So, in our last lecture, we have dealt with set theory. And, at the time when set theory began to influence the other mathematical branches, many "contradictions", the so-called "paradoxes", were discovered. Generally, a paradox is a puzzling conclusion we can easily derive, but who still are highly counterintuitive. * Megahertz has joined #lecture Among those, there are a large amount of paradoxes of a logical nature that have teased even professional logicians, and, sometimes, even for several millenia! Here, we will restrict to the "logical paradoxes", which is a more heterogeneous family. homogeneous* We'll begin with a greek philosopher, called Zeno of Elea. Nowadays, we still know him through the "Zeno's paradoxes". * Megahertz has quit IRC (Quit: ) * J0EL has quit IRC (Quit: ) The most well-known of his paradoxes is certainly : Achilles and the Tortoise. Let us recall what this paradox is about. * Pixi3_1103 has quit IRC (Quit: ) Basically, Achilles was challenged by a Tortoise to a race. But, the thing is, while the Tortoise was covering 10 meters, Achilles covered 5 meters, if the tortoise covered 20 meters, achilles covered 10 meters. I believe that you got the point : Achilles covered each time half less than the tortoise. But, you'll wonder, where is the paradox? So, let's write in a 'mathematical' way what Achilles is covering : 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... so, you see that there is an infinite amount of terms in this sequence * Pixi3_1103 has joined #lecture so, what does it mean? Achilles will never reach the end line? * qwertydawom sets mode: -m Huh. yes :P The point is is that it is a paradox.. He must reach the line, but mathematically he can't. is the reason he is slower then the tortoise is cos of his tendon? lol ok LMFAO! :) yes, IceDane is right on that point oo o.o this is good stuff to know... they make us take a class on this stuff in college but! that is why, in mathematics, we introduced the notion of "limit" Damn, I'm good. * Narada has quit IRC (Quit: Chatzilla 0.9.69.1 [Firefox 1.5.0.1/2006011112]) Oi. Of course. and, actually, this 1/2 + 1/4 + 1/8 ... adds up to.. 1! :) So, our good old Achilles would have reached the end line and won the race ;) 0.5 + 0.25 + 0.125 != 1. did I ever say it stopped to 1/8? Oh.. Of course. that is the point of a limit yeah to evaluate infinite sums like that lol It's like 1/11. 0.999_ forever. Yeah Was discussing this other day. Yep, and, while we're on it, I assume you've heard about this : 0.999.. = 1? :) i have * Cyph3r looks at porn instead Well, for those who haven't, let me show what is meant by that, and introduce a "proof" of this result. Yeah. So, we have : 1/3 = 0.333... .999 = 1, because the .000_1 can't be there if .999_ is infinite. If we multiply by 3 on both sides, on the LHS we'll get : 3*1/3, i.e. 1 and, on the RHS, we get : 0.333... * 3 = 0.999... Hence, we've showed that : 0.999... = 1 it might seem a bit paradoxical at first sight, eh? it is still really confusing to try and grasp the concept that one number equals another like that But, it is, however, mathematically correct. Yeah. so 1.999 not repeating doesnt equal 1? 0.999* I guess and, no, it doesn't :) like, does 1.9 equal one the same as 1.99999999999999999999999999999999999999999999 equals one? oh The thing is If you split 1 by 3. You get an infinite amount of 3s. then, you multiply them by 3 And you get an infinite amount of 9s. 0.9999... goes on forever. yep And because the 9s never stop The difference, 0.000....1 can't be there. but you would assume because the 9's never stop that it never quite reaches 1 Normally, 0.99 + 0.01 = 1. But the 9s are infinite. Yeah Exactly. No, wait. Haha The thing is The difference there between so its 1.09infinite Doesn't exist. Because of the 9s being infinite. so there is no number that fills the gap? between 0.99999 and 1? Yeah Exactly. If the 9s are infinite. As in, REALLY infinite. Then the 9s never stop And nothing can fill the gap. you know tho if that is the case wouldnt you assume then that the same goes for every other number as well? Not really. Because 0.11 = 0.11. 1.9999999999... =2? yeah. yes :) The paradox only applies to numbers with infinite decimals. so you are saying that 79.9999...... = 80 aswell? or it is only 1? yes it is not only 1 79.999... = 80 Just every number with an infinite number of 9s. alright =) Got any more logic theory for us ? :P I like this stuff. well, just before I go on, Elda_Winslacks will explain you how this paradox can be mathematically shown using sequences. ;) hehe who is elda? ok so, quertydawom told us that 0.9999...=1 only if there is a infinity of decimal and it can be shown with a sequence * b0re has quit IRC (Quit: bbl) in fact 0.99999...= 9 (10^-1 + +10^-2+ 10^-3+...+10^-n) and 10^-1+10^-2+...+10^-n is the sum a geometrical sequence Ah. Niceþ Infinity is such an abstract concept. xx. x.x so when you use the formul for the sum of such a sequence and you simplifie you find 1 if n tend to infiny aye :) got it? Yeah um... yes :) thank you elda! very nice It's just a way to prove the theory, right ? Because no matter if you go to infinity with 10^-n, it never reaches 1.. you're welcome um, yes And thanks, yeah. that's, once again, the concept of the limit who's striking here :) whoops, wrong window :D When's the next logic lecture? dude we aren't done OH HAHA he is loving this guys... :) talk forever, and keep him entertained :P Yeah. Please. Well, now, for IceDane's pleasure and mine too! and also for Pixi3 who can't handle anymore math :p HAHA! amen! we shall focus on "philosophical logic paradoxes" :) Wooot. ooh.. i know these! on ward! First of all, the well-known.. paradox of the liar! :) Pixi3, will you make us the honor to tell us what it is? ;) oh no no, go on... i will remember with time :D Hah, ok, well, simply, if I say : "I'm lying", that's a paradox, does anyone see why? Because you're telling the truth. yes You're saying you're lying as you're telling the truth. hm... dont remember this one. very good tho! Yes, basically, if this sentence is false, then it can't be true, and if it false that it's true, then it can't be false. if it is* Yeah :P i am thinking tho saevar, they are not encouraging you to lie, just telling you that when you are you are actually telling the truth :) no lying kids Haha * Cyph3r has left #lecture :) Rofl. Scared/Bored most people off. Haha Well, unlike our first paradox, this one can't be "solved"... Now, let us see another one, the paradox of the barber! :) (by the way) yes? :) (there is a book of logik like that called "the book who make mad") Really? I'mma get that one. (it's funny, but just in french I think) OH haha, gonna have a hard time with that one :P i have googled the barber paradox! very interesting got the definition if someone needs it :) go hard ;) The Barber paradox is a paradox that relates to mathematical logic and set theory. The paradox considers a town with a male barber who shaves daily every man who does not shave himself, and no one else. Such a town cannot exist:* If the barber does not shave himself, he must abide by the rule and shave himself.* If he does shave himself, according to the rule he will not shave himself.Thus the rule results in an impossible ation. :) ation? o.o situation :P oh i like this one :) So yeah, if a barber in a town shaves every man in the town who does not shave himself and no one else but no one else yeah he must shave himself, but cant :) yeah. Haha This is nice :P hairy men! yes, the problem is : who shaves the barber?! :D I didn't know this one well he is supposed to do it, but spends too much time on everyone else :P hairy men! --> this one was especially dedicated to IceDane (he'll know why :p) Yeah :P ewww.... Damn frenchies. x'D LOL! i see now :P we were talking about that the other day in fact! lOL! :) well, now, let me explain the part where they say it relates to "set theory" Here, there are two sets : - the one of the men who shave themselves - the one of the men shaved by the barber but, the barber can't belong to any of them... oh cool! i didnt see that :) Ah Of course. This paradox was introduced by the english philosopher Bertrand Russell, in the 20s. Hmm. Ah I See. haha VERY nice :p Can anyone think of a "solution" to this paradox? Someone else shaves him. Or he's shaved by another barber. Or he just sticks his head in an oven and thus scorches it off. :p HAHAHA LOOL Someone else shaves him. --> impossible, remember : he shaves all those who don't shave themselves ;) he gets really hairy Ah. I see. wait wait! he was bald and doesnt need it! Then it's the oven, I think. :) well, I think that you all got the point that, we need additional informations! :) indeed, although one solution would be to deny the existence of the barber Some people proposed other solutions, e.g. : -The barber's lying (<- no paradox here ;)) -The barber's coming from another town nah, cause if he was lying, he would really be telling the truth :D Lmao. -The barber is not a man, but a woman, a robot or an extraterrestrial Rofl. oooh... aliens barber aliens even just to throw a quick reference in : http://en.wikipedia.org/wiki/Category:Paradoxes very interesting stuff, lists lots of paradoxes * Elda_Winslacks has quit IRC (Ping timeout) elda has left the building! Well, now, we'll see another paradox (dunno if it's listed at Pixi3's link) tell me and i will look :) The crocodile's paradox nah, dont see it ok ;) well, a crocodile kidnaps a baby (poor him! :p) and says to the mother (of the baby :)) : * Elda_Winslacks has joined #lecture "If you guess what I'm gonna do, I give you the baby back, else, I eat it" what would you answer? :) eat it Yeah. cause either way you are screwed i am thinking :P And, indeed, the answer of the mother was : "you're going to eat it!" and, hehe, we've reached a dilemma :D why? well : -if the crocodile eats the baby, then the mother guessed it well, and the crocodile is forced to give her back the baby... Yeah :P -if the crocodile doesn't eat the baby, then the mother was wrong, and the croco. has to eat the baby.. yup yup :) :) Haha, nicey :p so, it is impossible to go through this dilemma tho it would not have helped i dont think had she had said something different in reply The correct thing to do in that situation Is pull an AK47 out of her ass, and blast the fucking croc to jersey. "dress the baby up and play house" would have gotten the kid eaten as well :P haha! if you wanna go that route the bitch should have been watching the kid from the beginning :P ooor the kid would have never been taken :P Tell the croc that the baby just swallowed nitro glycerin. LMFAO! LOOL And that was the reason he was left alone Danger of explosion. haha, you should be a sci-fi novelist :P or you can just NOT talk to the animal at all... cause normally they dont talk back unless you have visited the padded room quite a few times, and the kid you think you had never really existed :P i really hope you understood that, cause if not it was alot of typing for nothing :P i dont know what you guys would call a mental institution, but nevermind... i give up :P Rofl. lmao Lmao Pixie made a joke :D High five, sistah. well, let's get back to rationality :p :) Did I miss anything? To this dilemma, Lewis Caroll proposed a pragmatical solution He said : If the crocodile eats the baby, then the mother is right, and the crocodile is a liar. If he gives her back the baby, the mother is wrong, and the crocodile is a liar. lol Lol. God damn compulsive liar. So, anyway, the crocodile is a liar. Yeah. But still not a solution nah... he was telling the truth the whole time... i talked to him and asked him :P Then, since there's no hope the crocodile will satisfy the honor code Because the baby will be dead either way. Liar or not :P we can't doubt he'll act according to his nature. this is lovely :D not as good as the barber one :) Barbers, crocodiles and greek heroes. Logic is strange. you love it :P * R4d30N has quit IRC (Quit: ) I do.þ :) So, let's see some "paradoxical sentences" ;) "This sentence is false" "It is forbidden to forbid" Haha. "All rules have exceptions" Lmao. "Never say never" is somewhat a paradox too :P oooh... yes ;) also : on the front cover of a book : The sentence on the back cover is true. on the back cover of the same book : The sentence on the front cover is false. :) Nice :P well boys, this has been thoroughly amusing i must say, and daniel, you did an excellent job, but i must go on to be productive elsewhere ... like my homework! hahaha ok ;) so everyone take care, and have a lovely time entertaining my icelandic friend :D elda, it was a pleasure to meet you aswell! Pixi3 : What IceDane says is false. buh bye ppls! thanks, it was also a pleasure for me IceDane : What Pixi3 says is true. What pixie says is false. Ah :P Never mind. what are you guys talking about? :P i am right all the time :P haha, just kidding! bye everyone * Pixi3_1103 has quit IRC (Quit: ) Haha. bye bye Paradoxes FTFW! :) Also, a big aspect in paradoxes is "self-reference" Let me show an example : This sentence contains thirty-eight letters. Ah. now, do you get the concept of self-reference? :) Yeah. If it had less letters, like thirty-six, the sentence would be kind of incorrect. Or well yep :) yeah. The concept of self-reference is mainly used in maths, philosophy, linguistics, and.. last but not least! computer programming :) And, self-reference can lead us to paradoxes :) Recursions ?:P Or hmm.. Does it apply to them ? Nah.. Never mind :P nope not recursions ;) just have a look at : http://www.cgl.uwaterloo.ca/~csk/washington/paper/discussion1.html if you're interested So, let me show an example of a paradox with self-reference : Alright :P "Your answer will be no." Oh I know one.. When you're saying what you're doing. Right ? why? :) As in oh, you meant a self-reference example, then yes When someone asks you what you're doing yeah :P but it's not a paradox ;) Hmm Maybe not. While saying "Your next word will be 'no'", is one ;) can you see why? Yeah okay :) So, through this lecture, we've seen that, logic was far from being perfect, and that we could find some "flaws" in it. :) Some ? x'D hehe It's buggy like microsucks software. This remark will end tonight's lecture :)