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From: Virgil on 3 Nov 2005 15:47 In article <1131020542.301492.298300(a)f14g2000cwb.googlegroups.com>, albstorz(a)gmx.de wrote: > You know that the constructable real numbers are denumerable infinite. > If all real numbers are nondenumerable, there must be nondenumerable > infinite many real numbers between every pair of constructable reals. > > Proof that this numbers are entities. No numbers are "entities" without the assumption of their existence, since no 'number' exists in the physical world. The easiest way to get started is to assume the naturals and a little set theory as a basis, and construct the rest upon that basis.
From: albstorz on 4 Nov 2005 06:19 David R Tribble wrote: > Tony Orlow wrote: > >> If it doesn't contain any infinite members, it's not infinite. Those terms > >> differ by more than a constant finite amount, but rather a rapidly growing > >> amount greater than 1. There is no way you have an infinite number of them > >> without achieving infinite values within the set. > > > > David R Tribble wrote: > >> Yes, you and Albrecht keep saying that repeatedly. Please demonstrate > >> why it must be so, because it's not. > > > > Albrecht Storz wrote: > >> Your argumentation is not fair, but I don't wonder about that. > >> _You_ has to show, that in the case of the whole set there is no > >> natural number as big as the whole set. > >> You argue: there is no infinite natural number since the peano axioms > >> don't allow an infinite natural number. > >> That's right. I agree with you. > > > > Tony Orlow wrote: > >> Alas, Albrecht, it is not true. There is nothing in the Peano axioms that > >> states any such thing. The inductive proof of the finiteness of the naturals is > >> flawed in that it applies an increment to each successor, noting that adding 1 > >> does not turn a finite into an infinite, but ignores the intrinsic nature of > >> inductive proof as a recursively defined infinite concatenation of logical > >> implications, over which an infinite number of increments does indeed produce > >> an infinite value as successor. Of course, this cannot be achhieved in any > >> finite or "countably infinite" (finite but unbounded) number of steps. > > On one side we have Tony, who believes infinite naturals exist but > that the set of finite naturals is not infinite. On the other side > we have Albrecht, who also does not believe the set of naturals is > infinite but also does not believe infinite naturals exist. > > They are obviously both wrong, but for different reasons. > It's amusing to see this kind of discussion. And in the middle there is David R. Tribble who is also not right but has no insight and thinks he is great oh, what a fake. Have fun Regards AS
From: albstorz on 4 Nov 2005 06:50 William Hughes wrote: > albstorz(a)gmx.de wrote: > > William Hughes wrote: > > > 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 > > > > Yes. If we accept infinite sets. > > > > > > > > > > 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. > > > > You think about Dedekind cuts? > > Yes, this is one of the standard constructions. > > > > > > > > At which step do we fail to have a set? > > > > > > -William Hughes > > > > > > All this don't proof anything. > > > > You know that the constructable real numbers are denumerable infinite. > > If all real numbers are nondenumerable, there must be nondenumerable > > infinite many real numbers between every pair of constructable reals. > > > > Proof that this numbers are entities. > > Since all of these numbers are of the form (A,B) where > A and B are sets of rationals they are clearly entities. > > -William Hughes Non sequitur. I see no relation to the discussed issue. Let's try it in another way if you want. I have some question: Do you think math should be able to map reality? Do you think reality consist only in discret aspects --> entities, quants, objects, unities? Do you think continua and sets are the same things? Do you think nothing in nature changes gradually, all changes are stepwise, discontinuous, discret? Again, do you think, continua consists of entities? Do you know what continua means? Okay. I consider you accept the concept of continuous changes. Now, is math able to map this aspects of nature if it don't supply continuous aspects? Since nature containes discret and continuous aspects as well, math should contain discret and continuous aspects as well. Do you think, sets are a good concept to talk about continua? Do you think it means anything to try counting the changes of a gradually changing system? Do Cantor himself not names the "set" of reals continuum? It's just a word or is it a concept? What should be the set of the reals? By what should reals be separated from each other? Do you think, thinking have any sense? Are you able to grasp philosophical concepts? Does all this connote nothing to math? .... There are a lot of questions more. Regards AS
From: William Hughes on 4 Nov 2005 10:00 albstorz(a)gmx.de wrote: > William Hughes wrote: > > albstorz(a)gmx.de wrote: > > > William Hughes wrote: > > > > 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 > > > > > > Yes. If we accept infinite sets. > > > > > > > > > > > > > > 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. > > > > > > You think about Dedekind cuts? > > > > Yes, this is one of the standard constructions. > > > > > > > > > > > At which step do we fail to have a set? > > > > > > > > -William Hughes > > > > > > > > > All this don't proof anything. > > > > > > You know that the constructable real numbers are denumerable infinite. > > > If all real numbers are nondenumerable, there must be nondenumerable > > > infinite many real numbers between every pair of constructable reals. > > > > > > Proof that this numbers are entities. > > > > Since all of these numbers are of the form (A,B) where > > A and B are sets of rationals they are clearly entities. > > > > -William Hughes > > > Non sequitur. I see no relation to the discussed issue. You obviously think of "entities" as having to be discrete. I note that any real number can be thought of as a set of discrete things. > > Let's try it in another way if you want. > > I have some question: > > Do you think math should be able to map reality? No, but the fact that it often does is useful. Note, that one does not need a perfect "map". A sufficiently accurate model is fine. > > Do you think reality consist only in discret aspects --> entities, > quants, objects, unities? To the extent that this question makes sense, yes. > > Do you think continua and sets are the same things? > No. "continua" obviously has something to do with order an completeness. One can have a set without considering an ordering on the elements of the set (nor must this ordering be unique, nor will every ordering produce a continuum. The reals with the usual ordering form a continuum. The reals with a well-ordering do not.) > Do you think nothing in nature changes gradually, all changes are > stepwise, discontinuous, discret? To the extent that this makes sense yes (I believe that everything is quantized at some level) > > Again, do you think, continua consists of entities? Assuming that by a continuum you mean a set with certain properties (e.g. an ordering in which given any two elements there is a third element between them and a topology to which the set is complete), Yes. The underlying set consists of entities. > > Do you know what continua means? > Yes, but I doubt that you do. > > Okay. I consider you accept the concept of continuous changes. > Not in physics (everything is quantized at some level). In mathematics yes. > Now, is math able to map this aspects of nature if it don't supply > continuous aspects? Map, perhaps not. Model to an arbitrary level of accuracy yes. (Note that we have continuous mathemaics modelling discontinuous physics, not the other way round.) > Since nature containes discret and continuous aspects as well, Your grasp of physics mirrors your grasp of math. >math > should contain discret and continuous aspects as well. Math should certainly be able to model discrete and continuous aspects. > Do you think, sets are a good concept to talk about continua? They are a good start. You are going to need some other concepts before you get to continua. > Do you think it means anything to try counting the changes of a > gradually changing system? No. But modelling the gradually changing system by increasingly accurate discretely changing system (where the changes can be counted) does mean something. > Do Cantor himself not names the "set" of reals continuum? > It's just a word or is it a concept? continuum is a concept and a useful one. > What should be the set of the reals? The completion of the rationals. >By what should reals be separated > from each other? By the fact that if two reals x, and y represents different pairs of sets (A,B) and (A',B') then x and y are different. Since there is no such thing as the next rational, it doesn't make sense to talk about how a rational is separated from the next rational. The same reasoning applies to the reals. > > Do you think, thinking have any sense? Yes > Are you able to grasp philosophical concepts? Yes. > Does all this connote nothing to math? You seem to be confused about what math is, what it should be able to do , and what is needed for math to be able to do this. The properties of infinite sets are not contradictory, but they are most certainly counter-intuitive. If you don't like infinite sets, do not use them. You can contruct the reals starting with a potentially infinite set of integers. (Pesonally, I don't think you will be doing anything other than using a lot more words to say the same thing.) If you want to stick to constructable real numbers go ahead (you are in good company). -William Hughes
From: Ross A. Finlayson on 4 Nov 2005 10:19
William Hughes wrote: > albstorz(a)gmx.de wrote: > > William Hughes wrote: > > > albstorz(a)gmx.de wrote: > > > > William Hughes wrote: > > > > > 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 > > > > > > > > Yes. If we accept infinite sets. > > > > > > > > > > > > > > > > > > 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. > > > > > > > > You think about Dedekind cuts? > > > > > > Yes, this is one of the standard constructions. > > > > > > > > > > > > > > At which step do we fail to have a set? > > > > > > > > > > -William Hughes > > > > > > > > > > > > All this don't proof anything. > > > > > > > > You know that the constructable real numbers are denumerable infinite. > > > > If all real numbers are nondenumerable, there must be nondenumerable > > > > infinite many real numbers between every pair of constructable reals. > > > > > > > > Proof that this numbers are entities. > > > > > > Since all of these numbers are of the form (A,B) where > > > A and B are sets of rationals they are clearly entities. > > > > > > -William Hughes > > > > > > Non sequitur. I see no relation to the discussed issue. > > You obviously think of "entities" as having to be discrete. I > note that any real number can be thought of as a set > of discrete things. > > > > Let's try it in another way if you want. > > > > I have some question: > > > > Do you think math should be able to map reality? > > No, but the fact that it often does is useful. > Note, that one does not need a perfect "map". > A sufficiently accurate model is fine. > > > > > Do you think reality consist only in discret aspects --> entities, > > quants, objects, unities? > > To the extent that this question makes sense, yes. > > > > > Do you think continua and sets are the same things? > > > > No. "continua" obviously has something to do with > order an completeness. One can have a set > without considering an ordering on the elements > of the set (nor must this ordering be unique, nor will > every ordering produce a continuum. The reals with > the usual ordering form a continuum. The reals with > a well-ordering do not.) > > > Do you think nothing in nature changes gradually, all changes are > > stepwise, discontinuous, discret? > > To the extent that this makes sense yes (I believe that > everything is quantized at some level) > > > > > Again, do you think, continua consists of entities? > > > Assuming that by a continuum you mean a set with > certain properties (e.g. an ordering in which > given any two elements there is a third element between > them and a topology to which the set is complete), > Yes. The underlying set consists of entities. > > > > > Do you know what continua means? > > > > Yes, but I doubt that you do. > > > > > Okay. I consider you accept the concept of continuous changes. > > > > Not in physics (everything is quantized at some level). > In mathematics yes. > > > Now, is math able to map this aspects of nature if it don't supply > > continuous aspects? > > Map, perhaps not. Model to an arbitrary level > of accuracy yes. (Note that we have continuous mathemaics > modelling discontinuous physics, not the other way round.) > > > Since nature containes discret and continuous aspects as well, > > Your grasp of physics mirrors your grasp of math. > > >math > > should contain discret and continuous aspects as well. > > Math should certainly be able to model discrete and > continuous aspects. > > > Do you think, sets are a good concept to talk about continua? > > They are a good start. You are going to need some > other concepts before you get to continua. > > > Do you think it means anything to try counting the changes of a > > gradually changing system? > > No. But modelling the gradually changing system by > increasingly accurate discretely changing system (where > the changes can be counted) does > mean something. > > > Do Cantor himself not names the "set" of reals continuum? > > It's just a word or is it a concept? > > continuum is a concept and a useful one. > > > What should be the set of the reals? > > The completion of the rationals. > > >By what should reals be separated > > from each other? > > By the fact that if two reals x, and y represents > different pairs of sets (A,B) and (A',B') > then x and y are different. Since there is no such > thing as the next rational, it doesn't make sense to > talk about how a rational is separated from the next > rational. The same reasoning applies to the reals. > > > > > Do you think, thinking have any sense? > > Yes > > > Are you able to grasp philosophical concepts? > > Yes. > > > Does all this connote nothing to math? > > You seem to be confused about what math is, what it > should be able to do , and what is needed for math > to be able to do this. > > The properties of infinite sets are not contradictory, > but they are most certainly counter-intuitive. > If you don't like infinite sets, do not use them. > You can contruct the reals starting with > a potentially infinite set of integers. (Pesonally, > I don't think you will be doing anything other than > using a lot more words to say the same thing.) > If you want to stick to constructable real > numbers go ahead (you are in good company). > > -William Hughes I really want to not misquote or incorrectly use his phrase, but Mati Meron calls that "coffee table book" physics. To measure the mass of subatomic particles more precisely, the measured value doesn't converge to a finite positive value, it appears to go to zero, ie, it's infinitesimal. I'm not much of a physicist, but the set-theoretical universe is infinite, and the Ding-an-Sich is from technical philosophy. That's a good point about the continua, except there are some complications in well-ordering the reals, ie, unless there are adjacent points in the normal ordering. Yes, I agree the reals are a field so between two of them defined as the closure of the integers to division and furthermore the value of any convergent sequence of those, between those there are more of them, the reals are dense in the reals. While that is so, there is also the conundrum of rationals being dense in the reals, and irrationals dense in the reals. If you well-order irrationals in an interval, generated a well-ordered set of ordered pairs of nested interval endpoints as you go, well, Cantor's first is said not to apply to irrationals because it's said not to apply to rationals as the convergent value of their sequence may lie outside the set. When you well-order the reals, if there are only countably many nested intervals then the reals are not shown uncountable by Cantor's first. if there are more, there are that many disjoint intervals, and as many rationals as those. Unless there are at least two adjacent points in the normal ordering, there are always more, and the reals are not a well-orderable set, which would be a contradiction in ZFC. Those are some good points about the continua. Heh, points about the continua. Ross |