asbestosman Posted January 30, 2012 Posted January 30, 2012 The new number, which you agreed with is best represented as ".1111..." Beacuse the new number can only be constructed with the same amount of rows, then there must be infinite rows that were counted in the number.Yes.That being the case, the last row that was counted to get the new number- ".1111..." also had a count of ".1111..." in it otherwise the proof is wrong.No. The words, "the last row" are nonsense. There is no last row.If the new number really is ".1111..." as you admit then one must also admit that it has counted an infinity of rows.Yes.If we are to give the diagonal count expressed as "n" and the count of rows as "x" then it stands to reason that n=x, they have the same cardinality.Yes, the cardinality of rows in your list is the same as the cardinality of the columns (amount of 1's) in the diagonal. In this case the cardinality is aleph_0 or countable infinity.But, just as there are an endless supply of integers (all of which are finite), there is an endless supply of of rows in your list which contain a finite number of 1's followed by an endless supply of 0's.
asbestosman Posted January 30, 2012 Posted January 30, 2012 Rob gave me an interesting idea. Next time you want to punish a kid, put him in a round room and tell him to go stand in the corner. Once he reaches there, tell him to count backwards starting at the last integer down to--say--the last integer minus 42.
Rob Osborn Posted January 30, 2012 Author Posted January 30, 2012 AS anyone can see, the proof I have given shows the absurdity of infinity. It has also shown the obvious contradiction that a new number is never actually created from a set of infinite numbers. The diagonal argument in that proof is a fraud. There is no new number formed not already in the list. What does this mean for math and set theory? That there is no ridiculous thing as one infinity bigger than another. Oh, and throw out the continuam hypothesis also because there isn't two sizes of infinity.
asbestosman Posted January 30, 2012 Posted January 30, 2012 As anyone can see, the proof Rob has given shows the absurdity of treating infinity as though it were finite. It has also shown the obvious contradiction that a new integer is never going to be found which is larger than all other integers and that trying to specify the largest integer makes as much sense as finding the corner of a circle.
Tarski Posted January 30, 2012 Posted January 30, 2012 (edited) AS anyone can see....Anyone can see? LOLWho does see it your way? Have you convinced anyone? Name one.In fact, anyone who has been following this can see clearly how humorously confused your arguments are.You are tilting at windmills.Note for those reading this that are in their right mind; This whole thing is perhaps an object lesson. It is impossible to make progress in making sense out of the notion of infinity if one is going to argue informally and rely on untutored intuition. Definition, axiom, lemma, theorem; that's the only way to go. Rob is mostly hung up at step one. He can't wrap his mind around some simple definitions; set, relation, equivalence class, function, bijection, cardinality, etc.It is even quite clear that he is entirely unaware of what goes into making sense of the idea of a real number (as opposed to a natural number or a rational number). He is unaware of the need for something like Dedekind cuts or equivalence classes of Cauchy sequences for even putting the reals on firm ground? Just what is a real number Rob? Have you been arguning about the reals without even knowing what they are?We haven't had a better example of someone not knowing what they are talking about in a long time. The absurdity of his incorrigible assumption that he can out-think the experts and giants in the field is matched only by those who think they can set climates scientists straight on climate change or set biologists straight about evolution. There is another lesson in this about having respect for expertise in a hard science. Apeal to authoritative concensus may not work as a step in a deductive argument, but it is a rational policy when trying to get the truth on complex issues that one hasn't the time or training to tackle on one's own. Edited January 30, 2012 by Tarski
Rob Osborn Posted January 30, 2012 Author Posted January 30, 2012 Tarski, if you haven't been taking notes, look back at the past two pages with the latest arguments being presented. With asbestosman he has been trying pretty hard to defend a simple contradiction which he thinks must be true but at the same time can'yt be true. I see that you yourself have had nothing to say on that matter. Does it have you stumped as well? I will spell it out real easy for you-Is the new number in Cantor's argument the same size as the amount of rows? They obviously must be the same size as one is made from the other. If this is so, is the new number without end? It certainly must be so. If that is the case, which it certainly must, then there is a basic contradiction with my new list I proposed and Cantor's argument.
Zeta-Flux Posted January 30, 2012 Posted January 30, 2012 Rob,You seem to be using "row" in two ways. Each number appears on a given row. Each number has an infinite *decimal* expansion. The number of *digits* is infinite- and you seem to refer to the digits as "rows" also. Note: the number of *non-zero* digits in any given number may be finite.
Rob Osborn Posted January 30, 2012 Author Posted January 30, 2012 Rob,You seem to be using "row" in two ways. Each number appears on a given row. Each number has an infinite *decimal* expansion. The number of *digits* is infinite- and you seem to refer to the digits as "rows" also. Note: the number of *non-zero* digits in any given number may be finite.I admit that I don't use math lingo very well. By "rows" I am speaking of the numbers in the list as it goes from top to bottom. The new number is made by taking the opposite of each number starting in the first column (columns move left to right/ decimal placement expansion) and moving to the right one column at a time as one also goes down one row each step. Thus, in the new number, the number of columns count will match the number of rows count/ the new number will have the same cardinality (counting each placement number left to right) of the amount of rows going top to bottom.I hope I explained that better.
Tarski Posted January 30, 2012 Posted January 30, 2012 Tarski, if you haven't been taking notes, look back at the past two pages with the latest arguments being presented. With asbestosman he has been trying pretty hard to defend a simple contradiction which he thinks must be true but at the same time can'yt be true. I see that you yourself have had nothing to say on that matter. Does it have you stumped as well? I will spell it out real easy for you-Asbestosman, Zeta-Flux and myself are all on the same page. We all agree. I don't see anything wrong with anything ABman has said. You say wrong or unclear things in every post.Is the new number in Cantor's argument the same size as the amount of rows? Each row (read left to right) gives rise to one digit in the decimal expansion of the new number. So, unless you mean something besides what I think you mean then yes, the new number is the same "size" as the "number of rows". To be precise, the rows form a countably infinte list. The number of digits in the new number is countably infinite as it is for all real numbers when represented fully in this form.They obviously must be the same size as one is made from the other. You are being imprecise about size here but I m assuming that you are referring to cardinality (aleph_0 in both cases)If this is so, is the new number without end?All real numbers have a full decinal expansion that is infinite since even for for those that seem to end, there isstill all those endless zeros.So yes the new number has an infinite decimal expansion in that sense but whether or not it is all zeros after a certain point, depends on the original list.It certainly must be so. If that is the case, which it certainly must, then there is a basic contradiction with my new list I proposed and Cantor's argument.Yet you don't say what the contradiction is. What is it?By the way, Cantor's proof can be seen as a proof by contradiction. Namely, the assumption that one CAN list all the real numbers leads to a contradiction. Therefore one cannot list all the real numbers. Do you realize this?
Rob Osborn Posted January 31, 2012 Author Posted January 31, 2012 Asbestosman, Zeta-Flux and myself are all on the same page. We all agree. I don't see anything wrong with anything ABman has said. You say wrong or unclear things in every post.Each row (read left to right) gives rise to one digit in the decimal expansion of the new number. So, unless you mean something besides what I think you mean then yes, the new number is the same "size" as the "number of rows". To be precise, the rows form a countably infinte list. The number of digits in the new number is countably infinite as it is for all real numbers when represented fully in this form.You are being imprecise about size here but I m assuming that you are referring to cardinality (aleph_0 in both cases)All real numbers have a full decinal expansion that is infinite since even for for those that seem to end, there isstill all those endless zeros.So yes the new number has an infinite decimal expansion in that sense but whether or not it is all zeros after a certain point, depends on the original list.Yet you don't say what the contradiction is. What is it?By the way, Cantor's proof can be seen as a proof by contradiction. Namely, the assumption that one CAN list all the real numbers leads to a contradiction. Therefore one cannot list all the real numbers. Do you realize this?In myoriginal list, the one that started counting at .0, .1, .2, .3, etc and went on accounting for every possiblity as it built going from left to right, this in principle would be bale to cover every number because every possibility would be covered for every new column left to right. So regardless of Cantor making his number from my list, putting it in the proper rules of going through each possibility for each new column first, he can never build a number not already in the list. So, I present this new list of "0's" and "1's" and set it up to where we are both building the same number and suddenly the rules change. Logically, in my list, I could represent a never ending string of .1111's in my list because it always goes on forever or converges to infinity along that pattern. The new number being created or "done at once" must finish in order to even be a number. So in reality, even though both numbers converge on or towards infinity, in fact they never really ever get there- they can't, they can be seen as just going on forever.The "contradiction" I am showing here is that if the new number being created is best represented as ".1111..." then it must have already gone through a number in my list that was infinite in length because they would have the same exact cardinality. The contradiction here is that if any or all of my numbers in my list is only finitely long in the "1" count pattern, then the new number can't be represented as ".1111..." which is the contradiction and thus the new number really is in my list and not a new number at all.
Rob Osborn Posted January 31, 2012 Author Posted January 31, 2012 Let's get one point or principle straight here. There is no "all" in a set of something of infinite count. This is really a matter of philisophical debate because mathematics cannot provide a proof to discuss infinity. We tend to keep speaking about an infinite set as being known or complete as if we were to give it some finite quality or value. It's like saying that we only alot so much time for a computer to count and when it gets to a certain point that is all and thus the complete set. Yet, we can't define infinity this way- in some finite sort of way. Nothing can count an "all" in infinity. It flat out doesn't exist, it can't exist. We can't give words that define limits to something defined as unlimitable. To thus say that one infinity is bigger than another automatically places a limit on the one in its ability to keep counting. But that is a basic and fundamentally flawed paradigm We break the very rules of number theory when we do such things. An infinite count never has any limits for waht it can count. It can literally count everything in the universe including any abstract number itself- even all the numbering systems combined- it can count them all and never climb to any percentage of being complete or full.
Analytics Posted January 31, 2012 Posted January 31, 2012 The "contradiction" I am showing here is that if the new number being created is best represented as ".1111..." then it must have already gone through a number in my list that was infinite in length because they would have the same exact cardinality. The contradiction here is that if any or all of my numbers in my list is only finitely long in the "1" count pattern, then the new number can't be represented as ".1111..." which is the contradiction and thus the new number really is in my list and not a new number at all.I don't understand what Cantor's diagnol argument has to do with the fact that 1/9 is not on this series of 0.1, 0.11, 0.111, etc. Going back to Cantor's diagnol argument proving that it is impossible to list all real numbers between 0 and 1, the argument goes like this.1- Assume that a list of all real numbers between 0 and 1 exists2- A number between 0 and 1 not on the list can be created by analyzing the diagnols3- The creation of the number in step 2 contradicts the assumption in step 1Mainstream mathematics claims that since their is a contradiction, the assumption (in step 1) must be false.Rob claims that since we already established in step 1 that all real numbers are on the list, the number proposed in step two must be there too.
Analytics Posted January 31, 2012 Posted January 31, 2012 We tend to keep speaking about an infinite set as being known or complete as if we were to give it some finite quality or value....Rather than discussing the real numbers as being "infinite", as if "infinity" were a real number, do you think we could say that they just keep on going on and on and on, in an orderly fashion, without end?
asbestosman Posted January 31, 2012 Posted January 31, 2012 (edited) Rob, I don't know if this helps (hope beyond hope) but one reason that the diagonal-derived number of your modified list (what we call Cantor's number) doesn't actually appear on your list is that this diagonal number is defined from the limit of a sequence. It has to be the limit since there is no end to the number of rows from which it is derived, and it is a single number which we wish to obtain as the answer. The limit of a sequence need not actually occur within the sequence. The sequence 0.1000000, .110000, .1110000, .11110000, . . . converges to 1/9, but none of the terms in the sequence will actually equal 1/9. The sequence 1/2, 1/4, 1/8, 1/16, 1/32, . . . converges to 0, but none of the terms are equal to 0. There is no smallest positive (greater than zero) rational number. There is no contradiction here. It's a bit counter-intuitive, but there is no problem with it.Now, it is true that mathematicians can't prove the existence of infinity. Instead we take it as an axiom (at least most do) which means most of us find the idea to be self-evident (and that it cannot be derived from the other axioms). Furthermore, we can show consistent results with it and find the concept to be very useful. Edited January 31, 2012 by asbestosman
Rob Osborn Posted January 31, 2012 Author Posted January 31, 2012 I don't understand what Cantor's diagnol argument has to do with the fact that 1/9 is not on this series of 0.1, 0.11, 0.111, etc.Going back to Cantor's diagnol argument proving that it is impossible to list all real numbers between 0 and 1, the argument goes like this.1- Assume that a list of all real numbers between 0 and 1 exists2- A number between 0 and 1 not on the list can be created by analyzing the diagnols3- The creation of the number in step 2 contradicts the assumption in step 1Mainstream mathematics claims that since their is a contradiction, the assumption (in step 1) must be false.Rob claims that since we already established in step 1 that all real numbers are on the list, the number proposed in step two must be there too.But do you honestly think it is fair that the rules are made in such a manner that the Cantor number being created doesn't have to go through every possible outcome for each new column before it moves on? My fist list showed that logically you could set up an ordered list that contained every possibility for every column nad that it would be impossible to generate a new number from the start that didn't already have that sequence. My last list with the 1's and 0's was made in such a manner that showed logically that the list would generate the same number as is being created in the new Cantor number. That is a contradiction.
Rob Osborn Posted January 31, 2012 Author Posted January 31, 2012 Rob, I don't know if this helps (hope beyond hope) but one reason that the diagonal-derived number of your modified list (what we call Cantor's number) doesn't actually appear on your list is that this diagonal number is defined from the limit of a sequence. It has to be the limit since there is no end to the number of rows from which it is derived, and it is a single number which we wish to obtain as the answer. The limit of a sequence need not actually occur within the sequence. The sequence 0.1000000, .110000, .1110000, .11110000, . . . converges to 1/9, but none of the terms in the sequence will actually equal 1/9. The sequence 1/2, 1/4, 1/8, 1/16, 1/32, . . . converges to 0, but none of the terms are equal to 0. There is no smallest positive (greater than zero) rational number. There is no contradiction here. It's a bit counter-intuitive, but there is no problem with it.Now, it is true that mathematicians can't prove the existence of infinity. Instead we take it as an axiom (at least most do) which means most of us find the idea to be self-evident (and that it cannot be derived from the other axioms). Furthermore, we can show consistent results with it and find the concept to be very useful.I understand what you are saying, it just seems that we have to break a few rules in the process but that somehow that is ok because after all- we are dealing with infinite sets which make their own unprovable rules.
supersnail Posted February 1, 2012 Posted February 1, 2012 http://i.imgur.com/OVzmY.jpgThe non-textual part of that isn't an artistic depiction. That is an image of a distant galaxy with numerous other actual galaxies in the background. "World" doesn't always mean "planet," but our growing astronomical and physical knowledge may provide some perspective. Keep in mind that matter isn't necessarily limited to what we can observe of this universe.If God and matter are both uncreated, maybe the worlds that God has created from matter are finite at any given time, but endless in the long run. Maybe "indefinite" or "limitless" would be clearer than "infinite."Such a God is still worthy of worship. Perhaps some questions are easier to answer or get over after one has overcome prior commitments to Aristotelianism and Platonism. I don't understand why the question of whether God has in some sense already created an infinity of worlds is particularly important. If God didn't have such power or characteristics, would He somehow be less worthy of worship? There is nothing like God--the Godhead of the Father, Jesus Christ, and the Holy Ghost.We weren't really aware of the existence of other galaxies even a hundred years ago, much less millions of galaxies. In fact, there are hundreds of billions of galaxies in this universe. Many "orthodox" Christians have a problem even with the idea of one other planet that could be inhabited. So-called orthodox Christianity for most of its existence tied its concept of God to this particular world, earth. Earth may be unique (Moses 7:36), but Mormonism is less bound that way--despite lacking traditional Christianity's devotion to Greek philosophy where worship of God's supposed infinite attributes can take the place of obeying God's commandments.
Tarski Posted February 1, 2012 Posted February 1, 2012 (edited) But do you honestly think it is fair that the rules are made in such a manner that the Cantor number being created doesn't have to go through every possible outcome for each new column before it moves on? My fist list showed that logically you could set up an ordered list that contained every possibility for every column nad that it would be impossible to generate a new number from the start that didn't already have that sequence. My last list with the 1's and 0's was made in such a manner that showed logically that the list would generate the same number as is being created in the new Cantor number. That is a contradiction.There is no contradiction in the way you think.Your list doesn't contain the Cantor number derived from the diagonal.You haven't accounted for all possibilties. You only think this but it is a confusion on your part.It doesn't even seem like you have. It is a complete mystery why you think you are accounting for all possiblilties.No reason to think you have.In fact, the Cantor argument shows that you haven't.Check it out, the Cantor number derived from the diagonal differs from the n-th number in the n-th place. It was made that way!Now let me ask one more time: What is a "real number" in the mathematical sense? Edited February 1, 2012 by Tarski
Analytics Posted February 1, 2012 Posted February 1, 2012 (edited) But do you honestly think it is fair that the rules are made in such a manner that the Cantor number being created doesn't have to go through every possible outcome for each new column before it moves on? My fist list showed that logically you could set up an ordered list that contained every possibility for every column nad that it would be impossible to generate a new number from the start that didn't already have that sequence. My last list with the 1's and 0's was made in such a manner that showed logically that the list would generate the same number as is being created in the new Cantor number. That is a contradiction.For every single one of the infinite number of numbers on your list, Cantor proves that his number is something different. His number isn't the first, it isn't the second, it isn't the third,....it isn't the hundred trillionth,.... You point to any number on your list, and it can be proved that his number is something different.In contrast, consider the list of rational numbers 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, etc. You can name any rational number, and I can point to exactly what spot that number is on the list. Likewise, you can name any natural number, and I can tell you what rational number is in that spot.Do you see anything insightful about how literally every single rational number can be mapped to a specific, identifiable corresponding natural number, while in your series of 0.1, 0.11, etc. you can't identify a specific point where the number 1/9 is located? Edited February 1, 2012 by Analytics
asbestosman Posted February 1, 2012 Posted February 1, 2012 (edited) I can do even better than just create a list of all rationals. I can create a list of rationals, + the square roots and cube roots of rationals as well as pi and e. I'll even include rational multiples of these. In fact the list will contain sums, differences, and powers of rationals and many irrationals and much more (say, several complex numbers and even quaternions). How can I do it? I'll just use the Goedel number of each one. A Goedel number exists for pretty much any irrational number we ever talk about, but there are far more irrational numbers than Goedel numbers. Edited February 1, 2012 by asbestosman
asbestosman Posted February 1, 2012 Posted February 1, 2012 (edited) We haven't had a better example of someone not knowing what they are talking about in a long time. The absurdity of his incorrigible assumption that he can out-think the experts and giants in the field is matched only by those who think they can set climates scientists straight on climate change or set biologists straight about evolution. There is another lesson in this about having respect for expertise in a hard science. Apeal to authoritative concensus may not work as a step in a deductive argument, but it is a rational policy when trying to get the truth on complex issues that one hasn't the time or training to tackle on one's own.You know, I've been thinking about this for a bit. I have to wonder whether this reasoning also applies to critics who think they can set believers straight about the witness of the Holy Ghost or who think they know better than what revelations we've received through our prophets teach us. Edited February 1, 2012 by asbestosman
Nofear Posted February 1, 2012 Posted February 1, 2012 We haven't had a better example of someone not knowing what they are talking about in a long time. The absurdity of his incorrigible assumption that he can out-think the experts and giants in the field is matched only by those who think they can set climates scientists straight on climate change or set biologists straight about evolution. There is another lesson in this about having respect for expertise in a hard science. Apeal to authoritative concensus may not work as a step in a deductive argument, but it is a rational policy when trying to get the truth on complex issues that one hasn't the time or training to tackle on one's own.Comment made me think of the recent article I read The Eccentric Crank Who Tried to Legislate the Value of π (interesting debunkery on the idea that it was a biblicaly motivated act).
Rob Osborn Posted February 1, 2012 Author Posted February 1, 2012 There is no contradiction in the way you think.Your list doesn't contain the Cantor number derived from the diagonal.You haven't accounted for all possibilties. You only think this but it is a confusion on your part.It doesn't even seem like you have. It is a complete mystery why you think you are accounting for all possiblilties.No reason to think you have.In fact, the Cantor argument shows that you haven't.Check it out, the Cantor number derived from the diagonal differs from the n-th number in the n-th place. It was made that way!Now let me ask one more time: What is a "real number" in the mathematical sense?As far as I know, a real number is best represnted as any point in a measuring system whereas a natural number is a number used for counting something like when I count sheep at night.IF I built a list where each new row was a different number how many rows could I build if it was possible to do it all at once. I know this is impossible but how many?
Rob Osborn Posted February 1, 2012 Author Posted February 1, 2012 For every single one of the infinite number of numbers on your list, Cantor proves that his number is something different. His number isn't the first, it isn't the second, it isn't the third,....it isn't the hundred trillionth,.... You point to any number on your list, and it can be proved that his number is something different.In contrast, consider the list of rational numbers 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, etc. You can name any rational number, and I can point to exactly what spot that number is on the list. Likewise, you can name any natural number, and I can tell you what rational number is in that spot.Do you see anything insightful about how literally every single rational number can be mapped to a specific, identifiable corresponding natural number, while in your series of 0.1, 0.11, etc. you can't identify a specific point where the number 1/9 is located?But his number is the in the list when you apply the correct rules.
Analytics Posted February 1, 2012 Posted February 1, 2012 But his number is the in the list when you apply the correct rules.You claim it is on the list when you apply the "correct" rules, but you can't identify where.
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