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Anyone have any really good paradoxes (other than how many fps do we see life at, and others on these boards)? Especially real-life stuff. No math. Even though I'm the fourth-ranked math brain in my state's government-issued math competition, I'm not at calculus yet :) Just learning high-end geometry. So, I would like some classical long discussions on theoretical paradoxes.

I don't really have any good ones, other than:

The statement below is true.
The statement above is fake.

Assuming we can use graphics to explain this corelation, each time we say that one is true, we go across one level. Each time one is false, we must go diagonally.

#1.......#2
1--------2
.\........./.
..\......./..
...\...../...
....\.../....
.....\./.....
......X......
...../.\.....
..../...\....
.../.....\...
../.......\..
./.........\.
3--------4

We will never end. So the statements are a true paradox.

Anyone have any good discussion-worthy topics?

"This statement is false." is better yet.

"this thread will be post helled" is true.

oh come on, ogre. >:(

And yes, "this stateent is false" is a classical one. But because I have elaborated on the last one, this one needs no ASCII ;)

SmellyOgre said:

"this thread will be post helled" is true.

And you're about to get yourself losered.

Heh, sorry....

My math skills suck BTW.

"The more terrible the prospect of thermonuclear war becomes, the less likely it is to happen."

"In order to cross the room, you must first reach the half-way point between the origin point and the destination point. In order to reach the half-way point, you must first reach the mid-way point between the origin point and the half-way point. In order to reach the mid-way point you must first reach the point between the mid-way point and the origin point. In fact there are an infinite number of half-way points you must reach before you can cross the room... but of course you cannot cross infinity so you will never cross the room."

SmellyOgre said:

"this thread will be post helled" is true.

Ogre takes the Post Hell challenge.

At any given moment, an arrow in flight occupies and equal amount of space as its volume. But any object that takes up an amount of space as its volume is at rest.

Mercury, at point A, sees a tortoise walking along at point B and runs to catch up to it. But by the time Mercury reaches point B, the tortoise has reached point C. When Mercury reaches point C, the tortoise has moved to point D. And so on.

However, all of these paradoxes ae defeated by enforcing a sense of time rather than eternally subdividing distance and time into smaller and smaller fractions.

Most instances of what people tend to call paradoxes are just due to the imprecision of most languages. If the same issues are re-expressed in terms of formal logic, they cease to be paradoxes, or just become contradictions.

Two of those "paradoxes" most often discussed in books on popular logic/mathematics/philosophy are the paradox of the hanged man and the paradox of the unexpected surprise. I won't bother restating them, but a web search will probably find them in some form.

deldelda said:

Anyone have any really good paradoxes (other than how many fps do we see life at, and others on these boards)?

That's as much an example of a paradox as having ten thousand spoons when all you need is a knife is an example of irony.

I can do anything I wanted to. So can I make a sandwich I cant eat all of? Would I truly be able to do anything I wanted to?

Going nuts when you're already insane, getting a free ride when you've already paid, and having your first air flight turn into a crashing plane, a little ironic, don't you think?

The statement below is floss,
The statement above is too.

We could go into the .999~ = 1 discussion (Which isn't a paradox) but I don't think there's anyone here stupid enough to argue with that (unlike the gamefaqs forums, ugh.)

Epyo said:

We could go into the .999~ = 1 discussion (Which isn't a paradox) but I don't think there's anyone here stupid enough to argue with that (unlike the gamefaqs forums, ugh.)

Eh? There are people who claim that the limit of 9 * (10^-1 + 10^-2 + ... + 10^-n) as n tends to infinity, is not 1?

Russell's naive set theory paradox owns me.

Epyo said:

We could go into the .999~ = 1 discussion (Which isn't a paradox) but I don't think there's anyone here stupid enough to argue with that (unlike the gamefaqs forums, ugh.)

The easiest way to prove them wrong (they won't understand or will refuse to accept the concept of a limit) is:

1 / 3 = 0.333...
0.333... * 3 = 0.999...
(1 / 3) * 3 = 1
1 = 0.999...

An example of a paradox in nature is the whale, which is the biggest creature on the planet, eats only krill, which are microscopic.

And just how is that a paradox? Let's go over the definition of the word (emphasis added).

paradox: n. 1. a tenet contrary to received opinion 2. a : a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true b : a self-contradictory statement that at first seems true c : an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises

I believe you were looking for irony.

DooMBoy:
An example of a paradox in nature is the whale, which is the biggest creature on the planet, eats only krill, which are microscopic.

Bloodshedder:
And just how is that a paradox? paradox:... 2. a : a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true...

I believe DooMBoy is right in this matter. Looking at the size of a whale one would at first assume it's food must be big aswell (common sense). Scientific reseach however shows whales eat very small organisms (contradictory).

I believe DooMBoy is right in this matter.

*time and space unravels, universe collapses, all matter blips out of existance*

Fredrik said:

1 / 3 = 0.333...
0.333... * 3 = 0.999...
(1 / 3) * 3 = 1
1 = 0.999...

We had this discussion at school years ago and I still disagree with it. 0.3333... can only approach the value of 1/3. It will never be exactly the same.

My reasoning behind this is this:

0.9 != 1
0.99 != 1
0.999 != 1
0.9999 != 1
...

I could make this list infinitly long and every statement would still hold true.

OK. Work with me here. "I love you so much I hate you...yet I hate you so much I love you." Would this be a paradox. Is it too emotional? I ask the latter because isn't it true that emotions aren't true reality? Don't paradoxs have to be reality? Give me some thoughts. (I know I sound like an idiot half the time...if not all.)

Arno said:

The concept of limits is needed. The way it is argued mathematically is (a bit simplified) that you can make it as close as you want to 1 by choosing a suitable number of decimal places (the n in my previous post).

Infinity is not a simple concept, and erroneous conclusions are normally reached when it is casually used as if it were an ordinary (if very large) number.

Arno said:

We had this discussion at school years ago and I still disagree with it. 0.3333... can only approach the value of 1/3. It will never be exactly the same.

It is exactly the same. Thats the point. With a finite number of 3s, you approach, but never reach 1/3.

The more 3s you add, the closer you get towards 1/3. It follows that with an infinite number of 3s you will reach 1/3. Its important to realise that infinity is not a number.

Grazza said:

The concept of limits is needed. The way it is argued mathematically is (a bit simplified) that you can make it as close as you want to 1 by choosing a suitable number of decimal places (the n in my previous post).

Infinity is not a simple concept, and erroneous conclusions are normally reached when it is casually used as if it were an ordinary (if very large) number.

I know that "0.999... = 1" is generally accepted as a mathematical rule. But I'm not satisfied with the explanation my teachers gave me.

I agree with your previous statement, namely "the limit of 9 * (10^-1 + 10^-2 + ... + 10^-n) as n tends to infinity is 1". But I think of a limit as an upperbound. In this case I would say this limit bounds the possible values, but no value will ever be exactly the same as the limit. It just seems impossible to me, even with n at infinity. But that's just the way how I perceive infinity.

fraggle said:

Its important to realise that infinity is not a number.

Yes, infinity is not a number, but 0.3333... IS a number. A number which can't represent the same value as 1/3, IMO. That would be like writing Pi numerically, which can't be done.

Here's a good site on paradoxes, including of course, time travel:
http://members.aol.com/kiekeben/para.html

Arno said:

I know that "0.999... = 1" is generally accepted as a mathematical rule. But I'm not satisfied with the explanation my teachers gave me.

I agree with your previous statement, namely "the limit of 9 * (10^-1 + 10^-2 + ... + 10^-n) as n tends to infinity is 1". But I think of a limit as an upperbound. In this case I would say this limit bounds the possible values, but no value will ever be exactly the same as the limit. It just seems impossible to me, even with n at infinity. But that's just the way how I perceive infinity.

Maybe you have a misconception about infinity.

It is true that no real value will ever be exactly the same as the upper limit of the expression 9 * (10^-1 + 10^-2 + ... + 10^-n) (which is 1). However, bear in mind that infinity is a concept, not a number. It is inquantifiable, and isn't bound by the number laws. Hence, the statement made by Grazza is perfectly valid.

// Edit
To test your perception of the concept of infinity, allow me to ask you a few questions :p

1. Does infinity remains as such when operated on by any real number?
2. What is the value of infinity divided by infinity?
3. What is the value of infinity divided by zero?

Arno said:

Yes, infinity is not a number, but 0.3333... IS a number. A number which can't represent the same value as 1/3, IMO. That would be like writing Pi numerically, which can't be done.

In the case of both "0.333... recurring" and pi, one is not saying that they are a decimal number that can be written down; one is really defining them as the limits of series with an "infinite" number of terms.

BTW, there is a difference between a upper/lower bound and a limit. With a limit, one is saying that the sum of the series (or whatever) can be made arbitrarily close to the limit by choosing to sum a sufficiently large number of terms. However small a number you choose for the difference between the sum and its limit, I can always choose a number of terms to sum that will bring it closer to the limit than that. The point is not that this can be done in one particular instance, but that it can be proved that it can be done no matter what value is chosen.

(Sorry if the previous paragraph is confusing; I tried to make it as clear as I could without using symbols, and stuff.)

Your willingness to question your teachers' sloppy definitions is probably very useful if you wish to study higher mathematics in due course. Getting to grips with the concept of infinity is far easier when everything is intuitively easy to grasp than when dealing with some complex problem in, e.g., quantum mechanics.

Infinity is a difficult concept. There are some areas of applied mathematics where professional mathematicians wrestle with the issue of cancelling quantities on both sides of an equation that are infinite (of course, there are arguments supporting this - it's not done randomly). This tends to be justified by "well, it works, so it must be OK". Pure mathematicians of course laugh their heads off when this approach turns out in the end not to have been valid.

Wow, Grazza. You specialised in Mathematics?