Thread: The Paradoxal Mathematic

Hello all Math-interested,

I am a high school student, 17 years of age. Interested in Maths and Nature Science.

Math projects at school have begun to bore me a lil' bit. I want to explore new concepts and ideas of Maths and lately, I crossed by this interesting paradox of set theory in my Discrete Math book:

A barber/hair stylisthas decided to shave everyone that doesn't shave himself in town.
Does this barber shave himself?



The above is also known as the Russell's paradox. Another one:



There is an infinite number of rooms at a hotel. At some point of time, guests keep coming 'til there is one guest without room.
Does this hotel really have an infinite number of rooms?




Mathematics represent Logics and structures. Isn't it a little mocking/funny/interesting/thrilling to find even the most logical subject involves paradoxes? I hope I can collect more paradoxes to write an article (nothin' academical of that sort)




Do share your knowledge about Mathematical paradoxes.

Originally Posted by T.T.T

Hello all Math-interested,

I am a high school student, 17 years of age. Interested in Maths and Nature Science.

Math projects at school have begun to bore me a lil' bit. I want to explore new concepts and ideas of Maths and lately, I crossed by this interesting paradox of set theory in my Discrete Math book:

A barber/hair stylisthas decided to shave everyone that doesn't shave himself in town.
Does this barber shave himself?



The above is also known as the Russell's paradox. Another one:



There is an infinite number of rooms at a hotel. At some point of time, guests keep coming 'til there is one guest without room.
Does this hotel really have an infinite number of rooms?




Mathematics represent Logics and structures. Isn't it a little mocking/funny/interesting/thrilling to find even the most logical subject involves paradoxes? I hope I can collect more paradoxes to write an article (nothin' academical of that sort)




Do share your knowledge about Mathematical paradoxes.

Russel's paradox signals the death of what is called naive set theory. The problem is that there are some things that are "too big" to be sets. The resolution (and there are still some issues) is axiomatic set theory.

The way you state it, in terms of the barber, captures the flavor, but is really more of a semantics problem, typical of self-referential sentences. But such self-referential sentences are at the heart of Russel's paradox and Goedel's incompleteness theorems.

You might want to "Google" the Banach-Tarski paradox, which is not a paradox at all, but rather a surprising theorem.

You might also want to look into the several Zeno paradoxes, which also are not paradoxes but rather examples of poor reasoning.

Originally Posted by DrRocket

The way you state it, in terms of the barber, captures the flavor, but is really more of a semantics problem, typical of self-referential sentences. But such self-referential sentences are at the heart of Russel's paradox and Goedel's incompleteness theorems.

You might want to "Google" the Banach-Tarski paradox, which is not a paradox at all, but rather a surprising theorem.

You might also want to look into the several Zeno paradoxes, which also are not paradoxes but rather examples of poor reasoning.

The way my text book states it, yes. I find this a both semantical and logical problem. Or should I say: illogical therefore a semantical problem? (I hope you don't find any Grammar mistakes)

Thanks for the tips. My appreciation. (No Thank button in this forum?!)

50 years ago, the Swedish math edu attempted to apply Set Theory into high school math. The result was immensely terrible, in the sense that the generation lost their will to study Maths. Too abstract. (Trivial to know?!)
I am also a bit curious about Maths edu in other countries :-D

Originally Posted by T.T.T

50 years ago, the Swedish math edu attempted to apply Set Theory into high school math. The result was immensely terrible, in the sense that the generation lost their will to study Maths. Too abstract. (Trivial to know?!)
I am also a bit curious about Maths edu in other countries :-D

A similar thing happened in the U.S. The real problem is that many high school mathematics teachers do not understand mathematics. Set theory is a necessary ingredient in mathematics, but what is needed can be covered very quickly, and then one should move on to more important things. You can see this done in any introductory book on real analysis. There is no good reason to flog that horse in a high school class.

If you are interested, Paul Halmos wrote a thin little book called Naive Set Theory. For anyone other than professional logician it is all the set theory you will ever need -- more in fact.

Set theory is not really all that abstract. But when it is presented as an end in itself by someone who does not understand mathematics it will appear to be irrelevant to nearly everything. That is a natural but false impression.

What a pity, isn't it? That we cannot apply Set Theory into Maths education. I had the luck to accustom myself with this bunch pretty early, grade 6 or so. Correct me if I am wrong but isn't Real Analysis a uni maths course?

Originally Posted by T.T.T

Correct me if I am wrong but isn't Real Analysis a uni maths course?

yep

Originally Posted by DrRocket

You might want to "Google" the Banach-Tarski paradox, which is not a paradox at all, but rather a surprising theorem.

You might also want to look into the several Zeno paradoxes, which also are not paradoxes but rather examples of poor reasoning.

Would you please explain the Banach-Tarski paradox for me? I understand that a ball if being divided into a finite number of pieces can be resulted into two balls. @@

my favorite is the unexpected hanging paradox

A judge tells a condemned prisoner that he will be hanged at noon on one day in the following year but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on dec 31, as if he hasn't been hanged on dec 30th, there is only one day left - and so it won't be a surprise if he's hanged on the 31st. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on the 31st.

He then reasons that the surprise hanging cannot be on the 30th either, because the 31st has already been eliminated and if he hasn't been hanged on the 29th, the hanging must occur on the 30th, making the hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on any day that year. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next year, the executioner knocks on the prisoner's door at noon on a randomly chosen day— which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.


BTW, what is the last day that the prisoner can be hanged on and it still be a surprise?

The last day that he can be hanged on, and still be surprised is 30th of December, as only if the date is 31st, he will be confident that the hanging will occur at the next day. However, as he has concluded that the hanging will not occur, even 31st will be a surprise.

he might be surprised on the 30th that the execution doesnt occur on the 30th but he most certainly wouldnt be surprised on the 31st since that is the only day left.

the 'last day' is simply 'unknowable'.

Ah, amusing. Thanks grandpa.

Without the execution, the prisoner is still trapped in his cell. To have an uneasy life or to die, which would be your choice since both are as miserable?

If the prisoner thinks he won't be executed then it will definitely be a surprise when they come for him! As I'm sure they would. Archilles overtakes the tortoise; don't hire Zeno as your running coach. If a tree falls, that tree falls and it doesn't matter whether someone hears it. Not sure about the barber - could be a real paradox there. Are we sure it's a man? The paradox only works if it is, so I suppose that would be given. But if it's possible to say that he shaves every man who doesn't shave himself, reality requires amendments to the 'shaves every man' description.

I'm not a mathematician but it looks to me that these paradoxes are based on problems of language more than of mathematics? Or is that just my lack of mathematical knowledge talking? The tree falling one maybe doesn't count as a mathematical conundrum - or is it that it doesn't work if you do count it as a mathematical conundrum?

Oh well, I'm just going to walk to the supermarket - according to Zeno the bus can't get there before me. Except we could get stuck perpetually in a paradox at the point where the bus runs over me.

I really like the one about the barber. It seems that there are two sets: those that shave themselves and those that don't. And then there's the barber and the question is asked as if he's a member of one of the sets when clearly he's not a member of either. Very clever!

so he neither shaves himself nor doesnt shave himself?

I'd say that which group he is a member of is unknowable.

@Ledger: u got it

A similar problem is:

A criminal says out loud at the court: I do lie.
Does he lie?

1. If he tells the truth => it is paradoxal to his statement that he lies.
2. If he lies => it is a truth that he does lie. He has told the truth and therefore he does not lie.

In a more casual situation, to say "I do lie" does not mean "I do always lie", as any socially competent person would understand, implicitly. This is just a purely logical problem.

The problem here is always the yes/no question, which offers us only two options. Neither of these options satisfy the given condition. This leads to a paradox, as seen.

@grandpa:

Back to your prisoner problem, first. Since this prisoner of ours must be surprised, which can be interpreted as the probability to be hanged that day equals 0.

The chance to be hang one day in the year is 1/365.
On 2nd Jan, the chance to be hanged will increase to: 1/(365-1) = 1/364
On 3rd Jan, the chance is 1/363
and so on.

This leads to: The 31st Dec that year is the most probable day that he will be hanged, if he hasn't been hanged the other days.

Because all day of that year, there is a probability that he is hanged, he will not get surprised at all. He will not be hanged at all 8)

Now, the barber problem! It has excited me immensely.

So this barber does not shave those who do not shave themselves.
1. If he shaves himself, he as a barber has shaved someone that shaves himself => He has violated the rule.
2. If he does not shave himself, he has to shave himself. He cannot, according to the rule.

Logics as a science can be seen in both Mathematics and Language. A friend told me that she has a professor, whose hobby is to examine the other professors. Their speech might sometimes include paradoxes or sayings that contradict themselves. :P

Originally Posted by granpa

so he neither shaves himself nor doesnt shave himself?

I'd say that which group he is a member of is unknowable.

More specifically, he isn't a member of either group.

deleted

I'm not awake yet

Originally Posted by Ken Fabos

If the prisoner thinks he won't be executed then it will definitely be a surprise when they come for him!

Oh well, I'm just going to walk to the supermarket - according to Zeno the bus can't get there before me. Except we could get stuck perpetually in a paradox at the point where the bus runs over me.

Very accurately stated! (I think I forgot to mention in my prior comment)

The Zeno's paradoxes is a proof of rationalism failure. In 17th-18th century, Copernicus, Galillei and Newton have proved Sciences need empirical proofs Our Solar System is not geocentric 8)

Rationalism + Empirism = Newton, Einstein and other geniuses.

I find it a bit funny that, according to wikipedia, The Zeno's paradoxes includes around 40 paradoxes (being found). All of them grounds on the same argument, so do all those 40 paradoxes are separate or should they be regarded as a product of an argument?!

According to the Zeno, there is no motion at all! So, don't worry being late for the bus

Originally Posted by granpa

the only day that he MUST be hanged on is the most improbable day that he will be hanged on?

I worry about your logic process.

hint:You are 'overthinking' it.

Its just 'unknowable'.

Ack, what have I written? -____- I mean the most probable day he is hanged is 31st Dec @@ if he is still alive by then.

I am supporting your "unknowable" conclusion by my prior knowledge about probability If there is a small fraction of chance that he is hanged everyday in that year => he will not be surprised => there is no day in that year he is hanged @@

Zeno's is not a paradox at all, just an improper application of the analysis.

You are trapped, blindfolded, in a hut, in the jungle. There are two exits to this hut, one leads to freedom, the other to an alligator pit. There is a guard at each door, one who always lies, and the other, who always tells the truth. You are allowed to ask ONE question to either guard, and you don't know which is which.
What question do you ask ???

I ask one of the gaurds: "If I ask the other guard if that exit (pointing) leads to safety, will he say yes?"

MigL - you ask "Which door would the other guard tell me leads to freedom?" Choose the opposite.
The capacity for language to express impossibilities allows for lots of possibilities. Absolutes are easy to state but don't necessarily describe a state correctly or completely.

Choose the opposite with my question as well. I.e., if the answer is no then it's the right one. And it doesn't matter which guard you're talking to, or, in my case, which door you're pointing at.

But, why does this question provide the answer. That's what's applicable to this forum.

Ledger, yes, essentially the same question to ask, but it's vital to interpret the answer correctly.

To the surprise execution question - surprise is a state of mind that arises when something unexpected occurs. If, logically, the condemned man, having survived past noon on the second last day concludes he can't be executed on the last day because the possibility of surprise is lost, he will indeed be surprised when they come for him on the last day.

Of course if he absolutely and pessimistically expects execution no matter what - him suspecting his executioners don't give a damn for the finer points of logic and simply picked a date at random from amongst all days up to and including the last - would they refrain?

the judge could just as well have said it will come on a day that he doesnt know.

if the prisoner isnt executed by the 31 then the judges orders simply cant be fully executed because the prisoner will know when the execution will occur.

the point is that there are days when the prisoner can be executed without breaking the judges orders
(they only have to choose a day at random)
and there is at least day that he cant be executed on without breaking the judges orders.
But the last day he can be executed on without breaking the judges orders is unknowable.
If you could know a certain day was the last day then the prisoner could know it to and therefore it would not be the last day.

Originally Posted by Ledger

I ask one of the gaurds: "If I ask the other guard if that exit (pointing) leads to safety, will he say yes?"

That will make 4 possibilities:

1. If you ask the liar:
Yes -> The other answers No -> No, that door does not lead to freedom.
No -> Yes -> Yes

2. If you ask the truth-teller:
yes -> No -> Yes
No -> Yes -> No

How will I know the door to freedom?!

@Ken Fabos:

Nice! Very clever!
1. The truth-teller has to point out which door the liar would advise me to take -> The wrong door.
2. The liar has to point out the wrong door anyway.
=> Choose the opposite door


@MigL: This is a funny brainteaser but I am afraid I am inadequate to see the paradox here.

Everyone has heard about the Physics/science troll, I assume 8)







What is wrong? Everyone knows pi does not equal 4, some of us might even know how pi has been found. But where in this reasoning does the logics break down?

Originally Posted by T.T.T

What is wrong? Everyone knows pi does not equal 4, some of us might even know how pi has been found. But where in this reasoning does the logics break down?

The fact that the piecewise linear curve approaches the boundary of the circle does not imply that the arc length of the piecewise linear curve approaches the arc length of the boundary of the circle.

Sorry Granpa, just having a bit of fun; I actually do get the gist of the argument and realise that imprecision in how it was phrased allowed that flippant interpretation. Except that how such a problem is phrased is usually the crucial element.

TTT, um, eventually an infinity of (almost) triangles with the hypotenuse representing the perimeter of the circle? A long time since I did trigonometry, most of it lost, and even less retention of calculus but I'm thinking at infinity it's those hypotenuses that end up as straight lines rather than the perimeter of the circle equaling the sum of the infinity of the other two sides of the triangles? Interesting to hear if that's on the right track.

Sorry, didn't mean to imply it was a paradox, Just asimple brain teaser using true/false logic. I can't remember, is it called Boolean logic ?

Originally Posted by MigL

Sorry, didn't mean to imply it was a paradox, Just asimple brain teaser using true/false logic. I can't remember, is it called Boolean logic ?

Really? I didn't know and I will certainly look it up Thanks for sharing! I appreciate all kind of funny logic problems

@DrRocket: There is a similar proof for pi=2 using semicircles. See the link:

http://plus.maths.org/content/puzzle-page-65

Digging my head into these problems, I realize how much I need to get back to 1st grade's maths, which appears to stand in the new light for me Formulas are tools, learning formulas is to learn how to use the tools. Trying to repeat the thinking process while developing/constructing formulas is to learn to think.

Two cars collide at a random point along a meter stick. Of all the points along that meter stick, what where the odds that the cars would collide at that particular point?

Originally Posted by Scifor Refugee

Two cars collide at a random point along a meter stick. Of all the points along that meter stick, what where the odds that the cars would collide at that particular point?

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