts of time. Since there is an infinity of those parts, the distance covered is infinity times p, hence infinity. I fail to see the paradox.



[B]Self-deception:

I can be a victim of another's deception but I can also be a victim of self-deception. You cannot succeed in decieving me if I know what you are up to. So how can I decieve myself? Won't I know what I am up to, and won't thing necessarily undermine the self-deception?

Humans generally aren't aware of everything that happens in their brains/minds. There are many layers. Human mind is not one whole totally self-conscious thing.

[PeterL]

The concept of American Dream confuses me.

Why? Is the idea of hundreds of millions of people having the same dream strange? Actually, it is just a particularization of dreams that all humans have.

[Lioness_Heart]

What do people think of paradoxical statements?

For example:

I am currently learning a Mozart sonata, and my piano teacher was not impressed by my attempts at the opening. She told me to view the detached crotchets as 'exploding pearls'. This struch me as a rather paradoxical (well, it's probably more of an oxymoron) statement, yet it made perfect sense...

Why do you think it is that paradoxes sometimes elucidate things more clearly than something straightforward?