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Paradoxes Answered

just a place to discuss paradoxes (an immpossible statement) like The statement to the left is false the statement to the left is true

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The Jamalam

9 years ago

The next sentence is a lie. I'm lying.

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The Jamalam

9 years ago

Go back in time and kill yourself? Amy winehouse going back and killing jabba the hutt? Instructables destroyed and me building knex guns now?

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lemonie

9 years ago

Here's an apparent paradox:
You have an infinite line, which is a closed-loop, like a circle.
It's all scrunched up in zigzags \/\/\/\/\/\/\/\/\/\/ but really close together, more like |||||||||||||||||. Within the middle of the loop/circle there is an area which can be calculated - the peaks and troughs average out.

Infinite border - finite area?

L

L

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kelseymhlemonie

Reply 9 years ago

Are you taking the limit of zero angle for each zig-zag? If so, I think you've got an array of pseudo-delta-functions with finite peak height, and therefore infinitesimal (read zero) area, but covering the plane. How close to the center do the peaks go? The enclosed area should just be the space strictly interior to the peaks. If I'm wrong, could you post the derivation? Feel free to use LaTeX and a GIF or PDF, since it's much easier and clearer than ASCII math :-/

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lemoniekelseymh

Reply 9 years ago

I used the word apparent because it's not a paradox. Figuring out what's wrong with it is the point, and I think you can see that.

L

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kelseymhlemonie

Reply 9 years ago

Yes, indeed! I wasn't arguing :-) I don't think you're wrong -- a fractal, after all, is a boundary of infinite length which encloses or covers a finite area. I am interested in the computational details, though -- how do you represent the zig-zags, and how are you doing the integration?

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lemoniekelseymh

Reply 9 years ago

I don't - if it's infinite the angles must be zero, so obviating the thickness of any lines drawn or represented the enclosed area must be zero. Or without the zig-zagging infinite. It looks better on paper, but I didn't seek it out and post an image because these things work better for confusion without something to look at. It took me a little while to work it out (but I like thinking). L

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kelseymhlemonie

Reply 9 years ago

But you know as well as I do that limits don't always come out intuitively. I'm interested in how you represent the zig-zag in the finite angle limit (i.e., an N-pointed star). Let's use the inner vertices of the star as the baseline. Give that a radius r, so a circumference of 2(pi)r.

For N points, each point has a base of 2(pi)r/N. In the small-angle limit (appropriate for large N), we can treat the sides as having a fixed, constant length l, equal to the height, and hence an area a = l×2(pi)r/N / 2. The area of the whole star is therefore

A = N×a + 2(pi)r
= N×2(pi)rl/N / 2 + 2(pi)r
= 2(pi)rl/2 + 2(pi)r

So, A = (2+l)(pi)r. Notice that the N cancelled out before we ever took the actual limit, so this result is the answer for your limiting case.

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lemoniekelseymh

Reply 9 years ago

You'd have N as infinity, rather than l - so you get a result. Hmmm, there must be some wonky-maths in here somewhere... L

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kelseymhlemonie

Reply 9 years ago

That's correct. By construction I kept the height of the zig-zigs finite (think of one of those kitschy '60's era wall clocks), but made them narrower and narrower (limit as N -> infinity). The wonky math is precisely in taking that limit :-)

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yourcat

9 years ago

Look up a ring oscillator on wikipedia.

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kelseymh

9 years ago

Okay, E-R-IC. Was that a sufficiently interesting discussion of a paradox for you?
Next up ought to be quantum entanglement and spooky action at a distance, but it's already been done.

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kelseymh

9 years ago

No, that's just nonsense. The naïve paradox (with correct punctuation and words) is

The statement to the right is false. The statement to the left is true.

Statements like that are just semantic mumbo jumbo. Far more interesting are the true logical (mathematical) paradoxes, such as Gödel's Indecidability Theorem.

Modern physics (both quantum mechanics and relativity) are replete with apparent paradoxes (such as entanglement) which arise from our attempts to interpret results in a classical way.

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E-R-ICkelseymh

Reply 9 years ago

well then how about this: nothing is imposible

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Lithium RainE-R-IC

Reply 9 years ago

Again with the semantic mumb jumbo (to borrow Kelsey's phrase). Or shall we start up again with oxymorons? Okay, my turn. Thoughtful guy! Polite evolution debater! Um...intelligent knexer! Oo, that last one might have been too much...>ducks barrage of non-lethal plastic toys<

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xACIDITYxLithium Rain

Reply 9 years ago

Creation science! Oo, that last one might have been too much...>ducks barrage of non-lethal bible verses<

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kelseymhxACIDITYx

Reply 9 years ago

It's not the verses you have to worry about. It's the sharp corners on all of the badly mistranslated books.

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kelseymhxACIDITYx

Reply 9 years ago

I think Mick and Keith are old enough that you can out-run them if you really have to.

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kelseymhE-R-IC

Reply 9 years ago

Ummm..you proposed paradoxes, self-contradictory statements or groups of statements, or situations which are logically or physically inconsistent. "Nothing is impossible" is simply a falsehood, at least in this universe.

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gmjhowekelseymh

Reply 9 years ago

"With God, all things are possible" I have that on a bookmark somewhere..

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xACIDITYxgmjhowe

Reply 9 years ago

"With Ignorance, all things seem possible"

Just a related quote; of absolutely no hold on your particular comment's content.

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gmjhowexACIDITYx

Reply 9 years ago

Due to your explanation i will not flame. What you say is also perfectly true, alot of people through ignorance think things are possible. Me and my faith, is not an example.

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kelseymhgmjhowe

Reply 9 years ago

Yeah, bookmarks are always a great source for Truth. But how, exactly, do you write the citation?

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Lithium Rainkelseymh

Reply 9 years ago

But what if I have a pair of entangled bookmarks? Every time I click on the one, the other changes!

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kelseymhLithium Rain

Reply 9 years ago

Unfortunately, they're only entangled as long as you don't click on them. Once you do, you project them out into orthogonal (or parallel, depending on the specifics of the entanglement) eigenstates, and after that they evolve independently according to the Schrödinger equation.

You do know you were asking for it, don't you?

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Rock SoldierE-R-IC

Reply 9 years ago

if nothing is impossible, that means everything is possible, meaning that it is possible to make something that is impossible, but, that would be impossible because everything is possible, except making something impossible even though it's possible because everything is not not possible.

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Lithium Rainkelseymh

Reply 9 years ago

Thank you!

Those stupid "paradoxes" have always annoyed me!

>Waits for the inevitable God-rock one<

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Lithium Rain

9 years ago

Strangely, it keeps saying Kelsey commented on this recently, but no news posts. That's happened several times over the past few hours.

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Rock Soldier

9 years ago

Um... no means yes, and yes means no.