Hilberts hotel

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
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.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
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 <i><b>for any given scale</b></i>. 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

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
<br />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
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

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.

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 ...?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br />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 ...?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
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

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br />

[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?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

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.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br />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 ...?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
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
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

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.