morriswalters wrote:At this point in time I'm prepared to say that my idea is crazy. I'm having no fun defending it.
Then let's drop that idea. Meanwhile there's an underlying misunderstanding; do you wish to address this? I'll assume "yes" for now.
morriswalters wrote:Sit at your desk and draw a circle. Get some tracing paper lay it over the top of your circle. On the top layer draw three dots so that one is in the center and the other two fall on the edges of the circle below. That's circle one.
No, those three points are not a circle, at least not as "circle" is understood by mathematicians. Similarly, two points are not a line. Nonetheless, those three points define
a circle, just like two distinct points define
a line, in that there is only one [circle|line] that passes through those points. If you know those points, you can determine all the other points that make up
the shape in question.
But it takes all the other points to actually be
(Also note that a circle does not include its inside
. That would be a disk. Similarly, a sphere does not include its inside; that would be a ball. The word "sphere" is often used to mean "ball"; the formula for volume is often stated as the "volume of a sphere", but it's really the "volume of a ball" or the "volume enclosed
by a sphere". Yeah, confusing terminology. Bad mathematicians! Bad! Bad!)
Yes, giving the definition of a circle I've not specified a radius. Upon specifying a radius and a center, I've defined the
circle I'm talking about. I can define that
circle several ways; "the circle centered at the origin (0,0) and having radius 2" is the same circle as "the circle containing the points (0,2), (0,-2), and (2,0)". This is key to undersanding; the circle defined by three points still contains all the other points satisfying the first definition; it's the same circle.This post
by Eebster the Great is where "extent" was first used. Gmalivuk didn't use it. His er... point... was that shapes are embedded in (n-dimensional) spaces, and lower dimensional shapes have zero thickness in the remaining dimensions. For example, a line has zero thickness; thickness only has meaning in more than one dimensional space. So if you consider a (one dimensional) line embedded in (three dimensional) space, it has zero thickness in two (orthogonal)
directions. It has zero extent in those directions.
"Extent" has a more specific mathematical meaning, but this will do for now (especially since I don't happen to know it offhand).
The point being, that by defining a circle (in general) or by defining this
circle (giving origin and radius, for example), I have not "given the extent". The extent is implicit in the fact that a circle is a line that curves in upon itself to form an endless loop. Lines have zero extent in all directions except one... that's what makes them one dimensional.
Now, some sets (i.e. cantor dust) have intriguing properties, and the B-T procedure uses these properties, because the decomposition isn't into contiguous pieces but into "interlocking infinite dust", whose properties are non-intuitive. Take a look at the Wiki article on cantor dust
for more info on these odd properties. You need to grok this before hitting the B-T procedure.
morriswalters wrote:On HH
The two sets we are talking about aren't congruent.
If by the two sets you mean "the even rooms" and "the odd rooms", then I'm not sure what you mean by "congruent", but they have the same cardinality
. That is, they have the same number of elements. There are just as many odd rooms as there are even rooms, and there are just as many even rooms as there are prime-numbered rooms.
"Infinity plus one equals infinity."
morriswalters wrote:(but we think 1+..........+1=c)
If by "c" you mean "the continuum
", then no, that is not true. The continuum (the cardinality of the reals) is not the same as (1+...+1), which is ℵ0
(the cardinality of the integers). It is thought that c=ℵ1
, but this is not proven.
Whether HH rooms have extent is not relevant; what's important is that they have a room number (which is an integer). (Whether they have extent is only important to the guest, who might like a spacious room or a cozy room, but the desk clerk only cares about which key to give xim).
While I'm here, don't get confused between ℵ, which is a cardinal (indicating quantity), and ω, which is an ordinal (indicating position).
The HH hotel has ℵ0
rooms. They are all labeled by finite integers.
The first "infinite integer" is ω, but the HH has no room labeled ω. No guest can ever go there, because room ω does not exist. There's no room just before ω that has an integer label. You can't get there from here.
The HH can be full, but there's always room for more. That's one of the funny things about infinity. And the B-T procedure uses these funny things about infinity. Best way to get a handle on it is to read up on the funny things infinity does, and get comfortable with them. Then the B-T procedure will make more sense.
Hope this helps.
(edit: Fix quote mustard)