Courage?
in [Logic]
|
Prev: arithmetic in ZF
Next: Derivations
From: Will Twentyman on 11 Apr 2005 11:10 Eckard Blumschein wrote: > On 4/8/2005 12:03 AM, Will Twentyman wrote: > >>>I dealt not just with such mediocre proponents of a wrong concept like >>>Zermelo and the remedies they introduced but also with Hilbert. You will >>>find a few comments of mine at >>> >>>http://iesk.et.uni-magdeburg.de/~blumsche/M280.html >>> >> >>Reading this, it seems that your objection is that Cantor's work does >>not correspond to your notions of infinity. > > > Admittedly, I am not aware of any other reasonable notion of infinity > than expressed > by Aristotele: Infinitum actu non datur, > by Spinoza: Infinity cannot be enlarged, > by Gauss: "so protestiere ich ... gegen den Gebrauch einer unendlichen > Grýýe als einer Vollendeten, welche in der Mathematik niemals erlaubt > ist" and > by all other serious (i.e. non Cantorian) mathematical literature. > Those, in particular Galilei and Leibniz, who did not yet entirely agree > on how to calculate with infinity, nonetheless fully agreed on > some basic rules like oo+a=oo and oo^oo=oo. > Everybody except for Cantor's fellows understands why we write x->oo but > not x=oo. 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. > 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. If you don't want to think of them as being different levels of infinity, then don't. What Cantor did was follow the logical consequences of the definitions he used. I don't think he was planning on the results. > 2) Why did he manage to find so much support? Because his results are consistent with the axioms and definitions he used. > I would like to abstain from answering this question because it relates > to human fallibility rather than mathematics. Warnings by Kronecker, > Poincarý and many others were ignored from Weierstrass and others. The > same Bertrand Russell who declared causality a relic of bygone time like > monarchy and who suggested a preventive nuclear stroke that would have > killed me wrote: > "The solution of the difficulties which formerly surrounded the > mathematical infinite is probably the greatest achievement of which our > age has to boast". Well, boast is the matching word if one tries to > judge how demagogically Hilbert, Fraenkel and others promoted Cantor's > transfinite numbers. The weaker the argument, the stronger the euphoria. > > > >>So be it, just worry about >>whether or not there are bijective maps between sets, > > > 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. > > Please notice, I am perhaps the first one who does not try to refute > Cantor's evidence. > This bings me to a fresh idea. There might ideed be a mathematical > aspect answering in part question 2): Cantor was like a matador. In > particular his second diagonal argument porovoked a lot of failed > attempts for refutation making him more and more famous. > I realized that the argument might be formally correct. All the attackes > felt that Cantor was wrong. Being, however, not aware of the true > location of Cantor's fallacy, they made the same mistake like Cantor > himself. > Please read M280. A translation into German is available elsewhere and > might hopefully a little bit more understandable in some decisive > details. Having originally stated that there are not more real than > rational numbers, I tried to exclude the possiblity to be mistaken by > writing: "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, what Cantor was fundamentally discussing is the existence of bijective maps between the reals and rationals. -- Will Twentyman email: wtwentyman at copper dot net
From: Eckard Blumschein on 11 Apr 2005 13:08 On 4/11/2005 4:06 PM, Matt Gutting wrote: > If any quantity is based on counting, how can one call anything not a > natural number a quantity? Of course, rational numbers can be derived from IN. > And I apologize, my second sentence contains an error. To correct that > error, let me ask: What do you mean by saying that "oo is just a quality?" This means, oo cannot be quantified. It is not a number. It cannot be enlarged, cannot be exhausted. It obeys rules different from those for numbers. >> Finite sets have the quality to allow comparison. If something exceeds >> any limit, then it does not offer this opportunity. > > Why not? Because exceeding any limit is not a quantitative measure. >> It was G. Cantor who introduced the principle of bijection in >> combination with complete induction in order to be surprized himself >> because not just the potentially infinite series of natural numbers but >> also the rational numbers are potentially countable. Realizing that the >> whole universe does mathematically seen not contain more points than a >> tiny linear interval, Cantor wrote: "Je le vois. Mais je ne le crois pas." >> > > Do you disagree that the principle of bijection does not answer the question > "Is this set as big as that one?" for finite sets in exactly the same way > as counting would? If you do (that is, if you can think of a pair of sets > which have the same count of objects but do not have a bijection between them, > or vice versa) I'd appreciate an example. If you do not disagree, then what > hinders one from extending this notion of bijection to infinite sets, as > long as it gives useful answers? The set IN is infinite. It is not a quantitative measure. > >> Intuition of stupid thinkers including G. Cantor and B. Russell goes >> wrong when it suggests that the infinite can be enlarged. The notion of >> infinity is best expressed by oo+a=oo. > > That is your opinion - but see my final comment below for a full response. Look into a good dictionary for infinity. >> Intuitively oo+a is bigger than just oo. However, oo cannot be enlarged. > > Intuitively, since oo is not a number, oo + a doesn't make sense - it's like > saying "apple + seventeen". No that is not the reason. oo+oo also equals oo. > And what, precisely (*very* precisely) do you > mean by "cannot be enlarged"? The property to have no limit. >>>>"Bigger" is the result of a quantitative comparison. But it has nothing >>>>to do with countable or not. >>> >>>"Bigger", as I said above, is not necessarily the result of a >>>quantitative comparison, nor need it be. If it is not particularly >>>helpful to mathematicians to consider it so, there is no reason to >>>ordain that it must be so. >> >> >> The word bigger is much elder than Cantorian nonsense. Everybody has a >> feeling for "bigger". However, nobody has a likewise correct feeling for >> the notion of infinite like something that cannot be enlarged except >> maybe for mathematicians, logicians, or engineers like me who developed >> a sufficient ability of abstract imagination. > > In the first place, infinity isn't defined as "something that cannot be > enlarged"; It is. in non-mathematical terms it might be described as "something > that cannot be exhausted" or "something that cannot end". Correct. > Even with that > issue cleared up, what people have a "feeling" for has no bearing on how it > is (or should be) used in mathematics. Mathematical usage is governed by > formal definition, agreed on by mathematicians, and by strict logical > consequences. It was Cantor who introduced his (wrong) feeling into theory. His thinking was not self-consitent. Poincarý spoke of an illness affecting mathematics. Spiniza's definition of infinity and the pertaining rules like oo+a=oo are still valid in mathematics. Eckard
From: Eckard Blumschein on 11 Apr 2005 13:45 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. 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. > >> 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. > 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. > 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. > I don't think he was planning > on the results. He was keen to create the most unusual. >> 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! >> 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. "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. > 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. Eckard
From: Matt Gutting on 11 Apr 2005 15:09 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. 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. > > >>>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. 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. > > >>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. I still don't understand why you say this. If Cantor defined infinity in a particular way and concluded that by his definitions, there were different orders or types of infinite sets, then he did in fact reveal different levels of infinity according to the definitions he used. 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. > > >>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. > > >>I don't think he was planning >>on the results. > > > He was keen to create the most unusual. > > > >>>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! > > > >>>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. > > > "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. > What constitutes a "compelling argument" depends highly on feelings - it depends on what one is willing to be compelled by. It is often the case that an argument which is not widely accepted is not compelling because it is simply incorrect. > >>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. What, exactly, do you mean by "approachable"? Matt > > Eckard > >
From: Matt Gutting on 11 Apr 2005 15:26
Eckard Blumschein wrote: > On 4/11/2005 4:06 PM, Matt Gutting wrote: > > >>If any quantity is based on counting, how can one call anything not a >>natural number a quantity? > > > Of course, rational numbers can be derived from IN. > True, but they cannot be counted. Real numbers can be derived by creating sequences of rational numbers, which are in turn derived from naturals. But you seem to be claiming that real numbers (or at least some of them) are not quantities. > > >>And I apologize, my second sentence contains an error. To correct that >>error, let me ask: What do you mean by saying that "oo is just a quality?" > > > This means, oo cannot be quantified. It is not a number. It cannot be > enlarged, cannot be exhausted. It obeys rules different from those for > numbers. > > > >>>Finite sets have the quality to allow comparison. If something exceeds >>>any limit, then it does not offer this opportunity. >> >>Why not? > > > Because exceeding any limit is not a quantitative measure. You are assuming that comparison requires a quantitative measure. One can define "comparison" in such a way that it does not require such a measure; I still don't see anything wrong with this sort of definition. > > > >>>It was G. Cantor who introduced the principle of bijection in >>>combination with complete induction in order to be surprized himself >>>because not just the potentially infinite series of natural numbers but >>>also the rational numbers are potentially countable. Realizing that the >>>whole universe does mathematically seen not contain more points than a >>>tiny linear interval, Cantor wrote: "Je le vois. Mais je ne le crois pas." >>> >> >>Do you disagree that the principle of bijection does not answer the question >>"Is this set as big as that one?" for finite sets in exactly the same way >>as counting would? If you do (that is, if you can think of a pair of sets >>which have the same count of objects but do not have a bijection between them, >>or vice versa) I'd appreciate an example. If you do not disagree, then what >>hinders one from extending this notion of bijection to infinite sets, as >>long as it gives useful answers? > > > The set IN is infinite. It is not a quantitative measure. What do you mean by saying that a set is not a quantitative measure? Of course a set is not a measure. If you want to say that the set has no quantitative measure, that's a different proposition; but even then, whether or not it has such a measure depends on how one defines "measure". It is possible to define "measure" to have meaning for the set of natural numbers. > > > >>>Intuition of stupid thinkers including G. Cantor and B. Russell goes >>>wrong when it suggests that the infinite can be enlarged. The notion of >>>infinity is best expressed by oo+a=oo. >> >>That is your opinion - but see my final comment below for a full response. > > > 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. > > > >>>Intuitively oo+a is bigger than just oo. However, oo cannot be enlarged. >> >>Intuitively, since oo is not a number, oo + a doesn't make sense - it's like >>saying "apple + seventeen". > > > 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)? > > >>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. > > > >>>>>"Bigger" is the result of a quantitative comparison. But it has nothing >>>>>to do with countable or not. >>>> >>>>"Bigger", as I said above, is not necessarily the result of a >>>>quantitative comparison, nor need it be. If it is not particularly >>>>helpful to mathematicians to consider it so, there is no reason to >>>>ordain that it must be so. >>> >>> >>>The word bigger is much elder than Cantorian nonsense. Everybody has a >>>feeling for "bigger". However, nobody has a likewise correct feeling for >>>the notion of infinite like something that cannot be enlarged except >>>maybe for mathematicians, logicians, or engineers like me who developed >>>a sufficient ability of abstract imagination. >> >>In the first place, infinity isn't defined as "something that cannot be >>enlarged"; > > > 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". > > in non-mathematical terms it might be described as "something > >>that cannot be exhausted" or "something that cannot end". > > > Correct. > > >>Even with that >>issue cleared up, what people have a "feeling" for has no bearing on how it >>is (or should be) used in mathematics. Mathematical usage is governed by >>formal definition, agreed on by mathematicians, and by strict logical >>consequences. > > > It was Cantor who introduced his (wrong) feeling into theory. His > thinking was not self-consitent. Poincarý spoke of an illness affecting > mathematics. > Spiniza's definition of infinity and the pertaining rules like oo+a=oo > are still valid in mathematics. > > Eckard > > > |