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From: albstorz on 2 Nov 2005 07:36 David Kastrup wrote: > I don't see anybody doing this an "infinite number of times". I don't > even see anybody doing it once. The Peano axioms state that each > element _has_ a successor, not that this successor needs to be > generated or has been generated in some manner. That's nothing than finickiness and pettyfoggery again. There is no difference between "generating" or "having" or "doing something an infinite number of times". A set of infinite many elements consists of infinite many elements putting them in step by step infinite often or putting in infinite many at once. Infinite often or infinite many: who is really able to see a difference between them in this respect? In math there is no respect to matter, time, space or number in this sense and connection. Regards AS
From: albstorz on 2 Nov 2005 08:01 David R Tribble wrote: > Albrech Storz wrote: > > Do you have a test on objects to proof if they are sets or not? Or is a > > set just that what you want to have to be a set? > > A set is a collection of entities taken as a whole. A good answer. Now, please explain how you proof if something is a entity. Is an emotion an entity? Can you have sets of emotions? Is an instant an entity? Can you have sets of instants? Is a wave an entity? Is a color an entity? ... > The entities > are members of the set. Of all the possible entities, there are those > that are in the set, and those that are not in the set. A very interesting point of view. Think about the set of all your possible entities. You know, you will have problems about it? > > > > What's your definition of sets? Is the water in a can a set of water? > > We can model the can as a set containing water molecules, but we must > be able to distinguish each molecule from every other molecule, i.e., > each molecule must have some unique numbering or comparison operator > defined for it. That's far away from my intention. I had thought about the water in the form we experience it as a continuum. I know that we can consider it as molecules. That's not very absorbing in this context So, please proof, whether a real number is an entity. Regards AS
From: albstorz on 2 Nov 2005 08:11 David R Tribble wrote: > David R Tribble wrote: > >> Consider the set of reals in the interval [0,1], that is, the set > >> S = {x in R : 0 <= x <= 1}. The elements of this set cannot be > >> enumerated by the naturals (which is why it is called an "uncountably > >> infinite" set). But all sets have a size, so this set must have a > >> size that is not a natural number. It is meaningless (and just > >> plain false) to say this set "has no size" or "is not a set". > > > > Albrecht Storz wrote: > > I'm not shure if the reals build a set in spite of you and Cantor and > > others are shure. > > A set is defined by consisting of discrete, distinguishable, individual > > elements. Now tell me: what separates a point on a line from the very > > next point on the line to be discrete? What separates sqrt(2) from the > > very next real number to be discrete? > > If you look only on individual points, you may have a set. But if you > > look on all of them? > > > > So, your above argumentation has no relevance to me. Proof the reals to > > be a set, then let's talk again. > > Well, for a start, the set of naturals (N) is a subset of the set of > reals (R), i.e., every member in N is also a member of R. If N is a > set, then it would appear that R is also a set. That's totally circular. Perhaps N is a set because R isn't a set? Perhaps the reals separate the natural numbers from each other and that's why N is a set? But this doesn't mean anything about the question if the reals are a set or not. > > Or are there entities that can have subsets taken from them but > themselves are not sets? What do you call those entities then? The reals are a subset of the natural numbers???? > > I can also form the set > S = {pi, 2 pi, 3 pi, 4 pi, ...} > which is obviously an infinite set containing real numbers. S does > not contain all the real numbers, of course, but it is a subset of > all the real numbers. This, too, implies that the set of all real > numbers is a set. It's easy to show that every element of S is separated from each other. So I don't have a problem with your set S. But what's about the reals? Pardon, but this proofs nothing. Regards AS
From: albstorz on 2 Nov 2005 08:18 William Hughes wrote: > Tony Orlow wrote: > > William Hughes said: > > > > > > albstorz(a)gmx.de wrote: > > > > > > > You misinterpret totally when you say, I think there must be an > > > > infinite natural number. I don't think so. I only argue that, if there > > > > are infinite sets, there must be infinite natural numbers (since nat. > > > > numbers are sets). > > > > > > > > > > OK. Make the substitutions, natural numbers = Greeks, sets = mortals > > > and > > > infintite = German [1] > > > > > > Then the statment "if there are infinite sets, there must be infinite > > > natural numbers (since nat. numbers are sets)" becomes "if there are > > > German mortals, there must be German Greeks (since Greeks are > > > mortals)". > > > Aristotle must be rolling in his grave. > > > > > > -William Hughes > > > > > > [1] Deutchland, Deutchland, ueber alles > > > > > > > > Come on William. If Albrecht is stating that sets are natural numbers and > > natural numbers are sets, he is not saying a natural number is a KIND of a set, > > like Greeks are a type of mortal. This is disingenuous. Sorry. > > Note that Albrecht makes the statment "since natural numbers are sets" > The two statements "since natural numbers are sets" and > "since natural numbers and sets are the same thing" are different. > If the second statement is meant, Albrecht should say so > explicitely. (Other statements by Albrecht are are contradictory on > this point, at times he seems to claim that any set > is also a natural number, at other times he backs away from this > claim.) > -William Hughes > > > -- > > Smiles, > > > > Tony > > http://www.people.cornell.edu/pages/aeo6/WellOrder/ You are not able to understand that we had to consider the axioms and faiths of ZF set theory to expose it's inconsistency. I don't back away from any claim I've done. Regards AS
From: albstorz on 2 Nov 2005 08:21
Tony Orlow wrote: > albstorz(a)gmx.de said: > > > > Robert J. Kolker wrote: > > > albstorz(a)gmx.de wrote: > > > > > > > > > > > > > > > I'm not shure if the reals build a set in spite of you and Cantor and > > > > others are shure. > > > > A set is defined by consisting of discrete, distinguishable, individual > > > > elements. Now tell me: what separates a point on a line from the very > > > > next point on the line to be discrete? > > > > > > There is no very next point under the ordinary ordering of reals. But > > > given a pair of reals they are either equal or not. Between any two > > > distinct real numbers there is always a third real different from the > > > two given (with respect to the standard ordering of the reals). > > > > > > Bob Kolker > > > > Do you have a test on objects to proof if they are sets or not? Or is a > > set just that what you want to have to be a set? > > What's your definition of sets? Is the water in a can a set of water? > > > > Regards > > AS > > > > > It's a set of molecules, measured in moles. > -- > Smiles, > > Tony > http://www.people.cornell.edu/pages/aeo6/WellOrder/ Yes. But I didn't asked about a set of molecules. I asked about a set of water. Regards AS |