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From: Will Twentyman on 11 Apr 2005 18:02 Eckard Blumschein wrote: > On 4/11/2005 5:10 PM, Will Twentyman wrote: >>Eckard Blumschein wrote: > >>>>>http://iesk.et.uni-magdeburg.de/~blumsche/M280.html > > I am trying to explain that infinity is correspondingly just a > >>>quality, not a quantity. >> >>I think you are missing a significant point: the cardinality of an >>infinite set is *different* from whether or not a set is infinite. >>Cardinality is a more precise and fundamentally different quantity from >>infinity. > > > You are correct in that meanwhile nobody asks for why and how > cardinality was introduced. This seems unlikely at best. Without looking at any notes, it seems like cardinality gives a meaningful method of comparing the "size" of sets and defining what we mean by "size". It is a natural extension of the cardinality of a finite set. > The underlying notion of infinite whole > numbers is undecided between infinite and numbers mutually excluding > each other. Cantor's thinking was correspondingly split. He did not > decide between the meaning oo of what he called Maechtigkeit and later > cardinality and his intention and pracice to use it as and like just a > number that served as an infinite or even more than infinite measure of > quantity. So it is different from infinity but definitely not more > precise. It is just nonsense. How can you say it is not more precise? You can use cardinality to compare the sizes of N, P(N), and P(P(N)). oo does not distinguish between them at all, and countable/uncountable does not distinguish between the last two. >>>There are two questions: >>>1) Why did Cantor ignore the only reasonable notion of infinity >>>including the pertaining rules how to handle it? >>>Presumably he articulated widespread confusion between "infinite" and >>>"very large". I noticed that even Poisson and Weierstrass used the >>>nonsense term "infinite number". Most likely Cantor was driven by an >>>insane ambition to create something new. Recall Dedekind's careless >>>opinion that in mathematics any creation is allowed. >> >>Cardinality is a way of establishing categories of sets and ordering >>them based on whether functions between them are surjective, injective, >>or bijective. > > > As far as I know, the only decisive question is whether or not an > infinite set is bijective to the set of natural numbers. In that case it > is calles countable infinite, else non-countable. P(N) and P(P(N)) are also standards. This is where aleph_0, aleph_1, aleph_2 get started. Both aleph_1 and aleph_2 are uncountable, but they are different cardinalities. >>If you don't want to think of them as being different >>levels of infinity, then don't. > > Cantor got (in)famous just because he claimed to have revealed different > levels of infinity. He was a bluffer. Most people seem to find it convenient as a measure of levels of infinity. I do. If you don't care to think of it that way, just focus on the definition as it applies to sets. >>What Cantor did was follow the logical >>consequences of the definitions he used. > > No. How did he define his infinite whole numbers? > Instead of serious work, he published what M280 contains a link to. Read > it and judge yourself. Unfortunately, it's in German and I can't claim I would be able to reliably translate it. Google was not helpful. >>I don't think he was planning >>on the results. > > He was keen to create the most unusual. Most unusual or most correct? I've seen several odd ways of looking at things that are useful for easily getting a correct result. >>>2) Why did he manage to find so much support? >> >>Because his results are consistent with the axioms and definitions he used. > > That is definitely not true. Read the original papers! See the note above regarding my skills with German. >>>I cannot see any reason for that. Cantor was correct in that rational >>>numbers are countable in the sense, they can be brought into a >>>one-to-one relationship to the infinite set of natural numbers. >> >>Because cardinality is defined in terms of mappings between sets. I >>think you are missing that detail. > > No. He just made the wrong assumption that the reals can be mapped. This > cannot work despite of AC. If you accept that the reals are numbers, then they can be mapped to *something*. If you don't accept that the reals are numbers (which I think is your position), then the entire topic of conversation is moot. > "There are neither more nor less nor equally many real numbers > >>>as compared to the rational ones." >> >>Whether you feel there are more or not, > > Do not deny compelling arguments. Neither my feeling nor yours matters. Which is why I work with the ZF version of set theory and accept the various infinite cardinalities. >>what Cantor was fundamentally >>discussing is the existence of bijective maps between the reals and >>rationals. > > Such map does not exist. There is no approachable well-ordered map of > the reals. AC asserts (without proof and non-constructively) a well-ordering of the reals. There is no bijective map between the reals and rationals precisely because they have different cardinality. -- Will Twentyman email: wtwentyman at copper dot net
From: Eckard Blumschein on 12 Apr 2005 03:09 On 4/11/2005 9:26 PM, Matt Gutting wrote: > But you seem to be claiming that real numbers (or at least some of them) > are not quantities. Rational and even natural numbers are thought to be embedded into the real ones. This is one reason while the majority of mathematicians keeps sticking on Cantor's guess that there are more reals than rationals. However, we should be aware that rationals can be approximate the continuum as close as one likes. So the theoretical difference between the rationals and the reals including all embedded numbers is just given by the theoretically complete loss of approachable identity for the latter, no matter whether they are irrational or embedded like e.g. 0.999999999999999.... You need infinitely many real numbers in order to constitute just a single urelement (mathematical atom). It is this fascinating peculiarity of the reals I am mainly interested in. Weyl spoke of continuum sauce. Stifel spoke in 1544 of fog. > You are assuming that comparison requires a quantitative measure. Doing so, I refer to a quantitative comparison. > It is possible to define > "measure" to have meaning for the set of natural numbers. Isn't a quantitative measure is a quantity to compare with? Quantities can be expressed by means of rational numbers. What rastional number does express the quantity of IN? >> Look into a good dictionary for infinity. > > I have (Webster's Collegiate, which I regard as a good dictionary). I saw > nothing even remotely involving addition of quantities to infinity. It should tell you that infinity cannot be enlarged. >> No that is not the reason. oo+oo also equals oo. > > If, as you say, infinity is not a number (a direct quote from you), how can one > add it to anything (including itself)? Do not mistake oo+a=oo like an equation of the same sort as it is valid for numbers. It is just a formalized description of the fundamental property to be infinite. >>>And what, precisely (*very* precisely) do you >>>mean by "cannot be enlarged"? >> >> >> The property to have no limit. > > That doesn't help to clarify the statement to me. Ordinary people understand it like a barrel without bottom. It is not just as wide as you like in the sense you may choose any huge number but there is no chance ever to say that's it. It is not a number. All numbers are finite. >> It is. > > No, you seem to be inferring that fact from the description of infinity as > something "inexhaustible". If you define infinity as "something that cannot > be enlarged", then you are speaking of a different concept than that which > mathematicians refer to by the name "infinity". A part of mathematics would go slippery when it would leave the original meaning of oo that is still in use in other parts. Eckard
From: Eckard Blumschein on 12 Apr 2005 04:15 On 4/11/2005 9:09 PM, Matt Gutting wrote: > Eckard Blumschein wrote: > >>>>>>http://iesk.et.uni-magdeburg.de/~blumsche/M280.html >> As far as I know, the only decisive question is whether or not an >> infinite set is bijective to the set of natural numbers. In that case it >> is calles countable infinite, else non-countable. > > Except that there appear to be ways of ordering sets by "size" which allow > different uncountably infinite sets to be placed at different points in the > order. One may define anything. However, I am not even aware of any attempt to actually use cardinalities in excess of aleph_1. Do you have any idea? >> Cantor got (in)famous just because he claimed to have revealed different >> levels of infinity. He was a bluffer. > > I still don't understand why you say this. If Cantor defined infinity in > a particular way Do you consider his infinite whole numbers a definition of infinity? Please read what he wrote. It was not logically consistent. I am calling it schizophren because he evaded the decision between the quality oo and the quality of a number when he vaguely introduced his "infinite whole numbers". The notion of a number contradicts to the notion of infinity. > and concluded that by his definitions, there were different > orders or types of infinite sets, No. He did not conclude it but it was his intention to show it. > then he did in fact reveal different levels > of infinity according to the definitions he used. Like a conjuring trick. > You are free to disagree with > his definitions, but you should then be clear in stating that the problem is > not with different levels of infinity, but with the meaning of infinity. The only reasonable notion of infinity does not let room for different levels of infinity. Cantor himself respected this notion like the devine Infititum aeternum increatum sive Absolutum". He called his is own creation "Infinitum creatum sive Transvinitum". However, he did not give a reasonable definition. He just presented a misleading evidence for his claim as usually do illusionists. >> No. How did he define his infinite whole numbers? >> Instead of serious work, he published what M280 contains a link to. Read >> it and judge yourself. >> Do not deny compelling arguments. Neither my feeling nor yours matters. >> > What constitutes a "compelling argument" depends highly on feelings - it > depends on what one is willing to be compelled by. Cantor's second diagonal argument was a seemingly compelling one, so far. Paradoxically, it was compelling not because people were willing to believe Cantor's claim but because the failed to disprove it, and consequently they felt forced to accept it. > It is often the case > that an argument which is not widely accepted is not compelling because > it is simply incorrect. What about my arguments, they are new ones. You might check them. It frequently happens that factually compelling arguments were ignored just because they are not welcome. >> Such map does not exist. There is no approachable well-ordered map of >> the reals. > > What, exactly, do you mean by "approachable"? Coining the term well-ordered, Cantor overlooked a serious problem. The continuum cannot be resolved into single points. So we may imagine a steadily ascending order of reals, while one fails to express this order in terms of numerals. In other words, while we do not have any reason to doubt that there are infinitely many different positions on a line, we nonetheless cannot take advantage of this abundance. Since it is impossible to completely write down all infinitely many numerals of just one single real number, it is also impossible to name its successor. Eckard
From: Torkel Franzen on 12 Apr 2005 04:06 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > Since it is > impossible to completely write down all infinitely many numerals of just > one single real number, it is also impossible to name its successor. The impossibility of naming the successor of a real number is indeed the central flaw in today's mathematics. Little can be done about it, I'm afraid.
From: Gerry Myerson on 12 Apr 2005 04:15
In article <vcb4qec7718.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > > > Since it is > > impossible to completely write down all infinitely many numerals of just > > one single real number, it is also impossible to name its successor. > > The impossibility of naming the successor of a real number is indeed > the central flaw in today's mathematics. Little can be done about it, > I'm afraid. Fortunately, no real number has ever died, so the problem of naming a successor has not arisen. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |