Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 5 Dec 2006 12:18 On 12/4/2006 8:07 PM, MoeBlee 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. > > What are fictitious real numbers? Fictitious real numbers are defined by DA2 or like fictitious limits. They do not have a numerical address in a field of discrete numbers. > Anyway, I'm not sure what bearing your question has on the fact that > RxR is the set of ordered pairs of real numbers. It put in question what you call a fact. Fictitional real numbers would be flexible. The xy pattern is rigid. > > MoeBlee >
From: Eckard Blumschein on 5 Dec 2006 12:49 On 12/4/2006 11:32 AM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> Notice, there is not even a valid definition of a set which includes >> infinite sets. Cantor's definition has been declared untennable for >> decades. > > That is simply not so. For example the set of integers. There is is. Perhaps, you are honestly bold. Believe me that Fraenkel admitted that Cantor's definition is untennable. The question is e.g. in case of the naturals whether they are considered one by one or altogether like an entity. While a set is usually imagined like something set for good, this point of view is unrealistic. It complements the contrary point of view seeing the same naturals incomplete like something countable. If we say the set of naturals is countable, we refer to this realistic so called potentially infinite point of view.
From: Eckard Blumschein on 5 Dec 2006 12:50 On 12/4/2006 9:47 AM, Virgil wrote: > In article <4573D4DA.4040709(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/3/2006 8:22 PM, cbrown(a)cbrownsystems.com wrote: >> > Tony Orlow wrote: >> > >> > Well, you used the term "set" four times in your above definition of >> > what we mean by a "set". That's why I said "this begs the question, >> > what do we mean, exactly, by a set of properties?". >> > >> > There's something that we intuitively seem to think of as a "set"; but >> > unless such a thing is carefully defined, we end up with the >> > contradictions of naive set theory: >> > >> > http://en.wikipedia.org/wiki/Naive_set_theory >> >> Is it really justified to blame an allegedly insufficient definition of >> the term set for obvious antinomies of set theory? > > > As "set" and "is a member of" are primitives in axiomatic set theory, > any "definition" of them is outside of set theory and irrelevant to it. Yes. The problems are shifted outside.
From: Tony Orlow on 5 Dec 2006 13:06 Eckard Blumschein wrote: > On 12/1/2006 9:59 PM, Virgil wrote: > >> Depends on one's standard of "size". >> >> Two solids of the same surface area can have differing volumes because >> different qualities of the sets of points that form them are being >> measured. > > Both surface and volume are considered like continua in physics as long > as the physical atoms do not matter. > Sets of points (i.e. mathematical atoms) are arbitrarily attributed. > There is no universal rule for how fine-grained the mesh has to be. > Therefore one cannot ascribe more or less points to these quantities. > > Look at the subject: Galileo's paradox: The relations smaller, equally > large, and larger are pointless in case of infinite quantities. > > Now, just a minute, Eckard. You're contradicting yourself, if these objects are infinite sets of points. They have different measure. Therefore they are different "sized" sets in that respect. Where we can establish an infinite unit of measure for the sets corresponding to a finite unit of measure for the objects, then we can easily distinguish the relative sizes of the sets, even if one cannot establish certain facts pertaining to finite sets, such as divisibility by some finite value. We can still say, for instance, if we have cubes A and B with the edges of A half the length of those in B, that cube A contains 1/8 as many points as cube B, that A's faces have 1/4 as many points as those of B, and that the combined points within A's edges is 1/2 the points in B's. The 8 remainign 0D points, of course, are the same in both figures. Of course, that's not standard. >> Sets can have the same cardinality but different 'subsettedness' because >> different qualities are being measured. > > I do not know a German equivalent to subsettedness. Wiki did not know > subsettedness or at least subsetted either. So I have to gues what you > possibly meant. Proper subset means included but not all-including. > While I consider cardinality a rather illusory notion, I distinguish > between counted (alias finite), countable (alias potentially infinite) > and uncountable (alias fictitious). Well, the counteds are a subset of > the countables. So far, the mislead bulk of mathematicians regard the > countables (i.e. genuine numbers) a subset of the uncountables, too. > They're all points on the infinite line, that's all. > TO>> You do? Do you mean that the addition of elements not already in a set >>> doesn't add to the size of the set in any sense? >> In the subsettedness sense yes, in the cardinality sense, not >> necessarily. In the sense of well-ordered subsettedness, not necessarily. >> >> TO seems to want all measures to give the same results, regardless of >> what is being measured. > > Standard mathematics may lack solid fundamentals. At least it is > understandable to me. However, I admit being not in position to likewise > easily understand what you mean with well-ordered subsettedness. > >
From: Eckard Blumschein on 5 Dec 2006 13:19
On 12/5/2006 4:01 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> >> Those who declared Cantor naive overlooked Dedekind. Learn German and >> enjoy his childish innocent style. > > He also gave a correct definition for the real numbers, too. I explained why I am objecting against his claims. > > There are other definitions which are equivalent. Yes. They are equally imprecise. > For example using > limit points of Cauchy sequences I would agree if you did add the original word "fictitious" to these sequences. of rational numbers which is the > topological closure of the rational number space with interval topology > and the usual metric. > > Bob Kolker Please do no longer keep me for as ignorant as you seems to be. |