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From: Randy Poe on 2 Nov 2005 08:45 albstorz(a)gmx.de wrote: > Yes. > But I didn't asked about a set of molecules. I asked about a set of > water. Your question does not make sense grammatically, let alone mathematically. A set is a collection. The word "of" in this context means "from", "taken from". What follows the word "of" is a descriptive word or phrase that describes a collection from which objects might be taken. "Water" does not fit there. "Waters" would, meaning perhas "bodies of water" or "brands of bottled water". So whatever point you think you wanted to make, the only point you actually made was that you can construct a question which has no answer because it has no meaning. But we knew that already, no need to emphasize the point. - Randy
From: Randy Poe on 2 Nov 2005 08:52 albstorz(a)gmx.de wrote: > 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. This is not subject to proof. It's more of a definition. Can you prove something is a noun? I suppose if one were to attempt to define an entity in this context, it would be something with a distinguishable identity, described by a singular (as opposed to plural) noun. I'm struggling with a definition a little bit because in the past I've had trouble with pinning down some entities, such as "every possible human being". > Is an emotion an entity? Can you have sets of emotions? Certainly. > Is an instant an entity? Can you have sets of instants? Yes. Instants can be assigned unique coordinates. > Is a wave an entity? Is a color an entity? ... Yes to both. > > 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? Yes, because this "set" itself is one of the entities, and then we get somebody's paradox (Russell's?). Sets can't contain themselves. That's part of the definition of set. - Randy
From: David Kastrup on 2 Nov 2005 10:13 albstorz(a)gmx.de writes: > 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". Certainly there is. The Peano axioms don't describe a procedure for generating naturals. They give a set of rules to check whether you have them. They provide identification, not blueprints. > 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" has no meaning. Could you repeat the steps aleph_1 times if you wanted to? Could you continue with another step after "infinite often"? Could you stop a step short of "infinite often"? No. You don't get the set of naturals by getting them step by step. You get them by the static property that no number of steps will exhaust them. > Infinite often or infinite many: who is really able to see a > difference between them in this respect? If you don't, then it is your fault if you can't come to sensible conclusions. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David R Tribble on 2 Nov 2005 14:41 David R Tribble wrote: >> A set is a collection of entities taken as a whole. > Albrech Storz wrote: > 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? ... We are talking about mathematical entities, which are mental concepts and abstractions. We are not talking about physical entities. >> 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? If you are talking about a set of all sets, yes, that is a problem. >> 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 But we can form a loose analogy between a quantity of "continuous" water and a continuous interval of real numbers. But it is only an analogy. > So, please proof, whether a real number is an entity. A real number is the same kind of entity as a natural number. Both are mathematical abtractions. There is no way to "prove" that a mathematical concept is an "entity".
From: William Hughes on 2 Nov 2005 14:55
albstorz(a)gmx.de 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". > > > 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. What is your difficulty with the standard definitions? Certainly the Integers { ... -3,-2,-1,0,1,2,3 ... } form a set Take the set of all pairs of integers (i,j) where the second integer is not 0. Take some equivalence classes of the above and we have the rationals. (Note the rationals are not discrete). Take paris of sets of rationals (A,B) where all the rationals in A are less than those in B and where A union B is all the rationals. Now we have the reals. At which step do we fail to have a set? -William Hughes |