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1234567890
360 Posts |
 Posted - 02 Mar 2004 : 18:22:11
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quote:
Originally posted by jrich
In pure math there is no time, only order. An infinite set is complete when it is defined even if its members are defined by the infinite iterations of some function. One may argue whether this makes mathematical infinities inapplicable to describing reality, but in the purely mathematical realm of this thought experiment those considerations do not apply.
To see how the infinite set of persons in the hotel may not include all persons, suppose that the rooms are numbered with the infinite set of positive non-zero integers [1,2,3,...] and that all the rooms are rented except room 1. So the set of rented rooms is [2,3,4,...]. Since the set of rented rooms is likewise infinite there must be an infinite set of corresponding guests. It makes no difference whether the cardinality of [1,2,3,...] is greater than that of [2,3,4,...] (it isn't), there must be the same number of guests as there are rented rooms. Just as the set of rented rooms is infinite and yet does not include 1 room, likewise, the set of persons who are guests is also infinite but may not include 1 person.
JR
If a completed infinity (if there is such an animal) does not include all possible elements then one can use the same argument Cantor used to prove the real number set uncountable to prove that the set of the naturals, represented by the rooms, is uncountably infinite with respect to the subset of the naturals, represented by the infinite persons in the room already since one can always generate a room number that the infinite number of persons had not occupied.
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north
Canada
518 Posts |
Posted - 02 Mar 2004 : 19:22:24
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quote:
Originally posted by Larry Burford
north,
OK, this starts on pg 29 in my copy.
[quote] ... on pg.19 he talks about the difference between infinite decimal fractions and arithmetical fractions 3/7 or 8/277 then he convertes into decimal fractions ... e.g.
LINE 1 >> 3/7= 0.428571 : 428571 : 428571 : 428571 : 4...
LINE 2 >> = 0.(428571)
What does this segment mean?
If you do the division of 3 by 7 in longhand out to 21 or 28 decimal places you will see that the quotient comprises six digits that repeat over and over. They are: 4 2 8 5 7 1.
(So the arithmetic fraction 3/7 converts to decimal as 0.428571428571428571428571 ... )
In LINE 1 (I added some lables to help here) he uses a vertical line (which you represented here with a colon) to visually separate the groups of "repeating digits".
In LINE 2 he places the six repeating digits within parentheses. This is a math shorthand notation that means "the digits within repeat over and over to infinity".
More later ...
LB
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Larry
thanks,it is apparent i'm ignorant of much, i do though, appreciate your time and effort! i tried to put as much down as i could to save you time, although i could go further,i relised one step at a time.
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jrich
still trying grasp what you were trying to get across,sort of there,maybe sort of!! thanks also for your time and effort!
north |
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jrich
USA
287 Posts |
Posted - 02 Mar 2004 : 21:04:10
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123,
As I said previously, I'm undecided about Cantor's ideas about infinities beyond aleph-0, which would include his diagonal proof. But neither do I accept the idea you are proposing of the incomplete infinity. An incomplete infinity is really no infinity at all, so it seems to me your argument isn't that the hotel is full or that there is noone who is not a guest, your argument is really that the premise of the question is impossible.
Most people believe that the axioms of mathematics should reflect as closely as possible our intuitive understanding of the natural world. These people may generally be divided into at least two camps (the names escape me). One camp believes that infinities are too counter-intuitive and that this requires a rejection of the infinities. They take your view that Zeno is resolved by accepting a finite reality and that a math devoid of infinities is sufficient to describe it. The opposite camp believes that Zeno is resolved by the idea of the continuum. They have embraced the infinities and attempted to provide the axiomatic foundation for them. They have not been completely successful in that the axioms have been proven to be insufficient to prove what the cardinality of the continuum is. I think that in the MM, the cardinality of the continuum is aleph-0, but most others believe it must be at least aleph-1. If anyone can be more definitive on this, please correct me.
JR |
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Larry Burford
USA
1337 Posts |
Posted - 03 Mar 2004 : 07:59:02
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quote:
[north]
He goes on to say: We have proved above that the number of all ordinary arithmetical fractions is the same as the number of all integers; so the number of all Periodic decimal fractions must also be the same as the number of all integers.
What has he done here to prove this?
Do you understand Cantor's rule for comparing two infinities? This is in the section "How to count infinities", and starts on page 25 in my book.
(He begins with the Hottentot and their "truncated number system" (he doesn't use this phrase), showing how a clever Hottentot could overcome this limitation to know which of two groups of objects is largest if both groups have more than three items. The basic idea is to lay an object from one group next to an object from the other group. Then repeat until you run out of objects in one group. If at that point the other group still has objects in it, the Hottentot knows that this group must be larger. He may not be able to say how much larger, or even how large each group is. But he does know that one group is larger, and he knows which group that is.
Note - the two groups could also turn out to equal in count. In this case, when he runs out of objects in the first group, he has no leftovers in the second group.
The rule is an extension of this idea.)
===
Using this rule he shows that several infinite number groups (including all integers and all ordinary arithmetic fractions) have the same number of numbers.
Let me know if we need to spend more time on this.
LB |
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jrich
USA
287 Posts |
Posted - 03 Mar 2004 : 11:42:29
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quote:
Originally posted by jrich
I think that in the MM, the cardinality of the continuum is aleph-0, but most others believe it must be at least aleph-1.
Before everyone objects to this statement let me try to clarify what I mean by it. The continuum is a geometric term. It is the set of all points having the same cardinality as the real numbers. A 1-dimensional line of any length is a continuum, as is a 2-dimensional plane and 3-dimensional space. I'm not positive (geometry's not my thing), but I think that the same is true for the dimensions in non-Euclidean geometry. In this way set theory is supposed to provide the axiomatic foundation for geometry. The problem is that set theory can show that the size of the continuum is greater than aleph-0, but it can't show that it is aleph-1, which is the next greater size, without adding another axiom. So right now the axiomatic foundation of geometry provided by set theory is incomplete. This means that when geometry is used to represent reality, we can't be logically certain that it is doing so correctly because the intuitively obvious axioms (that supposedly follow directly from nature) are insufficient.
Now my point is that in the MM with the addition of the scale dimension, the classical continuum is no longer needed for 3-dimensional space. The cardinality of the set of points in 3-dimensional Euclidean space becomes aleph-0 for any given scale. Furthermore, the number of points in a finite volume of space is finite at any given scale. The scale dimension essentially allows finite subsets of the continuum. Space becomes quantized at every scale and thus may be treated as finitely divisible at that scale.
To summarize, I believe that in the 5-dimensional MM, only the scale dimension and perhaps time are a continuum. The other 3 dimensions of space are countably infinite.
OK, everybody, I'm ready for the onslaught. But please be gentle.
JR |
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1234567890
360 Posts |
Posted - 03 Mar 2004 : 15:20:57
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quote:
Originally posted by jrich
123,
As I said previously, I'm undecided about Cantor's ideas about infinities beyond aleph-0, which would include his diagonal proof. But neither do I accept the idea you are proposing of the incomplete infinity. An incomplete infinity is really no infinity at all, so it seems to me your argument isn't that the hotel is full or that there is noone who is not a guest, your argument is really that the premise of the question is impossible.
Most people believe that the axioms of mathematics should reflect as closely as possible our intuitive understanding of the natural world. These people may generally be divided into at least two camps (the names escape me). One camp believes that infinities are too counter-intuitive and that this requires a rejection of the infinities. They take your view that Zeno is resolved by accepting a finite reality and that a math devoid of infinities is sufficient to describe it. The opposite camp believes that Zeno is resolved by the idea of the continuum. They have embraced the infinities and attempted to provide the axiomatic foundation for them. They have not been completely successful in that the axioms have been proven to be insufficient to prove what the cardinality of the continuum is. I think that in the MM, the cardinality of the continuum is aleph-0, but most others believe it must be at least aleph-1. If anyone can be more definitive on this, please correct me.
JR
No, there are many mathematicians who do not buy Cantor's idea
of completed infinities. In fact, any person with any sense of
logic would not buy it. But if you are going to introduce the idea
of a completed infinity, it makes absolutely no sense to not include every possible element. Anyone who believes in completed infinities should be put to the task of locating one irrational number. That's right- just one.
Take out the irrationals in the real number set and it becomes aleph 0, using Cantor's criterion for aleph 0. But it's not that there is not enough naturals to count the irrationals but rather the fact that you can't even locate one irrational number to count. The irrationals demonstrate quite clearly the absurdity of completed infinities. |
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Larry Burford
USA
1337 Posts |
Posted - 03 Mar 2004 : 17:43:51
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jrich,
"Space becomes quantized at every scale and thus may be treated as finitely divisible at that scale."
You lost me here. Perhaps an example of what you mean ...? |
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jrich
USA
287 Posts |
Posted - 03 Mar 2004 : 18:36:26
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quote:
Originally posted by Larry Burford
jrich,
"Space becomes quantized at every scale and thus may be treated as finitely divisible at that scale."
You lost me here. Perhaps an example of what you mean ...?
I think what I was trying to say is that this change in the cardinality of Euclidean space allows the math to be congruous with finite forms. Within a finite volume of space and a finite range of scale, there would only be a finite set of forms. In fact, if scale were a continuum, this would not be true, since any finite range of scale would itself contain an infinity of scales. So now I modify my conjecture to state that the cardinality of scale is also aleph-0. Of course, over the infinite range of scale the same finite volume of space will still contain infinite forms in accordance with MM.
JR |
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north
Canada
518 Posts |
Posted - 04 Mar 2004 : 00:15:34
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quote:
Originally posted by Larry Burford
[quote][north]
He goes on to say: We have proved above that the number of all ordinary arithmetical fractions is the same as the number of all integers; so the number of all Periodic decimal fractions must also be the same as the number of all integers.
What has he done here to prove this?
Do you understand Cantor's rule for comparing two infinities? This is in the section "How to count infinities", and starts on page 25 in my book.
(He begins with the Hottentot and their "truncated number system" (he doesn't use this phrase), showing how a clever Hottentot could overcome this limitation to know which of two groups of objects is largest if both groups have more than three items. The basic idea is to lay an object from one group next to an object from the other group. Then repeat until you run out of objects in one group. If at that point the other group still has objects in it, the Hottentot knows that this group must be larger. He may not be able to say how much larger, or even how large each group is. But he does know that one group is larger, and he knows which group that is.
Note - the two groups could also turn out to equal in count. In this case, when he runs out of objects in the first group, he has no leftovers in the second group.
The rule is an extension of this idea.)
===
Using this rule he shows that several infinite number groups (including all integers and all ordinary arithmetic fractions) have the same number of numbers.
Let me know if we need to spend more time on this.
LB
______________________________________________________
Larry
i think i get this part.just thought i would give a little back ground of myself,this might help in understanding a little better where i'm coming from, from a math point of view.i went back to high school when i was 25,got my grade 13,then went to university for science and philosophy. physics was one course,but lacking geometry there was no way i could do this course,ever tried learning geometry and physics at the same time! impossible, to much other homework.also took chemistry,tried 3 times failed 3 times,lack of math again.for example i went to see a prof. for help, while waiting, a girl came for help as well out of curiosity i asked what she would do, she told me,i knew right then and there i was in trouble she was thinking in terms i never even came close too. this however never stopped me from enjoying physics,chemistry and others, could usually understand the concepts just not the math. i've tried always to find books on these subjects without math,i have some but they are hard to find,by the way if you or anybody knows of a book on quantum physics like this i'd sure would be interested. but as Einstein said "concept first then the math". just thought it best you know this, math has always been hard for me. funny though, some day would to get my masters in math,it would be so darn handy!!
thanks,north. |
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1234567890
360 Posts |
Posted - 04 Mar 2004 : 02:21:35
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quote:
Originally posted by jrich
quote:
Originally posted by Larry Burford
jrich,
"Space becomes quantized at every scale and thus may be treated as finitely divisible at that scale."
You lost me here. Perhaps an example of what you mean ...?
I think what I was trying to say is that this change in the cardinality of Euclidean space allows the math to be congruous with finite forms. Within a finite volume of space and a finite range of scale, there would only be a finite set of forms. In fact, if scale were a continuum, this would not be true, since any finite range of scale would itself contain an infinity of scales. So now I modify my conjecture to state that the cardinality of scale is also aleph-0. Of course, over the infinite range of scale the same finite volume of space will still contain infinite forms in accordance with MM.
JR
It's obvious from Dr. Flandern's usage of the infinite series that sums to 1 as resolution to Zeno's paradox that the continuum in MM is aleph 0 since the series is represented by a bunch of rational numbers. Too bad Zeno wasn't around when Cantor gave his diagonal proof for the reals. Gamot had it easy. Try proving a finite distance is equal to the sum of its divisions by enumerating its divisions using the real numbers. |
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jrich
USA
287 Posts |
Posted - 04 Mar 2004 : 10:52:49
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quote:
Originally posted by 1234567890
It's obvious from Dr. Flandern's usage of the infinite series that sums to 1 as resolution to Zeno's paradox that the continuum in MM is aleph 0 since the series is represented by a bunch of rational numbers. Too bad Zeno wasn't around when Cantor gave his diagonal proof for the reals. Gamot had it easy. Try proving a finite distance is equal to the sum of its divisions by enumerating its divisions using the real numbers.
That was my thought too. It was originally one of my objections to Tom's using the series argument since Zeno assumed the continuum. But I think that in light of my recent arguments, the infinite series wasn't inappropriate after all. I know this won't satisfy your fundamental objections, but it does clarify how Zeno might be resolved in MM.
JR |
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1234567890
360 Posts |
Posted - 05 Mar 2004 : 08:59:49
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quote:
Originally posted by Jan
123,
quote:
But I think it's more correct to say that there cannot ever be an infinite number of persons in the room already since an infinite set cannot ever be completed.
In other words, for every customer arriving at the hotel there is a room available. I have yet to see such hotel.
It's more like the hotel is ever in construction. Whenever a new guest arrives, a new room is added to the hotel. The hotel never had infinite capacity, only the potential for it. The hotel you have never seen is the very one described by Hilbert. The two permanent occupants are himself and Cantor. Last time I visited, it was more like a mental institution so we checked out early. |
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MarkVitrone
USA
386 Posts |
Posted - 05 Mar 2004 : 22:11:46
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I had a reservation but they wouldn't take my Diner's Club so I went to the Holiday Inn where my room was more than finite. In fact, only a skilled Yoga master could sit on the toilet and reach the barely three dimensional toilet paper which was located directly behind and below to the sitter's right hand, as far as possible from useful. :) I made many complaints to the travel agent and she assured me that nothing would be remedied.
I try to separate math truth and reality truth. The questions of infinity and eternity in math and reality are travel agent questions. Meaning that we must visit the number or place to define that number or place. The statement that space and time are infinite is enough. Attempting the quantify infinity especially when mathematics asks us to place that quantity within bounds (like all infinite numbers between 1 and 2 and in essence problems like Zeno, etc). These attempts bring frustration because the solution is unsatisfying. Defining and understanding the relationships of spacial scale is more satisfying because analogy can be made. What comes to mind is the grain of sand encompassing a whole universe type of reasoning. I seem to find that analogy easier to grasp and more fulfilling. Mathematics helps us attempt the approach of reality, its raw pursuit however will lead to unhappiness when debate ensues beyond the point of diminishing returns. Hilbert makes me feel this way because the model doesn't lead to understanding it leads to dissatisfaction similar to greasy roadside diner food when no antacid is present. My question to the panel is WHY? Keeping in mind that my ability to wield the pencil in the world of advanced math is limited, yet my point in questioning the accessibility and utility of this model comes into the realm of its approximation of the reality we know and love. Cheers, MV |
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1234567890
360 Posts |
Posted - 06 Mar 2004 : 02:40:28
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quote:
Originally posted by MarkVitrone
I had a reservation but they wouldn't take my Diner's Club so I went to the Holiday Inn where my room was more than finite. In fact, only a skilled Yoga master could sit on the toilet and reach the barely three dimensional toilet paper which was located directly behind and below to the sitter's right hand, as far as possible from useful. :) I made many complaints to the travel agent and she assured me that nothing would be remedied.
I try to separate math truth and reality truth. The questions of infinity and eternity in math and reality are travel agent questions. Meaning that we must visit the number or place to define that number or place. The statement that space and time are infinite is enough. Attempting the quantify infinity especially when mathematics asks us to place that quantity within bounds (like all infinite numbers between 1 and 2 and in essence problems like Zeno, etc). These attempts bring frustration because the solution is unsatisfying. Defining and understanding the relationships of spacial scale is more satisfying because analogy can be made. What comes to mind is the grain of sand encompassing a whole universe type of reasoning. I seem to find that analogy easier to grasp and more fulfilling. Mathematics helps us attempt the approach of reality, its raw pursuit however will lead to unhappiness when debate ensues beyond the point of diminishing returns. Hilbert makes me feel this way because the model doesn't lead to understanding it leads to dissatisfaction similar to greasy roadside diner food when no antacid is present. My question to the panel is WHY? Keeping in mind that my ability to wield the pencil in the world of advanced math is limited, yet my point in questioning the accessibility and utility of this model comes into the realm of its approximation of the reality we know and love. Cheers, MV
Hilbert's Hotel is a direct consequence of Cantor's treatment of
infinite sets so if Cantor is right then the hotel is an accurate analogy. The two concepts Hilbert was trying to demonstrate was that you can add indefinitely to an infinitely large integer subset without increasing its cardinality and also the fact that the set of reals are so much bigger than the integers that the major domo had to close down the hotel when requested to check in those "types" of guests. These follow from Cantor's proof showing that infinite subsets of the integer set and the set of rationals all have the same size, which he called aleph0, and that real numbers have a larger size he called aleph1.
If Cantor's results give you indigestion, you are in good company.
The large majority of mathematicians today think there should be more sizes for infinite sets (they think the Continuum Hypothesis is false) or don't believe in the notion of completed infinities altogether (though this is more a minority).
My rule of thumb is that if something doesn't make a whole lot of sense for the average educated person, it's probably wrong. |
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Hilberts hotel
