Cable Guy Paradox

IdleRich

IdleRich

IdleRich
"You might assume that you would want to change your bet at every point after 8am, but you cannot be certain, as there is a point infinitesimally soon after 8am that you could not be reasonably aware of being 8am+t, for obvious reasons."
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Well, practically you wouldn't be aware of it but that doesn't change the thought experiment does it?
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
Consider the set of all sets: clearly it is a set, and therefore a member of itself. If we like, we can consider a set of all sets that are members of themselves.
Consider the set of all things that are not sets: this is also a set, so it is not a member of itself.
Now consider the set of all sets that are not members of themselves: is this set a member of itself? If it is, then it is not; if it is not, then it is.

*Crocodile Dundee-esque flourish*

Now THAT'S a paradox.
 
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mixed_biscuits

mixed_biscuits

_________________________
Mr. Tea said:
Consider the set of all sets: clearly it is a set, and therefore a member of itself. If we like, we can consider a set of all sets that are members of themselves.
Consider the set of all things that are not sets: this is also a set, so it is not a member of itself.
Now consider the set of all sets that are not members of themselves: is this set a member of itself? If it is, then it is not; if it is not, then it is.

*Crocodile Dundee-esque flourish*

Now THAT'S a paradox.
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Makes brain hurt, thus *reported* ;)

PS - this is Russell's paradox: http://plato.stanford.edu/entries/russell-paradox/
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
mixed_biscuits said:
Hmm yes I was going to take that line of action ('meta-set,' 'list,' 'collection'? etc) before I gave up on myself.
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Probably for the best. You know that melancholic piano music they use on the soundtrack to documentaries about both mathematics and mental illness? It's the same for a reason.
 
E

Edward

+
I can't see anything paradoxical about this.
You haven't answered the question - what if he comes at 8am.
And time isn't necessarily infinitely divisble, particularly in terms of human thought. There's a minimum time it takes neurons to fire and the repair guy could arrive before they get the chance.



Anyway, in reality, the 8 hours would be up and he still wouldn't have turned up and then you'd have to ring up and have a go at the cable company etc.
 
IdleRich

IdleRich

IdleRich
"I can't see anything paradoxical about this.
You haven't answered the question - what if he comes at 8am."
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The set-up of the question says that he has to come after eight - if you allow him to come at eight then the paradox doesn't arise.

"Anyway, in reality, the 8 hours would be up and he still wouldn't have turned up and then you'd have to ring up and have a go at the cable company etc."
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Another of modern life's paradoxes.
 
E

Edward

+
Oh well if he HAS to come AFTER 8 then from the way you have specified the problem, the first period is just under 4 hours and the 2nd period is exactly 4 hours so the second period is longer and again there is no paradox.
 
IdleRich

IdleRich

IdleRich
No, the first period is from after eight until twelve, the second period is from after twelve up to but not including four - in the mathematical notation I read the problem in the intervals are written (8,12] and (12,4) - the first period is in fact longer as it contains one of its end points while the second contains neither (but I didn't mention this as it has no bearing on the problem).
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
OK, get this, suckers - this paradox occurred to me as I was falling asleep last night.

Consider an open-topped conical container, like a funnel only instead of a thin open-ended tube at the bottom, it just ends in a conical vertex. An ice-cream-cone shape, I guess. Now suppose I pour some water in it and leave it out in the sun; clearly, the water will evaporate. It seems reasonable to assume that the rate of evaporation, in unit volume per unit time, is proportional to the surface area of the remaining body of water. Because of the containers' conical shape, the water's surface area is proportional to the square of its depth. Thus the evaporation rate is proportional to the square of the remaining depth - so as the depth of remaining water approaches zero, so the evaporation rate approaches zero - more over, the evaporation rate approaches zero faster than the remaining depth. So the water never, ever fully evaporates. :cool:
 
IdleRich

IdleRich

IdleRich
"It seems reasonable to assume that the rate of evaporation, in unit volume per unit time, is proportional to the surface area of the remaining body of water."
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Is that right or is it like friction where although the common sense viewpoint suggests that it ought to be proportional to the surface areas that meet it is in fact completely independent of this?
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
IdleRich said:
Is that right or is it like friction where although the common sense viewpoint suggests that it ought to be proportional to the surface areas that meet it is in fact completely independent of this?
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To be honest I'm not sure. But clearly if you have some water lying around in a shallow dish and the same volume in a high-ball glass, the water in the dish is going to evaporate much quicker, right? If it's not directly proportional to area, I bet it's pretty closely correlated.

Edit:
Surface area: A substance which has a larger surface area will evaporate faster as there are more surface molecules which are able to escape.
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http://en.wikipedia.org/wiki/Evaporation#Factors_influencing_the_rate_of_evaporation
 
IdleRich

IdleRich

IdleRich
Yeah, looked and in a number of places it states that the size of the surface area does affect the rate of evaporation but nowhere does it state that it is proportional so I guess the paradox doesn't arise.
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
But why shouldn't it be proportional? The greater the area, the greater the opportunity for water molecules with sufficient kinetic energy to leave the surface of the water and escape.
And the paradox is still there if we make this assumption...
 
IdleRich

IdleRich

IdleRich
"But why shouldn't it be proportional? The greater the area, the greater the opportunity for water molecules with sufficient kinetic energy to leave the surface of the water and escape.
And the paradox is still there if we make this assumption..."
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Yes, intuitively it should be proportional, but I can't find any reference to this relationship and I'm sure that such a handy rule of thumb would be mentioned somewhere if it was close to being the case. So, I guess the answer to "why shouldn't it be proportional?" would be "because it isn't".
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
N

noel emits

a wonderful wooden reason
Mr. Tea said:
Thus the evaporation rate is proportional to the square of the remaining depth - so as the depth of remaining water approaches zero, so the evaporation rate approaches zero - more over, the evaporation rate approaches zero faster than the remaining depth. So the water never, ever fully evaporates. :cool:
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There's some truth to this, but it's not really a paradox.

First of all, and most importantly, water is always evaporating. The Sun's heat will make it evaporate quicker, and a body of water with a larger surface area will evaporate at a greater rate, but liquids evaporate anyway.

In addition to this, in the real world there would also likely be further transfer of heat from the Sun through the medium of the cone, and the air above the water, heating up the now smaller quantity of water which would thus require less energy to achieve a given temperature increase and therefore a given rate of evaporation.
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
noel emits said:
There's some truth to this, but it's not really a paradox.

First of all, and most importantly, water is always evaporating. The Sun's heat will make it evaporate quicker, and a body of water with a larger surface area will evaporate at a greater rate, but liquids evaporate anyway.

In addition to this, in the real world there would also likely be further transfer of heat from the Sun through the medium of the cone, and the air above the water, heating up the now smaller quantity of water which would thus require less energy to achieve a given temperature increase and therefore a given rate of evaporation.
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Yeah, yeah, obviously there are no paradoxes in 'real life', are there? You could also say that water isn't a continuous fluid, and that eventually you'd get down to individual molecules. It just struck me as something that seems to be paradox is you make some simple, but seemingly sensible, approximations - which, after all, is how science is usually done, and people wouldn't do it if it didn't (usually) work.
 
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