Mar 11 14:06:36 So, in our last lecture, we have dealt with set theory. Mar 11 14:07:36 And, at the time when set theory began to influence the other mathematical branches, many "contradictions", the so-called "paradoxes", were discovered. Mar 11 14:09:11 Generally, a paradox is a puzzling conclusion we can easily derive, but who still are highly counterintuitive. Mar 11 14:09:53 * Megahertz (Megahertz@shytb0x-B7D3E839.d-ip.magma.ca) has joined #lecture Mar 11 14:11:32 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! Mar 11 14:12:20 Here, we will restrict to the "logical paradoxes", which is a more homogeneous family. Mar 11 14:13:51 We'll begin with a greek philosopher, called Zeno of Elea. Mar 11 14:14:39 Nowadays, we still know him through the "Zeno's paradoxes". Mar 11 14:15:00 * Megahertz has quit (Quit: ) Mar 11 14:15:31 * J0EL has quit (Quit: ) Mar 11 14:16:01 The most well-known of his paradoxes is certainly : Achilles and the Tortoise. Mar 11 14:16:47 Let us recall what this paradox is about. Mar 11 14:17:25 * Pixi3_1103 has quit (Quit: ) Mar 11 14:17:30 Basically, Achilles was challenged by a Tortoise to a race. Mar 11 14:18:41 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. Mar 11 14:19:07 I believe that you got the point : Achilles covered each time half less than the tortoise. Mar 11 14:19:20 But, you'll wonder, where is the paradox? Mar 11 14:20:07 So, let's write in a 'mathematical' way what Achilles is covering : Mar 11 14:20:22 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... Mar 11 14:20:42 so, you see that there is an infinite amount of terms in this sequence Mar 11 14:20:42 * Pixi3_1103 (pixiestix1@shytb0x-C11C46A4.dhcp.clmb.ga.charter.com) has joined #lecture Mar 11 14:21:02 so, what does it mean? Achilles will never reach the end line? Mar 11 14:21:26 * qwertydawom sets mode -m #lecture Mar 11 14:21:39 Huh. Mar 11 14:21:39 yes :P Mar 11 14:22:01 The point is is that it is a paradox.. He must reach the line, but mathematically he can't. Mar 11 14:22:03 is the reason he is slower then the tortoise is cos of his tendon? Mar 11 14:22:10 lol ok Mar 11 14:22:11 LMFAO! Mar 11 14:22:19 :) Mar 11 14:22:26 yes, IceDane is right on that point Mar 11 14:22:29 oo Mar 11 14:22:31 o.o Mar 11 14:22:38 this is good stuff to know... they make us take a class on this stuff in college Mar 11 14:22:48 but! that is why, in mathematics, we introduced the notion of "limit" Mar 11 14:22:52 Damn, I'm good. Mar 11 14:22:56 * Narada has quit (Quit: Chatzilla 0.9.69.1 [Firefox 1.5.0.1/2006011112]) Mar 11 14:23:01 Oi. Mar 11 14:23:02 Of course. Mar 11 14:23:06 and, actually, this 1/2 + 1/4 + 1/8 ... adds up to.. 1! :) Mar 11 14:23:42 So, our good old Achilles would have reached the end line and won the race ;) Mar 11 14:23:45 0.5 + 0.25 + 0.125 != 1. Mar 11 14:24:00 did I ever say it stopped to 1/8? Mar 11 14:24:01 Oh.. Mar 11 14:24:02 Of course. Mar 11 14:24:09 that is the point of a limit Mar 11 14:24:13 yeah Mar 11 14:24:21 to evaluate infinite sums like that Mar 11 14:24:38 lol Mar 11 14:25:09 It's like 1/11. 0.999_ forever. Mar 11 14:25:30 Yeah Mar 11 14:25:33 Was discussing this other day. Mar 11 14:25:50 Yep, and, while we're on it, I assume you've heard about this : 0.999.. = 1? :) Mar 11 14:25:58 i have Mar 11 14:26:09 * Cyph3r looks at porn instead Mar 11 14:26:43 Well, for those who haven't, let me show what is meant by that, and introduce a "proof" of this result. Mar 11 14:26:53 Yeah. Mar 11 14:27:02 So, we have : 1/3 = 0.333... Mar 11 14:27:27 .999 = 1, because the .000_1 can't be there if .999_ is infinite. Mar 11 14:27:34 If we multiply by 3 on both sides, on the LHS we'll get : 3*1/3, i.e. 1 Mar 11 14:27:48 and, on the RHS, we get : 0.333... * 3 = 0.999... Mar 11 14:28:00 Hence, we've showed that : 0.999... = 1 Mar 11 14:28:12 it might seem a bit paradoxical at first sight, eh? Mar 11 14:28:27 it is still really confusing to try and grasp the concept that one number equals another like that Mar 11 14:28:33 But, it is, however, mathematically correct. Mar 11 14:28:34 Yeah. Mar 11 14:29:01 It is correct if and only if there is an infinite number of 9s. Mar 11 14:29:13 Paradoxes aren't to be understood, really. There are always 2 proveable things. Mar 11 14:29:21 Yeah. Mar 11 14:29:22 so 1.999 not repeating doesnt equal 1? Mar 11 14:30:12 0.999* I guess Mar 11 14:30:17 and, no, it doesn't :) Mar 11 14:30:19 like, does 1.9 equal one the same as 1.99999999999999999999999999999999999999999999 equals one? Mar 11 14:30:23 oh Mar 11 14:30:28 The thing is Mar 11 14:30:38 If you split 1 by 3. Mar 11 14:30:43 You get an infinite amount of 3s. Mar 11 14:30:49 then, you multiply them by 3 Mar 11 14:30:57 And you get an infinite amount of 9s. Mar 11 14:31:02 0.9999... goes on forever. Mar 11 14:31:06 yep Mar 11 14:31:07 And because the 9s never stop Mar 11 14:31:20 The difference, 0.000....1 can't be there. Mar 11 14:31:33 but you would assume because the 9's never stop that it never quite reaches 1 Mar 11 14:31:34 Normally, 0.99 + 0.01 = 1. But the 9s are infinite. Mar 11 14:31:43 Yeah Mar 11 14:31:44 Exactly. Mar 11 14:31:49 No, wait. Mar 11 14:31:50 Haha Mar 11 14:31:53 The thing is Mar 11 14:31:57 The difference there between Mar 11 14:31:58 so its 1.09infinite Mar 11 14:32:01 Doesn't exist. Mar 11 14:32:06 Because of the 9s being infinite. Mar 11 14:32:19 so there is no number that fills the gap? Mar 11 14:32:24 between 0.99999 and 1? Mar 11 14:32:25 Yeah Mar 11 14:32:28 Exactly. Mar 11 14:32:32 If the 9s are infinite. Mar 11 14:32:38 As in, REALLY infinite. Mar 11 14:32:42 Then the 9s never stop Mar 11 14:32:50 And nothing can fill the gap. Mar 11 14:32:52 you know tho Mar 11 14:32:56 if that is the case Mar 11 14:33:00 wouldnt you assume then Mar 11 14:33:08 that the same goes for every other number as well? Mar 11 14:33:13 Not really. Mar 11 14:33:18 Because 0.11 = 0.11. Mar 11 14:33:22 1.9999999999... =2? Mar 11 14:33:24 yeah. Mar 11 14:33:44 yes :) Mar 11 14:33:47 The paradox only applies to numbers with infinite decimals. Mar 11 14:34:12 so you are saying that 79.9999...... = 80 aswell? or it is only 1? Mar 11 14:34:20 yes it is Mar 11 14:34:25 not only 1 Mar 11 14:34:32 79.999... = 80 Mar 11 14:34:33 Just every number with an infinite number of 9s. Mar 11 14:34:42 alright =) Mar 11 14:34:57 Got any more logic theory for us ? :P Mar 11 14:35:13 I like this stuff. Mar 11 14:35:46 well, just before I go on, Elda_Winslacks will explain you how this paradox can be mathematically shown using sequences. ;) Mar 11 14:35:47 hehe Mar 11 14:35:54 who is elda? Mar 11 14:36:13 ok Mar 11 14:36:58 so, quertydawom told us that 0.9999...=1 only if there is a infinity of decimal Mar 11 14:37:34 and it can be shown with a sequence Mar 11 14:37:37 * b0re has quit (Quit: bbl) Mar 11 14:38:40 in fact 0.99999...= 9 (10^-1 + +10^-2+ 10^-3+...+10^-n) Mar 11 14:39:38 and 10^-1+10^-2+...+10^-n is the sum a geometrical sequence Mar 11 14:39:53 Ah. Mar 11 14:39:54 Niceþ Mar 11 14:40:56 Infinity is such an abstract concept. xx. Mar 11 14:40:57 x.x Mar 11 14:41:46 so when you use the formul for the sum of such a sequence and you simplifie you find 1 if n tend to infiny Mar 11 14:43:05 aye :) Mar 11 14:43:07 got it? Mar 11 14:43:42 Yeah Mar 11 14:43:42 um... yes :) thank you elda! very nice Mar 11 14:43:51 It's just a way to prove the theory, right ? Mar 11 14:44:06 Because no matter if you go to infinity with 10^-n, it never reaches 1.. Mar 11 14:44:15 you're welcome Mar 11 14:44:18 um, yes Mar 11 14:44:26 And thanks, yeah. Mar 11 14:44:30 that's, once again, the concept of the limit who's striking here :) Mar 11 14:44:32 whoops, wrong window :D Mar 11 14:44:37 When's the next logic lecture? Mar 11 14:44:41 dude Mar 11 14:44:44 we aren't done Mar 11 14:44:47 OH Mar 11 14:44:48 HAHA Mar 11 14:44:52 he is loving this guys... Mar 11 14:45:00 :) Mar 11 14:45:01 talk forever, and keep him entertained :P Mar 11 14:45:08 Yeah. Mar 11 14:45:13 Please. Mar 11 14:45:24 Well, now, for IceDane's pleasure Mar 11 14:45:33 and mine too! Mar 11 14:45:36 and also for Pixi3 who can't handle anymore math :p Mar 11 14:45:42 HAHA! amen! Mar 11 14:45:56 we shall focus on "philosophical logic paradoxes" :) Mar 11 14:46:06 Wooot. Mar 11 14:46:11 ooh.. i know these! on ward! Mar 11 14:46:28 First of all, the well-known.. paradox of the liar! :) Mar 11 14:46:43 Pixi3, will you make us the honor to tell us what it is? ;) Mar 11 14:47:22 oh no no, go on... i will remember with time :D Mar 11 14:47:54 Hah, ok, well, simply, if I say : "I'm lying", that's a paradox, does anyone see why? Mar 11 14:48:05 Because you're telling the truth. Mar 11 14:48:09 yes Mar 11 14:48:17 You're saying you're lying as you're telling the truth. Mar 11 14:48:59 hm... dont remember this one. very good tho! Mar 11 14:49:46 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. Mar 11 14:49:56 if it is* Mar 11 14:50:11 Yeah :P Mar 11 14:50:12 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 Mar 11 14:50:23 Haha Mar 11 14:50:30 * Cyph3r (opera@shytb0x-97721158.karoo.kcom.com) has left #lecture Mar 11 14:50:36 :) Mar 11 14:50:41 Rofl. Mar 11 14:50:48 Scared/Bored most people off. Mar 11 14:50:55 Haha Mar 11 14:51:16 Well, unlike our first paradox, this one can't be "solved"... Mar 11 14:52:02 Now, let us see another one, the paradox of the barber! :) Mar 11 14:52:15 (by the way) Mar 11 14:52:19 yes? :) Mar 11 14:52:48 (there is a book of logik like that called "the book who make mad") Mar 11 14:52:53 Really? Mar 11 14:52:57 I'mma get that one. Mar 11 14:53:06 (it's funny, but just in french I think) Mar 11 14:53:09 OH Mar 11 14:53:15 haha, gonna have a hard time with that one :P Mar 11 14:53:26 i have googled the barber paradox! very interesting Mar 11 14:53:35 got the definition if someone needs it :) Mar 11 14:53:45 go hard ;) Mar 11 14:53:53 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 Mar 11 14:54:46 ation. Mar 11 14:54:48 :) Mar 11 14:54:53 ation? Mar 11 14:54:54 o.o Mar 11 14:54:57 situation :P Mar 11 14:55:01 oh Mar 11 14:55:23 i like this one :) Mar 11 14:55:24 So yeah, if a barber in a town shaves every man in the town who does not shave himself Mar 11 14:55:30 and no one else Mar 11 14:55:33 but no one else Mar 11 14:55:34 yeah Mar 11 14:55:43 he must shave himself, but cant Mar 11 14:55:46 :) Mar 11 14:55:47 yeah. Mar 11 14:55:48 Haha Mar 11 14:55:51 This is nice :P Mar 11 14:55:53 hairy men! Mar 11 14:55:55 yes, the problem is : who shaves the barber?! Mar 11 14:55:56 :D Mar 11 14:56:04 I didn't know this one Mar 11 14:56:11 well he is supposed to do it, but spends too much time on everyone else :P Mar 11 14:56:16 hairy men! --> this one was especially dedicated to IceDane (he'll know why :p) Mar 11 14:56:18 Yeah :P Mar 11 14:56:28 ewww.... Mar 11 14:56:29 Damn frenchies. Mar 11 14:56:30 x'D Mar 11 14:56:32 LOL! Mar 11 14:56:34 i see now :P Mar 11 14:56:46 we were talking about that the other day in fact! lOL! Mar 11 14:56:51 :) Mar 11 14:57:15 well, now, let me explain the part where they say it relates to "set theory" Mar 11 14:57:24 Here, there are two sets : Mar 11 14:57:39 - the one of the men who shave themselves Mar 11 14:57:51 - the one of the men shaved by the barber Mar 11 14:58:05 but, the barber can't belong to any of them... Mar 11 14:58:15 oh cool! Mar 11 14:58:16 i didnt see that Mar 11 14:58:21 :) Mar 11 14:58:46 Ah Mar 11 14:58:48 Of course. Mar 11 14:58:50 This paradox was introduced by the english philosopher Bertrand Russell, in the 20s. Mar 11 14:58:56 Hmm. Mar 11 14:59:01 Ah Mar 11 14:59:03 I See. Mar 11 14:59:05 haha Mar 11 14:59:07 VERY nice :p Mar 11 14:59:14 Can anyone think of a "solution" to this paradox? Mar 11 14:59:22 Someone else shaves him. Mar 11 14:59:37 Or he's shaved by another barber. Mar 11 14:59:58 Or he just sticks his head in an oven and thus scorches it off. :p Mar 11 15:00:02 HAHAHA Mar 11 15:00:03 LOOL Mar 11 15:00:33 Someone else shaves him. --> impossible, remember : he shaves all those who don't shave themselves ;) Mar 11 15:00:46 he gets really hairy Mar 11 15:00:47 Ah. Mar 11 15:00:48 I see. Mar 11 15:00:56 wait wait! he was bald and doesnt need it! Mar 11 15:00:59 Then it's the oven, I think. Mar 11 15:01:04 :) Mar 11 15:01:18 well, I think that you all got the point that, we need additional informations! :) Mar 11 15:01:38 indeed, although one solution would be to deny the existence of the barber Mar 11 15:01:57 Some people proposed other solutions, e.g. : Mar 11 15:02:09 -The barber's lying (<- no paradox here ;)) Mar 11 15:02:23 -The barber's coming from another town Mar 11 15:02:35 nah, cause if he was lying, he would really be telling the truth :D Mar 11 15:02:55 Lmao. Mar 11 15:02:56 -The barber is not a man, but a woman, a robot or an extraterrestrial Mar 11 15:03:04 Rofl. Mar 11 15:03:11 oooh... aliens Mar 11 15:03:17 barber aliens even Mar 11 15:03:24 just to throw a quick reference in : http://en.wikipedia.org/wiki/Category:Paradoxes very interesting stuff, lists lots of paradoxes Mar 11 15:03:49 * Elda_Winslacks has quit (Ping timeout) Mar 11 15:04:26 elda has left the building! Mar 11 15:05:27 Well, now, we'll see another paradox (dunno if it's listed at Pixi3's link) Mar 11 15:05:51 tell me and i will look :) Mar 11 15:05:57 The crocodile's paradox Mar 11 15:06:15 nah, dont see it Mar 11 15:06:22 ok ;) Mar 11 15:06:43 well, a crocodile kidnaps a baby (poor him! :p) Mar 11 15:07:02 and says to the mother (of the baby :)) : Mar 11 15:07:13 * Elda_Winslacks (loulou@shytb0x-993496FC.w83-192.abo.wanadoo.fr) has joined #lecture Mar 11 15:07:26 "If you guess what I'm gonna do, I give you the baby back, else, I eat it" Mar 11 15:07:33 what would you answer? :) Mar 11 15:07:56 eat it Mar 11 15:08:04 Yeah. Mar 11 15:08:21 cause either way you are screwed i am thinking :P Mar 11 15:08:35 And, indeed, the answer of the mother was : "you're going to eat it!" Mar 11 15:08:48 and, hehe, we've reached a dilemma :D Mar 11 15:08:55 why? Mar 11 15:08:56 well : Mar 11 15:09:44 -if the crocodile eats the baby, then the mother guessed it well, and the crocodile is forced to give her back the baby... Mar 11 15:10:00 Yeah :P Mar 11 15:10:14 -if the crocodile doesn't eat the baby, then the mother was wrong, and the croco. has to eat the baby.. Mar 11 15:10:22 yup yup :) Mar 11 15:10:24 :) Mar 11 15:10:29 Haha, nicey :p Mar 11 15:10:35 so, it is impossible to go through this dilemma Mar 11 15:10:38 tho it would not have helped i dont think had she had said something different in reply Mar 11 15:11:09 The correct thing to do in that situation Mar 11 15:11:26 Is pull an AK47 out of her ass, and blast the fucking croc to jersey. Mar 11 15:11:31 "dress the baby up and play house" would have gotten the kid eaten as well :P Mar 11 15:11:34 haha! Mar 11 15:11:38 if you wanna go that route Mar 11 15:11:48 the bitch should have been watching the kid from the beginning :P Mar 11 15:11:49 ooor Mar 11 15:11:53 the kid would have never been taken :P Mar 11 15:12:00 Tell the croc that the baby just swallowed nitro glycerin. Mar 11 15:12:05 LMFAO! Mar 11 15:12:10 LOOL Mar 11 15:12:19 And that was the reason he was left alone Mar 11 15:12:24 Danger of explosion. Mar 11 15:12:46 haha, you should be a sci-fi novelist :P Mar 11 15:12:53 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 Mar 11 15:13:52 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 Mar 11 15:14:01 Rofl. Mar 11 15:14:05 lmao Mar 11 15:14:13 Lmao Mar 11 15:14:18 Pixie made a joke :D Mar 11 15:14:21 High five, sistah. Mar 11 15:14:25 well, let's get back to rationality :p Mar 11 15:14:38 :) Mar 11 15:14:46 Did I miss anything? Mar 11 15:14:59 To this dilemma, Lewis Caroll proposed a pragmatical solution Mar 11 15:15:45 He said : Mar 11 15:16:08 If the crocodile eats the baby, then the mother is right, and the crocodile is a liar. Mar 11 15:16:33 If he gives her back the baby, the mother is wrong, and the crocodile is a liar. Mar 11 15:16:39 lol Mar 11 15:16:46 Lol. Mar 11 15:16:52 God damn compulsive liar. Mar 11 15:16:54 So, anyway, the crocodile is a liar. Mar 11 15:17:04 Yeah. Mar 11 15:17:11 But still not a solution Mar 11 15:17:13 nah... he was telling the truth the whole time... i talked to him and asked him :P Mar 11 15:17:17 Then, since there's no hope the crocodile will satisfy the honor code Mar 11 15:17:24 Because the baby will be dead either way. Mar 11 15:17:28 Liar or not :P Mar 11 15:17:36 we can't doubt he'll act according to his nature. Mar 11 15:18:01 this is lovely :D not as good as the barber one Mar 11 15:18:08 :) Mar 11 15:18:44 Barbers, crocodiles and greek heroes. Logic is strange. Mar 11 15:18:53 you love it :P Mar 11 15:18:56 * R4d30N has quit (Quit: ) Mar 11 15:18:59 I do.þ Mar 11 15:19:17 :) Mar 11 15:19:41 So, let's see some "paradoxical sentences" ;) Mar 11 15:19:50 "This sentence is false" Mar 11 15:19:57 "It is forbidden to forbid" Mar 11 15:20:06 Haha. Mar 11 15:20:06 "All rules have exceptions" Mar 11 15:20:09 Lmao. Mar 11 15:20:23 "Never say never" is somewhat a paradox too :P Mar 11 15:20:24 oooh... Mar 11 15:20:27 yes ;) Mar 11 15:20:38 also : Mar 11 15:20:55 on the front cover of a book : The sentence on the back cover is true. Mar 11 15:21:10 on the back cover of the same book : The sentence on the front cover is false. Mar 11 15:21:14 :) Mar 11 15:21:17 Nice :P Mar 11 15:21:19 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! Mar 11 15:21:29 hahaha ok ;) Mar 11 15:21:46 so everyone take care, and have a lovely time entertaining my icelandic friend :D Mar 11 15:21:55 elda, it was a pleasure to meet you aswell! Mar 11 15:22:05 Pixi3 : What IceDane says is false. Mar 11 15:22:05 buh bye ppls! Mar 11 15:22:16 thanks, it was also a pleasure for me Mar 11 15:22:17 IceDane : What Pixi3 says is true. Mar 11 15:22:19 What pixie says is false. Mar 11 15:22:31 Ah :P Mar 11 15:22:32 Never mind. Mar 11 15:22:35 what are you guys talking about? :P Mar 11 15:22:45 i am right all the time :P Mar 11 15:22:48 haha, just kidding! Mar 11 15:22:50 bye everyone Mar 11 15:22:52 * Pixi3_1103 has quit (Quit: ) Mar 11 15:22:54 Haha. Mar 11 15:22:55 bye bye Mar 11 15:23:41 Paradoxes FTFW! Mar 11 15:24:02 :) Mar 11 15:24:20 Also, a big aspect in paradoxes is "self-reference" Mar 11 15:25:43 Let me show an example : Mar 11 15:26:28 This sentence contains thirty-eight letters. Mar 11 15:26:43 Ah. Mar 11 15:26:46 now, do you get the concept of self-reference? :) Mar 11 15:26:49 Yeah. Mar 11 15:27:09 If it had less letters, like thirty-six, the sentence would be kind of incorrect. Mar 11 15:27:14 Or well Mar 11 15:27:15 yep :) Mar 11 15:27:17 yeah. Mar 11 15:28:10 The concept of self-reference is mainly used in maths, philosophy, linguistics, and.. last but not least! computer programming :) Mar 11 15:28:38 And, self-reference can lead us to paradoxes :) Mar 11 15:28:44 Recursions ?:P Mar 11 15:29:07 Or hmm.. Does it apply to them ? Mar 11 15:29:20 Nah.. Mar 11 15:29:27 Never mind :P Mar 11 15:30:00 nope Mar 11 15:30:04 not recursions ;) Mar 11 15:31:04 just have a look at : http://www.cgl.uwaterloo.ca/~csk/washington/paper/discussion1.html if you're interested Mar 11 15:32:00 So, let me show an example of a paradox with self-reference : Mar 11 15:32:12 Alright :P Mar 11 15:32:16 "Your answer will be no." Mar 11 15:32:39 Oh Mar 11 15:32:40 I know one.. Mar 11 15:32:47 When you're saying what you're doing. Mar 11 15:32:56 Right ? Mar 11 15:33:07 why? :) Mar 11 15:33:26 As in Mar 11 15:33:28 oh, you meant a self-reference example, then yes Mar 11 15:33:31 When someone asks you what you're doing Mar 11 15:33:33 yeah :P Mar 11 15:33:36 but it's not a paradox ;) Mar 11 15:33:39 Hmm Mar 11 15:33:42 Maybe not. Mar 11 15:34:29 While saying "Your next word will be 'no'", is one ;) Mar 11 15:34:58 can you see why? Mar 11 15:35:03 Yeah Mar 11 15:35:17 okay :) Mar 11 15:35:55 So, through this lecture, we've seen that, logic was far from being perfect, and that we could find some "flaws" in it. :) Mar 11 15:36:00