asbestosman Posted January 27, 2012 Posted January 27, 2012 You know, I've had to reflect on my behavior on this thread. Am I committing one of the Cardinal Sins? 1
Tarski Posted January 27, 2012 Posted January 27, 2012 You know, I've had to reflect on my behavior on this thread. Am I committing one of the Cardinal Sins? Wasting time on something hopeless?
Rob Osborn Posted January 27, 2012 Author Posted January 27, 2012 You know, I've had to reflect on my behavior on this thread. Am I committing one of the Cardinal Sins? Well, we are told that God can forgive men of all their sins because of his Infinite mercy 1
Rob Osborn Posted January 27, 2012 Author Posted January 27, 2012 Because it is obvious and everyone here sees it but you, including 3 mathematicians.1/3 has an infnitely decimal representation while none of your rows is infinite. All of the numers on your list differ from 1/3.To be 1/3 is not the same as being almost 1/3. Equal is equal.I don't see where I am not seeing it. I am saying basically the same thing all along. If my list isn't infinite then neither can be Cantor's- they are the same list. Holy cow!It is also ridiculous for the reason that you are trying to make something pedantic out of the poetic writtings of some ancient bronze age priests.It is almost as if I said to my wife that I asked her countless times to not burn the toast and then she starts talkning about cardinality to prove me wrong.It is all silly no matter how you look at it.It is silly to bring up mathematical notions in connection with those scriptures but then when you do it anyway, you do it crazy wrong.Your fear of perfectly finite and exact notions of infinity seems to be driving you into some kind of batty denialism.I can one up you in nuttiness. Watch this.:Hey Rob, ya know what? Not only is there no such thing as infinity, there is no such thing as three. All I see are some apples, some dollars and some shapes on a piece of paper. No one has every actually seen or proved "three" itself so it doesn't exist.Hey, it isn't me who has the ridiculous notion that one infinity can be bigger than another. Cantor was up in the night and so are all other mathematicians who can't see that folly.
asbestosman Posted January 27, 2012 Posted January 27, 2012 Rob, read this slowly: Cantor. Does. Not. Make. A. List.
asbestosman Posted January 27, 2012 Posted January 27, 2012 Wasting time on something hopeless?Sloth, then? I was thinking maybe gluttony--as in becoming a glutton for punishment. Heaven forbid it might be the universal sin starting with a P.
mfbukowski Posted January 27, 2012 Posted January 27, 2012 You know, I've had to reflect on my behavior on this thread. Am I committing one of the Cardinal Sins? No, clearly it is an Ordinal Sin. 1
Analytics Posted January 27, 2012 Posted January 27, 2012 Rob, one thing you need to realize is that a list with an infinite number of rows has no "infinitieth" row (in other words, no last row). If any number you claim is on your list occurs on the infinitieth row, it actually isn't on your list at all. That's why we claim that 1/3 isn't on your list even though there's a perfectly sensible way to represent 1/3 in decimal notation. Yes, it has an infinite decimal expansion of 3's.This gets to the point of it. When mathematicians say a list is infinitely long, they simply mean that it has no end. So, if Rob were to concede that there is no finite distance down his list that contains exactly 1/3, he would be conceding that 1/3 is not on his list at all.
Rob Osborn Posted January 28, 2012 Author Posted January 28, 2012 Rob, read this slowly: Cantor. Does. Not. Make. A. List.But he sure seems to be able to make a number that somehow finishes out of an infinite list of numbers......Thats ilogical.
Rob Osborn Posted January 28, 2012 Author Posted January 28, 2012 This gets to the point of it. When mathematicians say a list is infinitely long, they simply mean that it has no end. So, if Rob were to concede that there is no finite distance down his list that contains exactly 1/3, he would be conceding that 1/3 is not on his list at all.Look everyone, I never said the perfect decimal expansion of 1/3 was in my list. Some keep arguing this ridiculous point. My list will contain every real number possible between 0-1. Give it up already folks. You guys are arguing a mute point that has nothing to do with what I am stating. Guess what? There are no octagons in my list either just as their are no horses or crabs.
JDave Posted January 28, 2012 Posted January 28, 2012 Look everyone, I never said the perfect decimal expansion of 1/3 was in my list. Some keep arguing this ridiculous point. My list will contain every real number possible between 0-1. Give it up already folks. You guys are arguing a mute point that has nothing to do with what I am stating. Guess what? There are no octagons in my list either just as their are no horses or crabs.But if you change your list to be in base 3, then the perfect expansion of 1/3 is in your list -- it's the first item in your list. So how can it show up in base 3, yet you say that it isn't even a possible real number?
Tarski Posted January 28, 2012 Posted January 28, 2012 Look everyone, I never said the perfect decimal expansion of 1/3 was in my list. Some keep arguing this ridiculous point. My list will contain every real number possible between 0-1. Give it up already folks. You guys are arguing a mute point that has nothing to do with what I am stating. Guess what? There are no octagons in my list either just as their are no horses or crabs.Huh??But 1/3 is a real number between 0 and 1. It isn't listed. Nothing in your list is simply 1/3. Making a list that include approximations to 1/3 isn't the same as making a list that actual includes 1/3.But the rationals ARE countably infinite anyway so there is apprarently no end to your silliness.A list that includes all rationals is possible and the fact that your supposed list of real numbers doesn't even include all the rationals is just a hilarious sidetrack. It only shows that Rob can't even demonstrate that the rationals can be listed --which really is possible.In summary,1) The rationals can be shown to be countably infnite (listable) but Rob can't even do that since he misses even 1/3.2) The reals between 0 and 1 are not countably infinite --no list exists. But Rob thinks his suggest list does this impossible task while at the same time admitting that 1/3 itself isn't really in the list but only approximations. His list doesn't even properly accomplish the possible let alone the impossible.
Rob Osborn Posted January 28, 2012 Author Posted January 28, 2012 (edited) But if you change your list to be in base 3, then the perfect expansion of 1/3 is in your list -- it's the first item in your list. So how can it show up in base 3, yet you say that it isn't even a possible real number?It is all an abstract- it is a way of conveying "language" so that we can define certain aspects of something. Let's take 1/3 for example. If I had a pie and I wanted to cut it into thirds I could use ddegrees of 120 each to cut the pie. But the numerical value of 120 won't show up in the list I previously defined of numbers between 0-1. If I was using weights in ounces and say our pie weighed 36 ounces I could weigh each piece until each weighed 12 ounces. But the numerical value "12" it being 1/3 of the pie, will not show up in my list either. The languages we use in the abstract (such as numbers) sometimes do not perfectly translate into other abstract math symbols very well even though every number will be represented in each abstract sytem we use. In the decimal system of base 10, this is just one case where 1/3 doesn't perfectly translate from one abstract sytem into another. We can come close- .3333333... but that is all. When i ask for a piece of pie and I want 1/3 of it I dont say "please give me. 333333...no more and no less". The person cutting the pie is going to probably translate that abstract into the abstract of degrees which best suits the situation. Thus he will, in his mind, cut 120 degrees of pie and give it to me.As this all relates, my list will have the number .3333... in it and it will also have it to whatever finite degree one wants to use it as for someone who wishes to use a base 10 system as it translates itself into 1/3. Edited January 28, 2012 by Rob Osborn
Rob Osborn Posted January 28, 2012 Author Posted January 28, 2012 Huh??But 1/3 is a real number between 0 and 1.It isn't listed. Nothing in your list is simply 1/3. Making a list that include approximations to 1/3 isn't the same as making a list that actual includes 1/3.But the rationals ARE countably infinite anyway so there is apprarently no end to your silliness.A list that includes all rationals is possible and the fact that your supposed list of real numbers doesn't even include all the rationals is just a hilarious sidetrack. It only shows that Rob can't even demonstrate that the rationals can be listed --which really is possible.In summary,1) The rationals can be shown to be countably infnite (listable) but Rob can't even do that since he misses even 1/3.2) The reals between 0 and 1 are not countably infinite --no list exists. But Rob thinks his suggest list does this impossible task while at the same time admitting that 1/3 itself isn't really in the list but only approximations. His list doesn't even properly accomplish the possible let alone the impossible.You should read my last post I just posted. I cover that abstract language there. But to summarize, my list of all possible numbers between 0-1 are certainly all there, it's just that some abstract math symbols do not translate themselves from one system "perfectly" to another abstract math symbol system perfectly. Your argument is pretty much this- "This sentence is false."
asbestosman Posted January 28, 2012 Posted January 28, 2012 But he sure seems to be able to make a number that somehow finishes out of an infinite list of numbers......Thats ilogical.He describes how to do it in principle if someone were to claim he had a description of such a list.
JDave Posted January 29, 2012 Posted January 29, 2012 As this all relates, my list will have the number .3333... in it and it will also have it to whatever finite degree one wants to use it as for someone who wishes to use a base 10 system as it translates itself into 1/3.Suffice it to say that the concept of achievable precision within engineering does not apply to mathematics and no mathematician will agree that .333333... carried out to N decimal places and then truncated is the same as .33333... with the last digit repeated.
Rob Osborn Posted January 29, 2012 Author Posted January 29, 2012 He describes how to do it in principle if someone were to claim he had a description of such a list.So....he doesn't really prove anything then? That's what I thought also.
Rob Osborn Posted January 29, 2012 Author Posted January 29, 2012 Suffice it to say that the concept of achievable precision within engineering does not apply to mathematics and no mathematician will agree that .333333... carried out to N decimal places and then truncated is the same as .33333... with the last digit repeated.No "infinite" math is ever used in the real world. That is why this topic is so ironic. We speak of infinity as if it exists...only- it neither doesn't nor can't exist. It reminds me of that stupid Hotel paradox with infinite rooms all filled with an infinite amount of people. Silly mind games they are, but in reality, infinity as an actual thing is nowhere to be found in anything we know or work with including solving math problems. How ironic is that? It's more probable that we will evolve into little blue smurfs than to actually prove the size of an infinity.
Zeta-Flux Posted January 29, 2012 Posted January 29, 2012 No "infinite" math is ever used in the real world. That is why this topic is so ironic. We speak of infinity as if it exists...only- it neither doesn't nor can't exist. It reminds me of that stupid Hotel paradox with infinite rooms all filled with an infinite amount of people. Silly mind games they are, but in reality, infinity as an actual thing is nowhere to be found in anything we know or work with including solving math problems. How ironic is that? It's more probable that we will evolve into little blue smurfs than to actually prove the size of an infinity.Be that as it may, our conceptualization of the infinite *has* led to math that is used in the real world. On a fundamental level it may simply involve the manipulation of a finite number of symbols, but the conceptualization itself has been quite helpful. You may think of them as "silly mind games" but in fact they represent fundamental truths about logic, and useful conceptualization for people to understand. Calculus has been one of the most successful innovations in terms of understanding the universe around us, and it is based on picturing the infinite. Maybe there is no such thing as the infinite, but I find your unwillingness to even try to picture such a notion, and your ridicule of those who do, to be closed-minded.Best wishes,Zeta-Flux
Rob Osborn Posted January 29, 2012 Author Posted January 29, 2012 Be that as it may, our conceptualization of the infinite *has* led to math that is used in the real world. On a fundamental level it may simply involve the manipulation of a finite number of symbols, but the conceptualization itself has been quite helpful. You may think of them as "silly mind games" but in fact they represent fundamental truths about logic, and useful conceptualization for people to understand. Calculus has been one of the most successful innovations in terms of understanding the universe around us, and it is based on picturing the infinite. Maybe there is no such thing as the infinite, but I find your unwillingness to even try to picture such a notion, and your ridicule of those who do, to be closed-minded.Best wishes,Zeta-FluxTrust me, I have tried to picture the notion of infinity on many levels. I even came up with my own infinity mind game which I am sure I have related on numerous accounts- the box of finite size containg ever decreasing balls of infinite count.My own ridicule of infinity comes from working out the problems of infinity in my own mind. One of the prevailent paradoxes for me is the size of the universe- does it truly go on forever or does it have an end or edge to it? How does God measure these things? Is his mind capable of an infinite quality to comprehend something actually numberless? I keep coming back to the same fundamental principles that make up the foundations of math and logic. Those principles tell me that there is no "infinite" that can be known so it is futile to ever conceive of a way to work within that framework. Numbers themselves only have value as they relate to other numbers of known values. The abstract of infinity has no case or merit in math other than establishing it as the capacity for numbers to be without limit.I understand that the scriptures use notions of infinity as a means to just say a really big number, or a number so large it may as well be numberless like as is spoken when referring to the sands on the seashore. In all forms of math, one can either believe or not believe in the actual infinite and it won't effect their ability to work within the numbering system whatsoever. There are no mathematical proofs for infnity to actually exist. We have proofs showing the numbering system can continue forever, but there is no proofs showing that numbering system to have limits or ends to it's size.When mathematicians speak of one infinity being of different size than another infinity it raises all kinds of paradoxial questions and endless contradictions. Dilbert's hotel paradox is but one example that is self contradicting. Sure it's amusing to think in such an abstract manner, even I enjoy a good paradox. But where the rubber meets the road, we have to put away such ludicrous notions about these imaginations that have no physical place in the reality of all that is known and tangable. Everyone knows that it is impossible to have a hotel with infinite rooms and then to have them all full is a contradiction. I know this is just to relate number abstracts but we also know that it is impossible to actually have a number that is infinite in numerical length. We all know that any number within any set has only finte attributes no matter how endlessly long we might think it to be.As for the natural counting numbers, we know it is an abstract of constantly, endlessly repeating powers of ten- it just repeats over and over into endless abstract. We may speak of "every" number within this system but we only do so in the manner of it being a set containing every limitless neverending count of ten that adds itself over and over forever. There is thus no actual size to natural numbers, they just go on forever having no end. At the same time, we all realize that every number conceivable within our numbering system is finite in length and will never actually reach a point of infinity.The universe can be constructed in much the same way- it may go on forever- it truly may, but we have no capacity to know all of it if this is the case. All we shall ever know, I propose, is that we will always have only a knowledge of finite distances and finite matter in the universe, even when we become Gods ourselves.
asbestosman Posted January 29, 2012 Posted January 29, 2012 So....he doesn't really prove anything then? That's what I thought also.He proves something is impossible in principle. If he can't prove it is impossible in principle with a thought experiment, why do you think you can prove something is impossible with your ball and box example?
Rob Osborn Posted January 29, 2012 Author Posted January 29, 2012 (edited) If you break down the principle of Cantor's diagonal argument, he can't ever stop making his number because the moment he does he loses, or, shows that infinity is really finite. So, all he is showing in reality is that something infinite continues on forever and can't be known. There is nothing new or novel about that. Mathematicians and philosophers knew that principle long before he came on the scene. There is nothing different with his argument than the two of us playing a game where you first choose a number and then I will think of another that is different. We could do that forever and never exhaust all the natural numbers. That principle, which is the same as Cantor's just shows you can always add one more number forever and ever. Suppose we were to look at Cantor's argument a little different- change the rules slightly but still end up with the same effect. Suppose that we were to build the list in turns. First a number is posted, and then Cantor chooses his. In the next turn we would try to duplicate Cantor's but as you can see, he always has the option of changing his after mine is placed so that his is always different. Let's just use 1's and 0's for this experiment because this will show the weakness of Cantor's argumnet.I choose .100000... Cantor chooses .0I choose .010000... Cantor chooses .00I choose .001000... Cantor chooses .000I choose .000100... Cantor chooses .0000I choose .000010... Cantor chooses .00000I choose ,000001... Cantor chooses .000000As you can see in the pattern Cantor's new number is .0... while my new number multiplys by the power of ten forever and ever. My number is always converging towards Cantor's also. Let's do it again but this timewe shall add one respectively each time and I shall always choose his previous number-I choose .000000... Cantor chooses .100000...I choose .100000... Cantor chooses .110000...I choose .110000... Cantor chooses .111000...I choose ,111000... Cantor chooses .111100...I choose .111100... Cantor chooses .111110...I choose .111110... Cantor chooses .111111...This pattern really shows how ridiculous this argument is and perhaps may be the nail in the coffin to his argument. Logic tells us that If I was to make my list "all at once" in this pattern I would have in my list this number- .1111... But, that is Cantor's new number- .1111... which was supposed to be a different number than any in my list!This last set really shows that all Cantor is doing is adding one more to his list than mine- he is just ahead one step in the number game than me. Doing it "all at once" shows the obvious contradiction and proves Cantor's diagonal argument to be indeed false. Now certainly there must be a contradiction in his argument. What say ye? Edited January 29, 2012 by Rob Osborn
asbestosman Posted January 29, 2012 Posted January 29, 2012 I choose .000000... Cantor chooses .100000...I choose .100000... Cantor chooses .110000...I choose .110000... Cantor chooses .111000...I choose ,111000... Cantor chooses .111100...I choose .111100... Cantor chooses .111110...I choose .111110... Cantor chooses .111111...This pattern really shows how ridiculous this argument is and perhaps may be the nail in the coffin to his argument. Logic tells us that If I was to make my list "all at once" in this pattern I would have in my list this number- .1111... But, that is Cantor's new number- .1111... which was supposed to be a different number than any in my list!This last set really shows that all Cantor is doing is adding one more to his list than mine- he is just ahead one step in the number game than me. Doing it "all at once" shows the obvious contradiction and proves Cantor's diagonal argument to be indeed false. Now certainly there must be a contradiction in his argument. What say ye?Tell me what row .11111... appears on in your list if it's made all at once. Then we'll take it from there.
Rob Osborn Posted January 30, 2012 Author Posted January 30, 2012 Tell me what row .11111... appears on in your list if it's made all at once. Then we'll take it from there.Aha! Now we are getting to the heart of the contradiction-my favorite part! With that set-.000000....100000....110000....111000....111100... ..111110...In that pattern cantor's number should be .1111... right? The problem is however that his number must be made from my list which of course follows the pattern given. Now of course we both know that nothing really counts to an actual infinity. Cantor's number though assumes that the set is finished or different than "every" number on the list. But, following the pattern we all know that one will never truly achieve an infinite number, only a finite number. So, Cantor's new number proves to be nothing more than just .1111... but carried out to only a finite limit according to my list. This thus proves the contradiction. In order for Cantor's number to be different, it must end in at least one more "1" than the list I gave. But, the list I gave has this pattern repeated forever- it can't end, so this translates into the false claim that his number can somehow be longer in repeating 1's than any number in my list! So, all Cantor's number shows is that he can't have it both ways. He can't claim his number is different because his number differs in nth place of nth row.Basically, Cantor never gets a number ".1111..." just as there is no ".1111..." in my list either. Yes, we both have the number .1111 carried out to the nth placement. It's kind of like saying- "who can make the longest string of this pattern- .11111. All that this shows is that we both are making the same number except that my operation happens before his operation which allows him to place one more "1" at the end. But wait...there is no end and thus- the contradiction. Like was said a few times before, this is the same exact thing as saying "This sentence is false."
asbestosman Posted January 30, 2012 Posted January 30, 2012 No Rob. Cantor's number would be .11111... (done instantly) while your list would never contain .1111... The reason your list does not contain .111... is that it would have to be the last row of your list, but there is no last row. I tried to explain that earlier. Analytics tried to bring it to your attention again. There is no infinitieth row (no last row). That's why we know your list doesn't contain .1111... even though Cantor's number derived from your list would be precisely .1111...Anyhow, I think we're back to our infinite loop.
Recommended Posts