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From: Tony Orlow on 31 Oct 2005 10:17 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. > 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 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. > > 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. > 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/
From: Tony Orlow on 31 Oct 2005 10:20 Randy Poe said: > > albst...(a)gmx.de wrote: > > 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. > > No, there musn't. There is no logical basis on which to draw > this conclusion from your premise. > > > But the definition of the nat. numbers with complete induction leads to > > the consequence, that there could not be an infinite nat. number. > > Correct. Your first "conclusion" is merely an assertion, not > based on axioms. The only rule you're using is "there must", > the same thing Tony Orlow does: shout loudly when you don't > have a mathematical basis for "must". > > There "must" based on WHAT? WHY must there? He has shown it inductively, as I have for months that seem like years. Of course, you cling to the notion that inductive proof only works for finite naturals, a rather circular and self-supporting assertion, that makes it conveniently unavailable for the topic at hand. > > - Randy > > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Tony Orlow on 31 Oct 2005 10:53 David R Tribble said: > 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. > >> 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. > > > > Albrect 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. > > No, I don't, since this has been established mathematics for well > over a hundred years. You are making claims that contradict > established mathematics, so it is your task to support your claims > with proof. > > But anyway, it is trivially easy to show that no natural can represent > the size of the set of naturals: > > Let N+ = {1,2,3,...}, the set of finite naturals greater than zero. > (We'll use N+ instead of N because it seems to appeal to your sense > of logic.) > > Suppose m in N+ represents the number of elements in N+. Since m > is a member of N+, it must be a finite natural. Since m represents > the number of elements in N+, each of which enumerates itself, m > must be greater than all the other members in N+. But every finite > natural has a finite natural successor, so m+1 must exist, and m+1 > must also be a finite natural, and thus a member of N+. But m+1 is > greater than m, which contradicts our supposition that m is greater > than all other members in N+. Contradiction. All this proves is that there is no size of the set, as Albrecht has been saying, since the set size is equal to its largest element, which doesn't exist. Great proof. Try again. This proves nothing about the infinitude of the set. > > This forces us to conclude that we cannot find any m in set N+ that > represents the size of N+. Which in turn forces us to conclude > that the size of set N+ is not a natural number. QED. Can you find the largest natural? No? Wouldn't that be the same as the size? Yes? (Yes, we have proven it inductively) So, you cannot find the size of the set. There is no contradiction, except when you assume that you CAN define the size of the set. That's the mistake, and what Albrecht is trying to explain to you. > > > > 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. > > You seem to be saying that infinite naturals cannot exist. > If you really are saying this, then apologies to you. > (But no apologies to Tony, who does not believe this.) I agree with Albrecht's argument. He resolves the glaring contradiction in standard theory by disallowing infinite values as you do, and noting as I do that the set is not actually finite. I resolve the contradiction by allowing infinite naturals, where I find no contradiction, thus enabling the infinite set of naturals. Either approach is valid, as I said months ago, but if you choose the first, you cannot claim to be talking about infinity. So, no apologies to you either. :) > > > > 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. > > Every set does have a specific number of elements. That's the > cardinality of each set. A set which does not end does not have a aboslutely measurable number of elements. Such sets can only be compared over value ranges formulaically or discretely. > > > > 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. > > Now you seem to be saying that infinite natural numbers do exist. No, he is saying that your assertion that set of naturals is infintiely leads inexorably to the conclusion that it contains infinite natural values, so the set is finite, in his opinion. > > If you mean that infinite natural numbers exist, then you must prove > this to be so. (Not infinite ordinals or infinite cardinals, since > we know that these exist, but infinite naturals.) > > Or if you are saying that there are no such things as infinite sets, > then you must prove this to be so. Prove that what everyone else > thinks is an infinite set (such as N) does not really contain an > infinite number of elements and is actually finite. That is exactly what Albrecht did, but you fellows have a built up immunity to anything that contradicts your system, and since that includes almost anything around you, your defenses are in tip top shape, and your denial response well honed. > > Either way, I withdraw my apology. What the heck is your aoplogy worth anyway? Albrecht, take it from me, it's worth taking too personally, though I understand it gets tiring to be called an idiot for seeing what others seem blind to. Laugh it off. They are dogs, barking at us flying birds. Don't be a squirrel trapped in a tree. > > > > 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, > > This holds for all finite sets of finite natural numbers. > This does not hold for infinite sets which have no largest member. it holds for every natural number, finite or infinite, that the number of elements less than or equal to that natural is equal to the value of that natural. I have proven this inductively, according to your own rules, which you refuse to acknowledge. > > > > if there are infinite nat. numbers > > there must be a infiniteth nat. number representing this set. > > That implies that there is a greatest member in the set that is an > infinite natural. But this infinite set does not have a greatest > member. So there is no reason for an infinite natural to exist. It doesn't mean just the greatest element, but the vast majority of elements are infinite. Wherever you have an actually infinite sequence of any kinds of elements, the finite subset will be an infinitesimal fraction of the whole. Once you have an infinite number of elements, you are into the territory of infinite values, which continues forever, literally, this time. > > > > But the definition of the nat. numbers with complete induction leads to > > the consequence, that there could not be an infinite nat. number. > > > > That's the contradiction. > > Your assumption that the infinite set contains an infinite natural > is what leads to your contradiction. Since this assumption is false, > there is no actual contradiction. He does not assume it without reason, and offers a graphical proof. I have offered several other proofs to the same effect, but standard set theory seems to have blinded you to anything but the most dry, axiomatic verbal statements. It's too bad. > > > > So either the definition of nat. numbers must be changed or there is no > > infinite set of natural numbers. > > Or your assumption must be changed. It's not an assumption, but an inductively proven fact. > > > > 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. > > "Infinity" has several meanings, and you're confusing at least two > of them. An "infinite set" has a size represented by an infinite > (transfinite) cardinal number. The limit of a series or sum that > "approaches infinity" is a limit value that is larger than any real > number. A "countably infinite" set is a set whose members can be > enumerated (bijected) with the natural numbers. An "uncountably > infinite" set is a set that has more members than the set of > natural numbers. > > There is nothing illogical or inconsistent about saying that some > sets contain more members than any finite number. No, but that's a bare beginning. > > > > All this is shown very expressive in my sketches at the start of this > > thread. > > > > 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. > > I don't think I'm misinterpreting what you're saying. I think what > you're saying is wrong. What Tony is saying is certainly wrong. What you are saying is beyond wrong. It's ludicrous. > > > > In this state there is no real problem with all this. aleph_0 is just > > onother symbol for infinity. > > One particular kind of infinity, yes. The particular kind which is actually finite. > > > > The problems occure in that moment if someone declares, that aleph_0 is > > a size, which is greater than any nat. number. > > But that is the definition of Aleph_0; it is the size of the set of > natural numbers (or any other set containing that amount of members). Prove it by something other than mere declaration of the fact or a similar fact regarding the ordinals. You can't. > > > > 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, > > The very fact that a bijection exists between set A and set B proves > that the two sets are the same size. This is true for all finite and > infinite sets. Bijection is the obvious way to show that sets are > the same size. It shows they are the same cardinality, but we have shown that cardinality is not a measure of size in any useable sense. > > > > since infinity is something sizeless, endless, countless. > > That's all. > > Well, infinity is certainly a kind of "endlessness", yes. But it's > not countless (if it's a countable infinity), and it's certainly > not sizeless (since infinity is greater than any finite). Given the endlessness of it, pure infinity is not measurable. Infinities can only be compared using bijection, including taking into account the exact mapping used, to get a precise comparison. > > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Randy Poe on 31 Oct 2005 11:15 Tony Orlow wrote: > David R Tribble said: > All this proves is that there is no size of the set, as Albrecht has been > saying, since the set size is equal to its largest element, which doesn't > exist. Hmmm. Adding the TO-axiom that "set size is equal to its largest element" leads to the conclusion that "sets without maximal elements have no size". Since without this axiom we have a perfectly self-consistent notion of cardinality (informally, "size"), there seems no point in adding this axiom to our system. I know, let's not make the assumptiont that "set size is equal to its largest element" and see where that goes. Oh gee. We already did that. - Randy
From: Tony Orlow on 31 Oct 2005 11:16
Randy Poe said: > > Tony Orlow wrote: > > David R Tribble said: > > > We are forced to conclude that there is no natural s that maps to > > > *N, and that therefore your mapping scheme is not a bijection > > > between *N and P(*N). > > You are not forced to conclude that this prevents a bijection. > > Um, yes you are. If a bijection requires that there exist such > an s, and no such s exists, you are forced to conclude that > you do not have such a bijection. And yet, the bijection between 2 unending sets does not require you to be able to name the end of the sets, does it? > > > The entire > > infinite set would be the final subset enumerated, > > Your usual red herring. But it's nonsense. This can clearly > be seen since your scheme CAN be used to establish a > bijection between *N and P(N), and no recourse to "last > subsets enumerated" is needed. It is provable that > EVERY subset of N is mapped, just as it is provable that > SOME subset of *N is not mapped, ever, by anything, in this > mapping. Look again at the bijection I offered. Your mapping between *N and P(N) is not valid in Bigulosity Theory. In the bijection I offered, no element is inthe set mapped to it, so the set of all elements not in the set they map to is the entire unedning set, and you want a natural that maps to the end of it? Heh! > > Actually, in a sane world, the fact that your mapping > produces a bijection from *N to P(N), using up all the > elements of *N, would be enough to convince you that > the enumeration doesn't go on to map the rest of P(*N). > But no, somehow you can believe both that your list > is bijection from *N to P(N), and the VERY SAME LIST > magically stretches when you want it to, to become a > bijection from *N to P(*N). It is the position of standard set theorists that infinite sets just stretch to infinity without end, and that this is a justification for declaring that any bijection means equality. I am simply pointing out that, given this line of thinking, despite the fact that the power set is obviously larger, one can still construct a bijection between a set and its power set, provided the set is ordered and unending. Bijection alone means very little. The actual mapping must be taken into account to get any real handle on relative infinities. > > - Randy > > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/ |