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From: albstorz on 2 Nov 2005 16:38 Tony Orlow wrote: > albstorz(a)gmx.de said: > > > > David R Tribble wrote: > > > David R Tribble said: > > > >> So I'm giving you set S, which obviously does not contain any > > > >> infinite numbers. So by your rule, the set is finite, right? > > > > > > > > > > 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. > > > > > > Yes, you and Albrecht keep saying that repeatedly. Please demonstrate > > > why it must be so, because it's not. > > > > > > 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. > > 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. It's the basic antinomie of infinity to find, that you can't come out of finiteness in spite of doing infinite steps or add infinite many elements or wait infinte time or count infinitely many times. You can't reach infinity since there is no connection between finity and infinity. The peano axioms implies not only a successor of every natural number, it implies also a predecessor of every natural number other than 1. The probem is, which predecessor should have the first infinite number which should be the last finite number. We could think the sequence of natural numbers in the only way of constant change only in the number but not in any other propertie. Or should the infiniteness arise gradually in increasing the number? At which number we will have 50% infinity? The solve of this problem in modern math is to declare infinity instead of construct it. And then to be astonished about the fantastic consequences which will occur by this act. To say "aleph_0" is nothing other than to set a new kind of unity and then, no wonder about, you can count "aleph_1, aleph_2, ...". Galileo Galilei wrote in 1638:"The attributes "equality", "to be bigger", "to be smaller" don't hold by infinite quantity rather only by finite quantity." > > > But that's no proof about sets. That's only an aspect of the definition > > which contradicts with the fact, that every set has a number of > > elements. > > > > 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). > > I don't say: there are infinite sets. You say: there are infinite sets > > and there is no infinite number. And I say: If there are infinite sets > > there must be infinite numbers. > So, your position is that there are no infinite sets, since there are no > infinite naturals/ Well, we agree on some things, and yet, in others the > standard nonsense is somewhere between us with a half baked notion of infinity. > Mine is ready for the frosting. :) My position is, that infinity leads to contradictions (in spite of the need of infinity in math). > > > > My argumentation is very easy: > > Every nat. number represents a set. If you look at the first 100 nat. > > numbers, the 100th nat. number "100" represents the set {1, ... , 100}. > > As this holds for every nat. number, if there are infinite nat. numbers > > there must be a infiniteth nat. number representing this set. > > But the definition of the nat. numbers with complete induction leads to > > the consequence, that there could not be an infinite nat. number. > Um, no, it doesn't. But do go on. > > > > That's the contradiction. > You resolve it by rejecting the infinity of the set. I resolve it by rejecting > the finitude of the elements. Sadly, there is no resolve. > > > > So either the definition of nat. numbers must be changed or there is no > > infinite set of natural numbers. > The Peano axioms can be adjusted to generate the infinites while generating the > finites. Tat's easy. > > > Or infinity must be interpreted in a completely other way. Not as a > > size like you do. Infinity is just an unability to count it with > > numbers because it runs out of all what we can know. > > > > All this is shown very expressive in my sketches at the start of this > > thread. > I thought so. Others don't always see what I see, though. > > > > Why do you misinterpret all the time? Maybe my ability to express my > > thoughts in english is too bad. > > But why do you misinterpret Tony also? I think he is native english > > speaker and you should be able to understand him. > LOL! :D Albrecht, while am indeed a native English speaker from New York with a > pretty decent command of the language, I also sometimes feel like I am from > Planet Xorxon, and find it difficult to communicate certain ideas to many of my > apelike family members. I don't think this is a language issue. Your diagram is > pretty language-independent. It's a matter of thinking visually, rather than > axiomatically and linguisitically. :-) > > > > In this state there is no real problem with all this. aleph_0 is just > > onother symbol for infinity. > > The problems occure in that moment if someone declares, that aleph_0 is > > a size, which is greater than any nat. number. > I agree wholeheartedly. Yes, herein lays the basic problem of modern math. > > > But there is no "greater" or "less than" or something like this. There > > is just something other, something out of the things we could measure, > > wigh or count. > > The possibility of bijection don't say anything about the size of > > infinity, since infinity is something sizeless, endless, countless. > > That's all. > But, Albrecht, wouldn't you say that the size of the set of naturals is twice > the size of the set of even naturals, since the latter comprise 1/2 of the > former? And wouldn't you consider [0,2) as containing twice as many points as > [0,1)? If you don't believe in infinity at all, how many reals ARE in [0,1)? > > > > Regards > > AS > > > > > > -- > Smiles, > > Tony > http://www.people.cornell.edu/pages/aeo6/WellOrder/ The problem, math has to deal with, is, that it needs actual infinite objects like the decimal expansion of irreal numbers or transcendent numbers or just rational numbers like 1/3, or the numbers of points which lays on a straight line or in an intervall on an straight line or the set of natural numbers, etc, pp. But this problems has to be solved without inconsitent systems like the cantorian, which leads to nonsense like transfinite numbers which don't have any sense. Consider the set of all thinkable objects. This set should be infinite since at least every natural number is a thinkable object. Consider the infinite list of the thinkable objects. Now consider an object which is different of the first object of the list, from the second object of the list, ..., of any object of the list. This object is the thinkable object with the property not to be on our list of the thinkable objects. Cantorologic leads to the consequence that the size of the set of thinkable objects is nondenumerable infinite. But since all thinkable objects must be describable and all describable objects are denumerable there occurs a contradiction. The second diagonal proof of Cantor works exact like this and the consequence of it is flawed exact in the same way. Regards AS
From: albstorz on 3 Nov 2005 06:58 albstorz(a)gmx.de wrote: > > The problem, math has to deal with, is, that it needs actual infinite > objects like the decimal expansion of irreal numbers or transcendent Pardon. Not "irreal number" but "irrational number" was meant > numbers or just rational numbers like 1/3, or the numbers of points > which lays on a straight line or in an intervall on an straight line or > the set of natural numbers, etc, pp. > But this problems has to be solved without inconsitent systems like the > cantorian, which leads to nonsense like transfinite numbers which don't > have any sense. > Regards AS
From: albstorz on 3 Nov 2005 07:22 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? > > 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. Regards AS
From: William Hughes on 3 Nov 2005 08:38 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
From: David R Tribble on 3 Nov 2005 13:22
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. |