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From: Eckard Blumschein on 13 Apr 2005 07:08 On 4/12/2005 9:03 PM, Matt Gutting wrote: > >>>I'm not sure what you mean by "loss of approachable identity". The >>>difference between the rationals and the reals is that every convergent >>>sequence of reals converges to a real, while not every convergent >>>sequence of rationals converges to a rational. >> >> >> In other words, numerical representations of rational numbers do not >> require infinitely many numerals. Convergency invites to restrict to a >> finite number of coefficients. In that case you do not reach a real >> number but are satisfied by an rational approximation instead. Real >> numbers are fictitious. >> > > Numerical representations of reals need not require infinitely many > numerals either. And whether even a rational number requires an infinite > number of digits to be represented depends on the method of representation > chosen. Such objections were already cleared away in many discussions. Do you really need resuming this issue? Please take into account that I (like Cantor) refer to a uniform repesentation of all reals together e.g. the decimal one. In this "system" the natural number 4 is embedded like 4.000000000... =3.999999999999999... Infinitely many numerals are essential even if we do not refer to an irrational number. > > I'm not sure what you mean by "Convergency invites to restrict to a finite > number of coefficients". Do you mean that to say "this sequence converges > to the real number r" is to say that "r can be represented as a number > which begins with the digits of one of the elements of this sequence"? > That is true. However, what r *is* and what r is *approximated by* are two > different things, and mathematicians keep that fact in mind. In this sense, > the real numbers are not fictitious. The real numbers exist mathematically in the sense they are fictions. >> Even embedded natural numbers cannot be numericall identified without >> all infinitely many numerals e.g. 3,99999999999999999999999999999... > > I thought you just said that "numerical representations of rational numbers > do not require infinitely many numerals"? Since 4 = 3.999999... is an integer > and therefore a rational number, you appear to be contradicting yourself here. No. See above. >> Did you refer to infinite sets? >> >> > > No, but neither did you. You were speaking here, as I understood it, > simply of how one compares one set to another ("Isn't a quantitative > measure a quantity to compare with?"). Yes. In that case I referred to finite sets. > My point is that one can describe a method of comparison between sets > which (i) works for infinite sets exactly the way it does for finite sets, > (ii) gives meaningful, well-defined comparisons for any two sets, (iii) > does not refer to "quantity" in any way, and (iv) yields exactly the same > results as a quantitative comparison does for those sets to which "quantitative > comparison" applies. Having found such a comparison, I wonder if you really found it. Concerning infinite sets I am only aware of the possiblity do decide whether or not there is a bijection being synonymous to countable. > I don't > see an immediate connection between "boundlessness" (something that cannot be > exhausted or limited) and "something that cannot be enlarged". Descriptions are indeed sometimes difficult. Try to describe the principle of a sewing machine: Using a needle with the eye on top. >> A definition should not contain the defined expression. > > I believe your native language is German? Yes. > You may be confusing "infinite" (in this sense, perhaps "unbegrenzt")and "indefinite" ("unbestimmt"). No. > The definition does not, in this case, contain the defined expression. It does. I refer to: >>>"Infinity: >>>2 an indefinitely great number or amount. in English, "indefinitely" has two differnt meanings: - without clear definition, vague e.g. a view - without limit, e.g. a value "2" relates to "without limit". > Perhaps I misinterpreted what you meant by saying "A part of mathematics would > go slippery..." Would you mind explaining that? Just an example. Children at school must not be taught Cantor's nonsense infinite whole numbers. There are no infinite numbers. Eckard
From: Eckard Blumschein on 13 Apr 2005 07:11 On 4/12/2005 8:32 PM, Matt Gutting wrote: > > I'm not sure exactly what you mean by "numerically identify". If you mean > that there is no symbolic representation for any real number, then you > are quite obviously wrong, since e (for example) is a finitely long symbolic > representation of a single, specific real number. If you mean that there > is no finitely long decimal expansion for a real number, this is certainly > true of some real numbers, but not all of them. And the same is true for > rationals. (Besides which, any rational, and thus a considerable part of > the reals, can be represented with a finite expansion given an appropriate > base.) I apologize for not explaining these simple fallacies again and again. E.
From: Eckard Blumschein on 13 Apr 2005 07:42 On 4/12/2005 11:21 PM, David Kastrup wrote: > And the core of > Cantor's argument is sound, It is fallacious. He introduces non existing infinite numbers in order to misinterpret the unquestionably different qualities of rational und real numbers. Just the latter are uncountable. Cantor's thinking is at best fallatious. He ignores that both the rational and the real numbers do likewise have the quality to be infinite, and infinite cannot be enlarged. Consequently, the missing possibility to represent the reals in a list cannot be attributed to a larger size (Maechtigkkeit, cardinality) of the reals but it relates to something else. >> So he tried to show that there are more reals as compared to the >> "size" of the set of the rationals. > > Nope. He showed that no bijection can be established. His second diagonal argument was an evidence by contradiction. Cantor misinterpreted it by claiming that there are more real numbers than his list contained. He overlooked the correct possiblity that his assumed list simply did not represent the reals. > And that means > that there is an order of cardinalities, where cardinalities are > considered as an indicator of surjectability of sets. Please indicate the pertaining pages where he claimed and proved that. >> I argue that such comparison lacks any basis. Infinity is a quality, >> not a quantity. > > You are just waffling around with stupid terms. The difference between quality and quantity is quite fundamental. > The existence and > non-existence of surjections is a _hard_ fact that has nothing to do > with any philosophy of "infinity". And it also is a hard fact that > being surjectable is a transitive and reflexive property. What do you think about surjection between IR and IR+? > If the reals obeyed the laws of ordinary numbers, > they would be structural equivalent to them and could be put into a > one-on-one correspondence with them. Yes. I agree with you that the reals do differ from ordinary numbers. > Cantor showed that this is impossible, and thus the reals fail to obey > the laws of integers and can't be brought in correspondence with them. I agree. > This is what his second diagonal argument is about. Exactly that, and > nothing else. So far I do not object. And who added the whole cardinality story? Infinite numbers are insane. Cantor got famous by means of a incredible misinterpretation. I am suggesting a less spectacular explanation that does not contradict to very fundamental rules. >
From: Eckard Blumschein on 13 Apr 2005 07:56 On 4/12/2005 6:34 PM, Chris Menzel wrote: >> Whether or not an infinite set is countable depends on its structure. > > Sets don't have structure. What you seem to have in mind is that a set > is to be considered countable or not depending on how it is *ordered*. You are correct. To me the real numbers only make sense together with their ascending order. I am avoiding the word well-ordered because it is burdened with expectations that cannot be fulfilled with the reals. > So do you think the set of rational numbers is uncountable when ordered > by the less-than relation and countable when well-ordered in some > familiar fashion? No. The set of rational numbers is countable because it consists of genuine numbers in the sense, each of them is approachable with a finite amount of numerals. > >> The reals are obviously not countable because one cannot even >> numerically approach/identify a single real number. > > It's not clear what it is to "numerically approach" a number -- > certainly what you say is false if we understand "approach" in terms of > limits -- but could you demonstrate your thesis with regard to, say, the > real number 2? Haven't I just identified it? The 2 is a natural number. After embedding into the reals it reads in decimal representation either 2.000000000... or 1.9999999999999... There is no chance to completely address its successor. Eckard Blumschein
From: Eckard Blumschein on 13 Apr 2005 08:03
On 4/12/2005 11:46 PM, Will Twentyman wrote: >> Cantor was mislead by his intuition. >> I do not attribute the difference between countable and non-countable to >> the size of the both infinite sets. > > What do you view the difference between them to be? The difference resides in the property of each single real number itself. Cantor assumed his list represents all real numbers. Actually, nobody can provide any list of real numbers, not even two subsequent of them can be named. >> Actually, infinity is not a quantity but a quality that cannot be >> enlarged or exhausted. Whether or not an infinite set is countable >> depends on its structure. The reals are obviously not countable because >> one cannot even numerically approach/identify a single real number. > > No, that is NOT the reason the reals are not countable. This was your statement. Where is your evidence for it or at least some justification? Simply tell me the successor of pi. Eckard |