Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 5 Dec 2006 11:15 On 12/4/2006 9:56 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> >> >> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > > But aleph-0 is a quantity. > > Bob Kolker To those who belive in the usefulness of that illusion.
From: Eckard Blumschein on 5 Dec 2006 11:16 On 12/4/2006 9:55 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> On 12/1/2006 7:04 PM, MoeBlee wrote: >> >> >>>The Cartesian plane is the set of ordered pairs of real numbers. >> >> >> Doesn't coordinate transform e.g. into circular coordinates require >> fictitious real numbers? Here engineers might have a better feeling for >> the categorical difference between number and continuum. > > All numbers are ficticious so call a number or a set of numbers > ficticious is redundant and conveys no information. > > Bob Kolker > 1,000,000.00 Mark were paid for spreading this fog.
From: Eckard Blumschein on 5 Dec 2006 11:19 On 12/4/2006 9:54 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> You are right if you consider each time the complete set. Neither the >> integers not the rationals are actually complete. So both complete sets > > Really? Tell me an integer or rational that is not in the set of > integers or rationals? Which ones did we miss? > > Bob Kolker Read carefully: I did not write set of integers, I just wrote the integers. The complete set of integers is something quite different.
From: Eckard Blumschein on 5 Dec 2006 11:30 On 12/4/2006 9:49 PM, David Marcus wrote: >> According to my reasoning, the power set is based on all elements of a >> set. > > "Based"? Yes. The power set algorithm does not change what mathematicians still used to call cardinality. 2^oo=oo. > >> In case of an infinite set, there are nor all elements available. > > "Available"? Yes. You cannot apply the algorithm until you have all numbers. > >> Nonetheless I can do so as if they would exist, and I am calling them a >> fiction. > > "Exist"? "Fiction"? Exist means, they have their numerical address within a rational order. Fiction means, they don't have it but it is reasonable to do so as if. > >> Fictions are uncountable. > > "Uncountable"? The continuum cannot really be resolved into countable elements. It can just be thought to consist of an actually infinite amount of fictitious elements. > >> So this power set has no chance but >> to be also uncountable. >> Try to get the title cardinal Kolker, or at least Bob the Builder and I >> will possibly convert. > > When you tire of religion, you could always learn some math. My topic is not relegion but outdated quasi-religious mathematics.
From: Tony Orlow on 5 Dec 2006 11:53
Lester Zick wrote: > On Sat, 02 Dec 2006 13:38:47 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Fri, 01 Dec 2006 11:36:00 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein >>>>> <blumschein(a)et.uni-magdeburg.de> wrote: >>>>> >>>>>> On 11/29/2006 6:37 PM, Bob Kolker wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> It has the same cardinality perhaps, but where one set contains all the >>>>>>>> elements of another, plus more, it can rightfully be considered a larger >>>>>>>> set. >>>>> Tony, you know we've been over this previously. All "infinite" means >>>>> is lack of definition for a particular predicate such as numerical >>>>> size. And when you add numerical finites to numerical infinites the >>>>> result is still infinite. >>>> When you add anything to anything, you have more than you had, eh? >>>> That's pretty basic. Let's try to keep that in mind. >>> Provided they have the same metric and you can just "add anything to >>> anything". Next you'll be trying to add apples and oranges. >>> >> I can do that, by the pound, or apiece. That's a different problem. > > Oh I agree. But if you add them as pieces of fruit for example that > requires a conversion of metrics between apples, oranges, and pieces > of fruit or anything else they have in common according to a common > metric. With a finite r and infinitesimal dr you don't have that > without some common basis of metric integration. > Is dr a change in r, or not? That's my understanding. How can a change in a value not rest on the same metric as the value? >>>>> This problem mainly arises I suspect because mathematikers insist on >>>>> portraying infinites as larger than naturals and somehow coming beyond >>>>> the range of naturals such as George Gamow's famous 1, 2, 3, . . . 00. >>>>> Then mathematkers try to establish certain numerical properties for >>>>> infinities by comparative numerical analysis and mapping with >>>>> numerically defined finites. However one cannot do comparative >>>>> numerical analysis and numerical analysis with numerically undefined >>>>> infinites anymore than one can do arithmetic. Infinites are neither >>>>> large nor small; they're just numerically undefined. >>>> Uh, what if you define them, and even work out a language for expressing >>>> them, and arithmetic that be performed on them, and they produce >>>> intuitive results that include measure, as well as count? Why do you >>>> claim that's impossible, because you don't like the idea? >>> I don't like the idea because you can't establish any metric for them >>> not because you work out all kinds of things you claim are intuitive. >>> >> You can establish a common metric, even if you can't describe one in >> terms of the other in a finite formula. > > Well you can say you have a common metric but that doesn't mean you > can show one. In my opinion taking the derivative of a straight line > such as r with respect to itself produces 0dr and shows definitively > that there is no metric in common between r and dr. > I still don't know what you mean by taking the derivative ofa line with respect to itself. Can you say y=3x+2, so the derivative of that line is 0 d(3x+2)/dx? With respect to x, the derivative is 3. The derivative of that is 0. I suppose that's what you mean.... >>> Points are no more units of measure than zero is a metric. Someone >>> wrote the other day that Cantor was surprized that cubes have the same >>> number of points as squares and I was tempted to reply that if he was >>> he really didn't understand what he was talking about because cubes >>> and squares certainly have different numbers of infinitesimals. >> They most certainly do, as well as having infinitesimals of different >> dimensions. How many infinitesimals would you say a cube has, compared >> to a cube? Can you express that relationship? > > Well that seems to be pretty easy. At least taking the derivative of a > cube like rrr with respect to r indicates a ratio of infinitesimals is > 3rr dr for its square and with respect to r would be 6r dr. > Let's assume r is the length of the edge? Okay, the volume is r^3, the surface area is 3r^2, the linear measure (edges), are 6 r, and the number of points is the derivative of that, 6. Wait, is that right? No, for we have used an incorrect unit of measure. Use the actual radius, from the center of the cube, to the middle of a face. Now, the 3D volume is (2r)^3=8r^3. The 2D surface area is then 24 r^2 (6 faces * (2r)^2/face). The 1D edges are the derivative of that, 48r, (24 edges * 2r/edge). But, that's wrong. It's twice as much as it should be, because each edge is shared by two faces. So, the edges number 12, and the measure is 24r. The derivative of that should yield the number of edges in similar fashion. The derivative would be 24, but there are only 8 vertices. The factor of 3 comes from the fact that each vertex is shared by three edges. So, with the right unit of measure, and a notion of overlap between features, we can view the surface measure of dimension x as the derivative of the surface feature of dimension x+1, and we can integrate 8 points into 12 lines, into six faces, into one volume. >>> How many points are there in a finite interval? > >> Big'un, times the length in number of units of measure, plus or minus >> some finite number. > > But I just said above that points are not units of measure any more > than zero is a metric, Tony. Do you intend to say how you avoid that > issue? Points cannot be integrated into lines despite Bob's contention > a year or two ago. > See above with points and lines. Where tow points cannot be distinguished they are in the same place. Integration adds dimension. >>> Technically when it >>> comes to arithmetic and comparison there are only the two points >>> defining the interval metric. >> Yes, two "defining" it, meaning "marking the ends of" it. Notice the >> "fin" in "define"? There are two endpoints, and in between, and infinite >> number of intermediate points. > > So exactly how do you expect to show any infinite number of points > apart from the finite number of points which define the figure itself? Using the axiom of internal infinity which pertains to all continuous spaces: Ex ^ Ey ^ x<y -> Ez ^ x<z ^ z<y > I mean you can have any number of points defined by intersection > within a line segment but they have nothing to do with definition of > the segment and certainly nothing to do with any kind of mechanically > integratable dr. > They exist within the space as defined, or do you really think there are only two points in a line? >> Similarly for cubes and squares. >> >> Yes, 2^2 and 2^3 "endpoints". >> >>> And >>> presumably there are as many points in a point as zeroes in a zero. >> One? Think again. > > How can you tell? There could be any number. > > ~v~~ It depends on how you define such things. |