Mathematical Logic...
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| Author | Topic: Mathematical Logic... |
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Shaper
Member # 32
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written Thursday, September 8 2005 09:40
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Alrighty, time to test your reasoning skills. Supply an answer to each of the following questions. In addition, post your reasoning for said answers. For your assistance, consider the following definitions: N = The set of Natural Numbers (i.e. {1,2,3,4,5...}) R = The set of all Real Numbers (0,1) = The set of all Real Numbers between 0 and 1, not including 0 or 1. Continuum Hypothesis(simplified): There is a set whose size lies between that of the Real Numbers and the Natural Numbers. Poll Information This poll contains 3 question(s). 49 user(s) have voted. You may not view the results of this poll without voting. function launch_voter () { launch_window("http://www.ironycentral.com/cgi-bin/ubb/ultimatebb.cgi?ubb=poll;d=vote;pollid=giNpoknMyaZP"); return true; } // end launch_voter function launch_viewer () { launch_window("http://www.ironycentral.com/cgi-bin/ubb/ultimatebb.cgi?ubb=poll;d=view;pollid=giNpoknMyaZP"); return true; } // end launch_viewer function launch_window (url) { preview = window.open( url, "preview", "width=550,height=300,toolbar=no,location=no,directories=no,status,menubar=no,scrollbars,resizable,copyhistory=no" ); window.preview.focus(); return preview; } // end launch_window   -------------------- Lt. Sullust Cogito Ergo Sum Polaris Posts: 2462 | Registered: Wednesday, October 3 2001 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Thursday, September 8 2005 09:53
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I think I'll wait until some more people have done this before posting my reasoning. Although I'd find proving my answer to the third one rather tricky. -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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Warrior
Member # 5886
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written Thursday, September 8 2005 09:54
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What do you mean by the size descriptors in the poll? Are you counting different degrees of unboundedness? Or by "larger" do you mean that one set is contained in another? :eek: :rolleyes: :cool: -------------------- My friend! Do not fear the flames of the demonic legion! Let the flames incinerate your weakness, strip away all neuroses, and transform you into a newly-forged brazen GOD! Then you will show the subhuman supplicants the true meaning of domination and the proper use of POWER! --Daemon Pavidum Posts: 52 | Registered: Friday, June 3 2005 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Thursday, September 8 2005 10:00
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If he's using the normal definitions, set X is larger than set Y if there is no way to choose a different thing in Y for each thing in X. -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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Warrior
Member # 5886
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written Thursday, September 8 2005 10:23
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Ok, going with that definition of size, I chose smaller for the first two questions, and true for the final question: rational numbers, integers, and whole numbers fit that. I guess in this case it is proof by example. :D [ Thursday, September 08, 2005 10:33: Message edited by: Aranea Hirsuta ] -------------------- My friend! Do not fear the flames of the demonic legion! Let the flames incinerate your weakness, strip away all neuroses, and transform you into a newly-forged brazen GOD! Then you will show the subhuman supplicants the true meaning of domination and the proper use of POWER! --Daemon Pavidum Posts: 52 | Registered: Friday, June 3 2005 07:00 |
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By Committee
Member # 4233
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written Thursday, September 8 2005 10:28
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Given that both sets are infinite, how can you compare infinities? Posts: 2242 | Registered: Saturday, April 10 2004 07:00 |
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Warrior
Member # 5886
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written Thursday, September 8 2005 10:37
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I see that I was the only one who said (0,1) was a smaller set than R. Getting nervous..... :( Ok, maybe the explanation is more subtle than that. Perhaps merely restricting the endpoints to finite values does not make a difference for real numbers because there is always another to choose from in this set, whereas with any type of integers, finite endpoints means a finite set. -------------------- My friend! Do not fear the flames of the demonic legion! Let the flames incinerate your weakness, strip away all neuroses, and transform you into a newly-forged brazen GOD! Then you will show the subhuman supplicants the true meaning of domination and the proper use of POWER! --Daemon Pavidum Posts: 52 | Registered: Friday, June 3 2005 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Thursday, September 8 2005 10:45
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quote: quote: [ Thursday, September 08, 2005 10:46: Message edited by: Khoth ] -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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...b10010b...
Member # 869
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written Thursday, September 8 2005 11:47
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Without giving away too many spoilers, there really should be a third option for the Continuum Hypothesis question. :P -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Shaper
Member # 32
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written Thursday, September 8 2005 12:02
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I agree completely; however, a neither option isn't very appealing... Also, there are infinite infinities, so saying one is 'larger' than the other is alright, if not slighly confusing... -------------------- Lt. Sullust Cogito Ergo Sum Polaris Posts: 2462 | Registered: Wednesday, October 3 2001 07:00 |
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Skip to My Lou
Member # 40
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written Thursday, September 8 2005 12:27
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This depends upon your perception of infinity and measurement. Personally, I say that if infinity is boundless then you cannot have two truly boundless things where one is less boundless than the other. The boundlessness, in fact would render irrelevant measurements of size or boundary since, being infinite, an object cannot be given a set measurement to compare its relative size to something else. However, while your x and y definition of size seems to show R is larger than N it depends upon how you correlate them. Going with the assumption that R is larger than N in that it would correlate R:N, x:y, 1:1, 2:2, 3:3...continuing to infinity but -1, -2, -3... would have no correlation in y than it is possible that R could be said to be larger than N. However, if the correlation does not need to be of matching numbers (for example -456:8, 4:90, instead of strictly 567:567, 24375:24375) then R is not larger than N because, both sets being limitless, it would be impossible to get to the point where you run out of N to correlate to R, meaning it no longer applies to the x:y definition of size. Thus I believe that all sets of infinite numbers are equal in size. Or, rather, it being impossible to compare the size of things that that cannot, by their definition, be given set measurements to compare, I believe that all things that are truly limitless are equally limitless in comparison to all things that are also truly limitless. [ Thursday, September 08, 2005 12:37: Message edited by: Archmage Alex ] -------------------- Take the Personality Test! INTJ 100% 75% 100% 44% Huzzah for the Masterminds! www.Keirsey.com for personality information. The Sloganizer! "Swing your Archmage Alex." Deep down, you wish you were a stick figure. Posts: 1629 | Registered: Wednesday, October 3 2001 07:00 |
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Law Bringer
Member # 2984
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written Thursday, September 8 2005 12:34
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All are the same size, ie. infinite. Or are we going to introduce different sizes of infinity? -------------------- The Encyclopaedia Ermariana <-- Now a Wiki! "Polaris leers down from the black vault, winking hideously like an insane watching eye which strives to convey some strange message, yet recalls nothing save that it once had a message to convey." --- HP Lovecraft. "I single Aran out due to his nasty temperament, and his superior intellect." --- SupaNik Posts: 8752 | Registered: Wednesday, May 14 2003 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Thursday, September 8 2005 12:52
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So far, out of ten people, four seem to have got the firs two questions right. -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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...b10010b...
Member # 869
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written Thursday, September 8 2005 13:18
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Some people in this topic need to read up on Georg Cantor and set theory. -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Law Bringer
Member # 2984
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written Thursday, September 8 2005 13:44
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quote:Am I one of them? Because I was basing my response on what I thought I had understood of that theory. I might have missed the point completely. -------------------- The Encyclopaedia Ermariana <-- Now a Wiki! "Polaris leers down from the black vault, winking hideously like an insane watching eye which strives to convey some strange message, yet recalls nothing save that it once had a message to convey." --- HP Lovecraft. "I single Aran out due to his nasty temperament, and his superior intellect." --- SupaNik Posts: 8752 | Registered: Wednesday, May 14 2003 07:00 |
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BANNED
Member # 6074
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written Thursday, September 8 2005 13:45
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I'm a bit slow, someone tell me the answers. :eek: -------------------- --------------------------------- Live long and prosper. Posts: 84 | Registered: Wednesday, July 6 2005 07:00 |
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Triad Mage
Member # 7
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written Thursday, September 8 2005 14:03
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Khoth gave the answers twice already. -------------------- "At times discretion should be thrown aside, and with the foolish we should play the fool." - Menander ==== Drakefyre's Demesne - Happy Happy Joy Joy desperance.net - We're Everywhere ==== You can take my Mac when you pry my cold, dead fingers off the mouse! Posts: 9436 | Registered: Wednesday, September 19 2001 07:00 |
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Master
Member # 4614
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written Thursday, September 8 2005 16:27
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Yeah, it just seems weird. I like to take the example of an infinite coordinate plane, split into its four quadrants. Quadrant I is infinite in two cardinal directions, it's only boundaries being x=0 and y=0. Is it then really a quarter of a plane, or, since it's infinite, are the whole plane and Quadrant I really the same size? But to my resoning, 4(∞) = ∞. So they're all the same. I don't know if that helped anyone more than previous posts have. :P -------------------- -ben4808 For those who love to spam: CSM Forums RIFQ Posts: 3360 | Registered: Friday, June 25 2004 07:00 |
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Warrior
Member # 5268
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written Thursday, September 8 2005 16:44
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A quick google suggests that all one can hope for on this quiz is consistent answers - not necessarily correct answers. Please disabuse me of this notion if it is wrong (its been over a decade since I did algebra) - Q1 has a clear answer, but the answers to Q2 and Q3 can merely be consistent not 'right'. Godel, Godel, Godel, I made it out of axioms Godel, Godel, Godel, Now Godel I shall invoke Posts: 148 | Registered: Tuesday, December 7 2004 08:00 |
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Shaper
Member # 32
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written Thursday, September 8 2005 18:55
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The first two questions have correct answers... -------------------- Lt. Sullust Cogito Ergo Sum Polaris Posts: 2462 | Registered: Wednesday, October 3 2001 07:00 |
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Law Bringer
Member # 2984
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written Thursday, September 8 2005 21:28
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quote:Wait! That means R *is* larger than N. All natural numbers are real numbers: N ⊂ R ⇔ R = N + A ⇒ |R| = |N| + |A| Where A is defined as the set of all real numbers that are not natural numbers, and |X| is the size of set X. If we now assume that |N| = |R| ⇔ |N| = |N| + |A| ⇔ |A| = 0 ⇔ A = ∅ However: 1.5 ∈ A ⇒ A ≠ ∅ Therefore A is clearly not empty, and therefore the statement N=R cannot be true. Neither can N>R be true, since N is a subset of R. Therefore, the statement N<R must be true, even though they are both infinite. --- Gah, you can tell it's been seven months since I last had a lecture on sets. :rolleyes: And that doesn't explain why it would be possible to prove |(0,1)| < |R| in the same way. Even though I am quite sure Cantor said they'd be equal. [ Thursday, September 08, 2005 21:31: Message edited by: Arancaytar ] -------------------- The Encyclopaedia Ermariana <-- Now a Wiki! "Polaris leers down from the black vault, winking hideously like an insane watching eye which strives to convey some strange message, yet recalls nothing save that it once had a message to convey." --- HP Lovecraft. "I single Aran out due to his nasty temperament, and his superior intellect." --- SupaNik Posts: 8752 | Registered: Wednesday, May 14 2003 07:00 |
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? Man, ? Amazing
Member # 5755
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written Thursday, September 8 2005 21:42
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I have no clue. The furthest I got in math was the following equation. Integral of e to the x = the function of u to the n. *this message sponsored by smaller, samer, nfc so true* Posts: 4114 | Registered: Monday, April 25 2005 07:00 |
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...b10010b...
Member # 869
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written Thursday, September 8 2005 23:28
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Aran, don't use addition on infinite sets. It gives misleading results. :P -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Law Bringer
Member # 2984
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written Thursday, September 8 2005 23:34
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Ah, okay. See how long it's been? I can't even remember that rule... but it's relieving to know where the mistake is. ;) -------------------- The Encyclopaedia Ermariana <-- Now a Wiki! "Polaris leers down from the black vault, winking hideously like an insane watching eye which strives to convey some strange message, yet recalls nothing save that it once had a message to convey." --- HP Lovecraft. "I single Aran out due to his nasty temperament, and his superior intellect." --- SupaNik Posts: 8752 | Registered: Wednesday, May 14 2003 07:00 |
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Shaper
Member # 32
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written Friday, September 9 2005 02:34
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Incidentally, the one that I believe is harder to see more people have gotten correct... [ Friday, September 09, 2005 02:35: Message edited by: Lt. Sullust ] -------------------- Lt. Sullust Cogito Ergo Sum Polaris Posts: 2462 | Registered: Wednesday, October 3 2001 07:00 |
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