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Benny
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Re: paradoxes
« Reply #50 on: May 6th, 2008, 10:15am » |
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Take 3 people : #1, #2 and #3
Suppose these 3 people have these preferences:
#1 : A > B > C
#2 : C > A > B
#3 : B > C > A
If we ask these 3 people to make a group choice, that is majority vote, between A and B, they would choose A, and a choice between B and C, then they would choose B, and between C and A, then they would choose C.
I expected this relation to be transitive, but it is not!
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towr
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Re: paradoxes
« Reply #51 on: May 6th, 2008, 10:50am » |
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Yes, I get that; that's exactly what you said in the previous post. Unfortunately it doesn't address my question.
You wanted the probability that there is a collective choice; I want to know what the condition is for a collective choice.
Suppose if we have a ranking ABC, that we thengive 2 points to A, 1 to B, and 0 to C.
Then if three people vote ABC, ABC and BCA respectively, A and B both get 4 points. Which would be a draw in this scheme.
Now consider another scheme, where given ABC, we gave A 3 points, B 1 and C 0.
Then if three people vote ABC, ABC and BCA (as before) respectively, A gets 6 points and B gets 5. So with this scheme A wins.
So the same votes give different results depending on the scheme for aggregating the votes.
Other schemes (not even necessarily using a point system) are possible.
So what I want to know, who wins in this case, if anyone. What is the criterion for a collective choice? Otherwise I won't be able to answer your question.
If you pick the first scheme, then the answer is a probability of 31/36.
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| « Last Edit: May 6th, 2008, 10:52am by towr » |
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Benny
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Re: paradoxes
« Reply #52 on: May 6th, 2008, 11:45am » |
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Sorry, you're right. I got stuck with the transitivity/non-transitivity aspect. I did not frame it properly to resolve it.
Are you suggesting that if we reduce the number of cases to analyze in order to consider what happens if two voters agree on their first choice?
If we frame it differently, then we don't have a paradox.
How about in real life?
would you change your vote if your favorite candidate is falling behind? If your vote isn't likely to matter a lot, in the sense of breaking a tie,
would you still vote for your favorite candidate? Many perceive that it would be wasting your vote to vote for a candidate who has a diminished chance of winning. It was reported on the news that people who vote for the democrat candidates are divided. If, for example, Sen. Obama gets the nomination, a number of Sen. Hillary Clinton's supporters would switch side and vote for the republican candidate.
I just find this type of dynamics fascinating.
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| « Last Edit: May 6th, 2008, 12:05pm by Benny » |
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towr
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Re: paradoxes
« Reply #53 on: May 6th, 2008, 12:16pm » |
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on May 6th, 2008, 11:45am, BenVitale wrote:| Are you suggesting that if we reduce the number of cases to analyze in order to consider what happens if two voters agree on their first choice? |
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I'm not quite sure what to make of this sentence.
I just want to know how you propose combining the results from the three voters, such that you get a result that either is or is not decisive.
For ABC, BCA, CAB, no scheme will give a decisive result; but for other votes, for example ABC, ABC, BCA, some schemes give a decisive result while others don't.
So just tell me; if the three voters vote ABC, ABC, BCA, who, if anyone, wins.
Quote:| If we frame it differently, then we don't have a paradox. |
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There isn't a paradox either way; non-transitivity is not a paradox.
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Benny
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Re: paradoxes
« Reply #55 on: Jul 4th, 2008, 5:10pm » |
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Murphy's Law says that if anything can go wrong, it will. But, this stupid law applies to itself: itself can go wrong, that is, there must be a situation where something can go wrong and it won't go wrong. So, Murphy Law is paradoxal.
Your thoughts, please.
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towr
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Re: paradoxes
« Reply #56 on: Jul 5th, 2008, 3:28am » |
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on Jul 4th, 2008, 5:10pm, BenVitale wrote:Murphy's Law says that if anything can go wrong, it will. But, this stupid law applies to itself: itself can go wrong, that is, there must be a situation where something can go wrong and it won't go wrong. So, Murphy Law is paradoxal.
Your thoughts, please. |
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It's not an actual law; it's a statement of pessimism.
In day to day life, obviously not everything that can go wrong goes wrong; you would have failed posting that message. In fact you'd have been long dead from an eating or breathing mishap.
It's insane to even consider this in a literal sense. However that doesn't mean it's stupid.
As for your claim that it's paradoxical; Murphy's law is not something that "goes", so "going wrong" doesn't apply to it. And it also doesn't apply to things that can't go wrong. (Which is kind of funny, because what it typically tries to say is that everything can go wrong; but in a literal sense it says nothign of the sort).
Also, again, you just plagarized from another site. If you're not going to use your own words, attribute them to their rightfull source: http://uncyclopedia.org/wiki/Murphy%27s_Law#Murphy.27s_Paradox
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| « Last Edit: Jul 5th, 2008, 3:36am by towr » |
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Benny
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Re: paradoxes
« Reply #57 on: Jul 5th, 2008, 10:54am » |
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I'm on the computer everyday, I play chess online, I'm on social network websites discussing politics, and someone sent me the Murphy's thing,
Quote:
Murphy's Law says that if anything can go wrong, it will. But, this stupid law applies to itself: itself can go wrong, that is, there must be a situation where something can go wrong and it won't go wrong. So, Murphy Law is paradoxal.
Your thoughts, please |
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and I thought it was cute, so i decided to post it here. It was not my intention to plagiarize. I know, I could have asked my virtual friend the source of this thing, or i could have checked the source myself, or i could mentioned on this thread how i got this thing. I did not do any of those things. I got sidetracked by other things, that's no excuse
I'm sorry. I respect you, guys, and I appreciate greatly for your courtesy and taking the time to answer to my posts. Please accept my apologies.
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| « Last Edit: Jul 5th, 2008, 10:55am by Benny » |
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ThudnBlunder
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Re: paradoxes
« Reply #58 on: Jul 5th, 2008, 5:46pm » |
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on Jul 5th, 2008, 10:54am, BenVitale wrote:
I'm sorry. I respect you, guys, and I appreciate greatly for your courtesy and taking the time to answer to my posts. Please accept my apologies.
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Fair enough. No harm done. It's just that towr likes to know exactly whose opinions he is grinding into the dust. LOL
But the other example required some surgical editing.
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| « Last Edit: Jul 6th, 2008, 4:22pm by ThudnBlunder » |
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towr
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Re: paradoxes
« Reply #59 on: Jul 6th, 2008, 7:33am » |
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on Jul 5th, 2008, 5:46pm, ThudanBlunder wrote:| Fair enough. No harm done. It's just that towr likes to know exactly who he is grinding into the dust. LOL |
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Hah, well, it's also because of that other thread, and I was starting to wonder if Ben was just a collection of quotes gathered from around the internet.
I suppose I also don't really care specifically which source it is, just that there is another source. Then I can always try google if I want to check it out. So anything from putting quotes around a quote, or adding "I read somewhere that" or "My friend send me this" is fine by me.
Of course, in this case, the fact it's from uncyclopedia gives a big hint on how seriously the "paradox" should be considered. Uncyclopedia is a parody version of wikipedia; so typically they aren't entirely serious.
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Benny
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Re: paradoxes
« Reply #60 on: Jul 6th, 2008, 2:09pm » |
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Thanks Towr and ThudanBlunder. I will be more thorough next time. After all, since I am aiming for a Master's degree in math, I need to be thorough in every thing. I noticed after re-visiting my own posts my English is not that great. I need to pay more attention to my syntaxes, grammar and spelling. Is there a spelling check button here?
I didn't know an Uncyclopedia existed.
Is there a reliable site for Theoretical Physics?
I really don't want to be labelled "reality-challenged" again.
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towr
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Re: paradoxes
« Reply #61 on: Jul 6th, 2008, 3:15pm » |
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on Jul 6th, 2008, 2:09pm, BenVitale wrote:| Is there a spelling check button here? |
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No, there isn't. But if you use firefox (2 or 3), it has a build-in spell check. If you travel around a lot, you could use firefox-portable and put it on a USB-stick. (Also convenient for having your own bookmarks and history etc wherever you go)
Quote:| Is there a reliable site for Theoretical Physics? |
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Possibly you can find a few things at http://ctp.lns.mit.edu/links.html, like the journal section (If it's good enough for MIT to link, it can't be all bad)
Quote:| I really don't want to be labeled "reality-challenged" again. |
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Well, if that is the only argument someone can give as to why he thinks you're wrong (or worse, simply to stop you asking questions), then at the very least the problem doesn't lie entirely with you. Challenging preconceived notions of what constitutes reality isn't necessarily the same as being reality-challenged. (Mind you, this is not a good argument to use against people, because it will just upset them.)
I suppose in a basic sense, being challenged by reality is what being a scientist is about: standing up to that challenge. Rather than letting it bully you around 
On the other hand there's a time and place for certain types of questions (and a way to ask them). So don't take this as an advice to interrupt class with questions your professor is unable and/or unwilling to answer or discuss. If it can't be answered in 5 minutes and is outside the curriculum, it's probably best asked after class or via email. (I figure this is about your physics professor and that question about QM interpretations, but of course I don't know the situation at all. So consider this a general shot in the dark type of advice.)
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Benny
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Re: paradoxes
« Reply #62 on: Jul 6th, 2008, 6:40pm » |
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Well, before the cat incident in class, the physics professor always welcomed our questions and encouraged us to think and discuss. He encouraged us to think outside the box. We usually have 5 minutes for discussions before we start with his lectures. So, the other day, I raised my hand and expressed my fascination with theoretical physics, I said that I was contemplating on the eleven dimensions, parallel universes, and a world made out of strings, and that the string theory that might hold the key to unifying the four forces of nature according to Brian Greene, the supersymmetry, and beyond the standard model.
Then I ask, "What's your take on the Schroedinger's cat experiment?"
His mood changed all of sudden. He got upset. I didn't understand it. Theoretical physicists have been spending time thinking about it. People like Penrose, Hawkings, Dr. Michio Kaku.
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Benny
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Re: paradoxes
« Reply #63 on: Jul 6th, 2008, 8:21pm » |
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Imagine that there is a scientific law that says "All crows are black." If only three or four crows are observed, then the law is weakly confirmed. If millions of crows are seen to be black, then it is strongly confirmed.
Professor Carl Hempel, who invented this paradox, believes that a purple cow actually does slightly increase the probability that all crows are black.
Source: http://www.geocities.com/CapitolHill/Lobby/3022/hempel.html
I am not totally satisfied with the explanation offered in the linked document.
Proposition 1: All crows are black
Double-negation gives the same proposition: All !(crows) are !(black) <--> All things that are not black are not crows
Observation 1: There exists a cow which is purple
Thus, Proposition 1 is consistent with Observation 1.
But, it is also consistent with Proposition 2: All crows are blue (Through the same method); thus Observation 1 supports two propositions which cannot be true at the same time. Thus, Observation 1 is inadmissible.
I asked myself: what do I make of this purple cow?
Then I get that there exists a cow which is purple (which is consistent with Proposition 1)
Doesn't a purple cow provides evidence for the hypothesis that all things that are not black are not crows, but it also provides evidence for, for example, all things that are not pink are not crows - or, All crows are pink. Thus this evidence is inadmissible.
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ThudnBlunder
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Re: paradoxes
« Reply #64 on: Jul 6th, 2008, 9:41pm » |
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If we consider multiple hypotheses simultaneously (eg. all ravens are black or all ravens are blue or all ravens are pink), it is true that any observation of a non-black/blue/pink object that is not a raven increases the probability that all ravens are black or all ravens are blue or all ravens are pink by a very, very small amount corresponding to the ratio of ravens to non-ravens. Also, the information that an object is not a raven removes the possibility of this object being a counterexample to the rule. And then there is the small matter that no blue or pink ravens have ever been observed, whereas black ravens are known to exist.
There is a 'little' more to it than that, but that's my take on the matter. 
Cf.
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| « Last Edit: Jul 6th, 2008, 10:30pm by ThudnBlunder » |
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towr
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Re: paradoxes
« Reply #65 on: Jul 7th, 2008, 12:59am » |
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on Jul 6th, 2008, 8:21pm, BenVitale wrote:Imagine that there is a scientific law that says "All crows are black." If only three or four crows are observed, then the law is weakly confirmed. If millions of crows are seen to be black, then it is strongly confirmed.
Professor Carl Hempel, who invented this paradox, believes that a purple cow actually does slightly increase the probability that all crows are black. |
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The general argument is that this is why "confirmation" is not a good approach to science.
You can "confirm" your theory by observing anything that it doesn't apply to. You study crows, then end up looking for cows.
It's one of the reasons why Karl Popper suggested that instead you should try to aim at falsifying theories; look for non-black crows. You can't prove a universal, except by examining every object in the domain; but proving it false takes finding just one counter-example. So a search directed at the latter makes more sense.
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rmsgrey
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Re: paradoxes
« Reply #66 on: Jul 7th, 2008, 7:06am » |
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Your sampling technique is significant when it comes to whether a purple cow supports the "all crows are black" hypothesis - if you go and collect non-crow objects and then look at their colours, that tells you nothing any more than collecting black objects and then checking whether they're crows does (though the latter does give you some evidence for/against the existence of black crows...)
If you go around looking at cows, that tells you nothing about whether all crows are black. If you go around looking at purple objects, then finding a bunch of cows, but no crows is weak evidence that all crows are black. Similarly, looking at a bunch of crows and seeing that they're all black is (relatively) strong evidence that all crows are black.
And "all crows are black" and "all crows are pink" are not necessarily mutually exclusive propositions - they only become mutually exclusive once it is known that some crows exist - at which point, checking the colour of any known crow will disprove at least one of the two.
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Grimbal
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Re: paradoxes
« Reply #67 on: Jul 7th, 2008, 8:04am » |
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The more purple cows you see, the more it confirms that there is no crow. And that supports that all crows are black.
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rmsgrey
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Re: paradoxes
« Reply #68 on: Jul 8th, 2008, 8:31am » |
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on Jul 7th, 2008, 8:04am, Grimbal wrote:| The more purple cows you see, the more it confirms that there is no crow. And that supports that all crows are black. |
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It still depends on how you're finding the purple cows - if you send a PhD student out to find you a thousand cows, and they don't bring back any crows, that doesn't tell you anything about the abundance of crows (regardless of colour) - there was no chance (assuming a competent PhD student) of getting any crows back anyway...
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Grimbal
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Re: paradoxes
« Reply #69 on: Jul 8th, 2008, 10:05am » |
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I assumed you wander idly and check what you see.
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Random Lack of Squiggily Lines
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Re: paradoxes
« Reply #70 on: Jul 9th, 2008, 9:58am » |
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Wait....
All cows ARE black.
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You can only believe i what you can prove, and since you have nothing proven to cmpare to, you can believe in nothing.
I have ~50 posts to hack a "R" into a "D". Which one?
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Benny
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Re: paradoxes
« Reply #71 on: Jul 19th, 2008, 10:38am » |
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I have a question about the Ross-Littlewood paradox, see link
http://www.suitcaseofdreams.net/Paradox_Infinity.htm
According to this document, the answer is none.
I don't get the same answer.
P1 + P2 + ... + P10) - P1 +
P11 + P12 + ... + P20) - P2 +
P21 + P22 + ... + P30) - P3 +
....
P10j-9 + P10j-8 + P10j-7 + ... + P10j - Pj+
...........
= SUM[i=1,i=10]Pi - P1
+ SUM[i=11,i=20]Pi - P2
+ SUM[i=21,i=30]Pi - P3 +
...
+ SUM[i=10j-9,i=10j]Pi - Pj + ...
= limj--> oo (SUM[i=1,i=10j] Pi - SUM[i=1, i=j] Pi
= limj--> oo (SUM[ i=j+1,i=10j] Pi = oo
I'm getting infinite elements.
did i go wrong somewhere?
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towr
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Re: paradoxes
« Reply #72 on: Jul 19th, 2008, 11:12am » |
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on Jul 19th, 2008, 10:38am, BenVitale wrote:| did i go wrong somewhere? |
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Yes. Just name one single term that is still in the sum, and I'll tell you exactly at what step it was removed.
There is no i, such that Pi was added but not later subtracted.
You could look up the "impish pixie" thread, or one of the similar ones.
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ThudnBlunder
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Re: paradoxes
« Reply #73 on: Jul 19th, 2008, 11:13am » |
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on Jul 19th, 2008, 10:38am, BenVitale wrote:I have a question about the Ross-Littlewood paradox...
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Ah, so that's what it is called.
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Benny
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Re: paradoxes
« Reply #74 on: Jul 19th, 2008, 11:49am » |
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Thanks. I didn't know about "impish pixie" thread. But before I go and read impish pixie's thread, I just need to show you the things I wrote:
Things with different rate of growth can't be subtracted to give zero.
What's the limit of a2 - a as a approaches infinity? We can sure see that both a2 and a approaches infinity as a approaches infinity. So is it that:
lim (a2 - a) = lim (a2) - lim (a) = 0 ?
a --> oo
Clearly not. There are two flaws. One: Things with different growth rates cannot have their limits spread over them in a linear fashion.
lim (SUM [i=1, i=10j]Pi - SUM [i=1,i=j]Pi is difinitely not SUM [i=1,oo]Pi - SUM [i=1,oo]Pi.
j --> oo
We have to do the the brackets first.
We cannot do oo - oo.
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