Your favorite paradoxes

[Seldon2639]

...there are none, right?

It's not so much a paradox as a logical fallacy. The first number not nameable in under ten words would itself be named as "the first number not nameable in under ten words" which is nine words. http://www.math.toronto.edu/mathnet/falseProofs/numbersDescribable.html

That site is wrong.
There is no fallacy, it's simply the case that every natural number can be described in 14 words or less. Yes, the phrase "The smallest natural number that cannot be unambiguously described in fourteen words or less." cannot be consistently applied to any natural number. That isn't a fallacy, that's the proof.

If it can't be consistently applied to any number, it can't be true of any number.

Is it possible you misread the site? It is, at the very least, a paradox, if not a fallacy in and of itself.

The only way the statement "every natural number can be described in 14 words or less" works (absent an actual proof) is by beginning with the assumption that it works. "If it can't be consistently applied to any number, it can't be true of any number" doesn't work in this context, because you're mixing the mathematical and English meanings (which is what all of these "weird math proofs" almost invariably do).

Your intention is that because "the smallest natural number that cannot be unambiguously described in fourteen words or less" cannot actually be said of any number (since it would create the paradox that the statement both applies as a description which is less than fourteen words, and thus cannot apply) would mean that there is no number which "cannot be unambigously described in fourteen words or less", but that's not a proper logical extension.

If I say "all numbers contain the letter "a" in their description", there must be a number for which that description doesn't apply, which would make it "the number which does not contain the letter 'a' in its description", which would mean that the number in question does have 'a' in its description, which means the aforementioned description cannot apply. Thus, by your logic, no letter contains the letter "a" in it, since I cannot apply the statement to all numbers.

You see the issue? The fact that I cannot show that the "smallest number that cannot be described in less than fourteen words" actually exists does not mean that there is no number which cannot be described in fourteen words or less. The negation of the full thought does not mean negation of all the component parts. Analogously: the statement "I am a human named Bill" does not become fully negated if my name is actually Andrew. The statement is on the whole false, but I'm still human.

 

[Maze1125]

The fact that I cannot show that the "smallest number that cannot be described in less than fourteen words" actually exists does not mean that there is no number which cannot be described in fourteen words or less.

Yes it does.

If you try and take the set of all natural numbers that cannot be described in 14 words or less, I can take the smallest one and describe it in 14 words or less, thereby proving it is not part of the set. So the actual set of all natural numbers that cannot be described in 14 words or less does not contain that number. So take the new set without that number and I can do the same with that set's smallest number, and I can do this every time. Therefore the actual set of natural numbers that cannot be described in 14 words or less must be empty.

If I say "all numbers contain the letter "a" in their description", there must be a number for which that description doesn't apply, which would make it "the number which does not contain the letter 'a' in its description", which would mean that the number in question does have 'a' in its description, which means the aforementioned description cannot apply. Thus, by your logic, no letter contains the letter "a" in it, since I cannot apply the statement to all numbers.

I don't understand what you're trying to prove there. For a start, it you assume "all numbers contain the letter "a" in their description" then it isn't true that there must be a number that the description doesn't apply.

 

[Good morning blues]

Here's a great one that reveals that even perfect logical thinking can sometimes fail to lead us to the truth.

Imagine that you're firing an arrow from a distance at a target. There will be a point at which the arrow is halfway between us and the target. There will be a point at which the arrow is halfway between that point and the target. Indeed, there will be an infinite number of such points. The arrow, however, cannot travel through an infinite number of points. As a result, we must logically conclude that it is impossible to shoot a target with an arrow.

 

[Maze1125]

Here's a great one that reveals that even perfect logical thinking can sometimes fail to lead us to the truth.

Imagine that you're firing an arrow from a distance at a target. There will be a point at which the arrow is halfway between us and the target. There will be a point at which the arrow is halfway between that point and the target. Indeed, there will be an infinite number of such points. The arrow, however, cannot travel through an infinite number of points. As a result, we must logically conclude that it is impossible to shoot a target with an arrow.

Of course perfect logical thinking can lead to false conclusion if you have your premises wrong.
And the unfounded premise in this case is the following "The arrow, however, cannot travel through an infinite number of points."

 

[Inverse Skies]

In deference to your friend, I'll assume he's never read 1984, and simply came to a very similar paradox to the phrase "freedom is slavery" (or was it "slavery is freedom") which occurred in the Orwell book. Though, yes, he's basically aping Big Brother's paradoxical phrases.

On the Hedberg question, it's not a paradox, simply a misstatement of the status of the relationship between the belt and the loops. If the belt loops are holding up the belt, it means that the pants are tight enough to be held up by friction alone, in which case the belt is not holding up the pants. If the belt is holding up the pants, it is actually the loops which are keeping the belt properly attached to the pants (either insofar as they keep the belt at the correct level to provide more normal force, and thus more friction, or through the loops actually being used to "hold up" the pants).

Only one of the statements between "my belt is holding up my pants" and "my belt loops are holding up my belt" can be true at one time. Analogously, it's not a "paradox" to say "it's nighttime outside, but the sun is up, I don't know what's going on". A contradiction in terms is not paradoxical.

I also hate Mitch Hedberg in general (his death notwithstanding), though, so I'm predisposed to have disdain for his "jokes"

I'm pretty sure he never has read 1984 (he would have mentioned it or encouraged me to read it if he had) but it wouldn't surprise me if he'd heard the phrase at some point in his life then re-parroted it without realising it. Still I like that line.

Lol, stop bringing friction into it! Physics was never my strong suite! Although I can very much understand what you're trying to say, I just thought it was quite a funny thing to write, that and another friend who absolutely loves him would never have forgiven me if I didn't mention him in some way.

 

[QuirkyTambourine]

Killing Ocelot in MGS3

I love the idea of how that would play out

 

[TheZapper]

I wish you wouldn't grant my wish.

That's all I can really think of right now.

 

[xplay3r]

Everything your mother says is true -dad
Everything your dad says is a lie -mom

I love paeadoxes.

 

[HappyPillz]

One bright day in the middle of the night,
Two dead boys got up to fight.
Back to back they faced eachother,
Drew their swords, and shot eachother.
One deaf cop, heard this noise,
And rushed right over to kill two dead boys.
Don't belive this lie, it's true,
Ask the blind man, he saw it too.

 

[Eclectic Dreck]

"the first number not nameable in under ten words"

...there are none, right?

There are plenty. For example, the number six-hundred and fourty-seven million, five-hundred and twenty-three thousand, one-hundred and one (647,523,101) contains eleven words (if you count hypenated words as a single word).

As a question that is not a paradox (and has a deciptively simple answer), what is the smallest number divisible by every number between one and one-hundred. One can find a number that meets the criteria simply by multiplying all of the required divisors. In this case, since the number is a perfect square (10^2 = 100) you find that you can divide this first number by the square root of 10 and achieve a second, smaller number.

 

[Maze1125]

"the first number not nameable in under ten words"

...there are none, right?

There are plenty. For example, the number six-hundred and fourty-seven million, five-hundred and twenty-three thousand, one-hundred and one (647,523,101) contains eleven words (if you count hypenated words as a single word).

The point of the paradox is that if you think you've found the first number not nameable in under ten words you can then call that number "the first number not nameable in under ten words". Which is a name that uses less than ten words.

The only resolution is to conclude that every number has some name that uses less than ten words.

As a question that is not a paradox (and has a deciptively simple answer), what is the smallest number divisible by every number between one and one-hundred. One can find a number that meets the criteria simply by multiplying all of the required divisors. In this case, since the number is a perfect square (10^2 = 100) you find that you can divide this first number by the square root of 10 and achieve a second, smaller number.

Yes, you can divide by 10 to get a smaller answer, but not because 100 is a perfect square. There are also a lot of other numbers than you can divide by to get a smaller answer too. It all comes down to prime factors.

 

[Thy-Art-Is-Awesome]

Mine are :
"New and Improved" Something has to be old to be improved.
and
"If you try to fail, and succeed, which have you done?"

 

[Biosophilogical]

I will not live long enough to die.
EDIT: Not a paradox but an oxymoron for certain; Military Intelligence. ;P

 

[J0k3]

Going to certain death; all deaths are certain

 

[malestrithe]

Everything I say is a lie. I am speaking the truth right now.

 

[Biosophilogical]

War is Peace.
Freedom is Slavery.

How is that in any way true. It's more of an oxymoron because it isn't true. I mean, war isn't peace, it is required to get peace (sort of like saying beating the batter IS cake), and freedom isn't slavery. If it somehow is please explain it to me as I think that went straight over my head.

 

[crimson5pheonix]

"I'm having a meaningful argument on the internet."

 

[Eclectic Dreck]

"the first number not nameable in under ten words"

...there are none, right?

There are plenty. For example, the number six-hundred and fourty-seven million, five-hundred and twenty-three thousand, one-hundred and one (647,523,101) contains eleven words (if you count hypenated words as a single word).

The point of the paradox is that if you think you've found the first number not nameable in under ten words you can then call that number "the first number not nameable in under ten words". Which is a name that uses less than ten words.

The only resolution is to conclude that every number has some name that uses less than ten words.

I seems to me that this problem might simply be poor communication rather than a paradox. Normally, we name numbers in a fashion that is unambiguous. Three-hundred and four is a name that correlates to precisely 304 - there is absolutely zero ambiguity. This is possible because all number names are based on a set of key words that are strung together acording to a set of conventions that have no regard for length. Rarely do you see people write out the words required to describe a large or complex number as the symbols generally do the job more succintly.

As such, we can assume that if there are infinite numbers, then there are an quantity of of these numbers that will meet an abritrary name length criteria. If you only regard numbers that make up the real integer set, there is in fact a single word that meets the critera of "the smallest number not namable in under ten words" and this phrase could unambiguously describe this value. There is a problem however with this plan. The moment that we have this value and describe it as "the smallest number not namable in under ten words" then we have the next value which is now the smallest. . . The result is the phrase has ambiguity as it would refer to an unlimited set of numbers.

If you include numbers from all number sets, you find that suddenly there is no smallest number in this set - the resulting set of numbers you get is infinite. Therefore, the phrase "the smallest number not namable in under ten words" cannot possibly be applied to any of them because it is now false.

Now, even if we limit ourselves to the real interger set where our rule originates, we find that the final assertion doesn't hold true. There are infinite numbers that meet the critera "greater than ten words". Unfortunately, even if all the languages in the world are combined you still have a finite set of options to apply to a finite set of spaces. Assuming you simply arbitrarily strung English words together to produce 10 word strings you still only have enough names for 9^1,000,000 numbers. While this is a very large number you'll find that any finite number is less than infinity no matter how large said finite number is.

One could attempt to keep the paradox alive by arguing that one could arbitraily make up words and thus choose from an infinite set and this would resolve the paradox. Unfortunately, in either scenario a new problem arises. One has created a new phrase that meets an arbitray condition of length but in the process has introduced amguity to the problem yet again.

 

[the_dancy_vagrant]

"I'm having a meaningful argument on the internet."

I lol'd.

 

[Maze1125]

As such, we can assume that if there are infinite numbers, then there are an quantity of of these numbers that will meet an abritrary name length criteria. If you only regard numbers that make up the real integer set, there is in fact a single word that meets the critera of "the smallest number not namable in under ten words" and this phrase could unambiguously describe this value. There is a problem however with this plan. The moment that we have this value and describe it as "the smallest number not namable in under ten words" then we have the next value which is now the smallest. . . The result is the phrase has ambiguity as it would refer to an unlimited set of numbers.

That's not true.
Once you find what you think is the smallest number not nameable in under ten words, you can then call it "the smallest number not nameable in under ten words" thereby proving that the number in question is not the smallest number not nameable in under ten words and therefore, even though we have proven that the number has a name that takes less then ten words "the smallest number not nameable in under ten words" is not that name.

So, there is always at most one candidate for the title of "the smallest number not nameable in under ten words" and so it is not ambiguous.

Also, I believe you mean the natural numbers, not the real integers, as this problem doesn't exist in the integers.