An uncountable countable set
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From: cbrown on 25 Sep 2006 22:09 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > > <snip> > > > >> I did give another curve with the same "Tlimit" as the staircase in the > >> limit, which produced an interesting result, giving weight to the notion > >> that an infinitesimal is something distinct from 0, whose square is not > >> distinct from 0. > > > > Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an > > infinitesimal. Then what you are saying means > > > > 1/B = 1/B > > 1*1/B = 1/B > > (B*1/B)*1/B = 1/B > > B*(1/B*1/B) = 1/B > > B*(1/B^2) = 1/B > > > > Since 1/B is infinitesimal, its square is not distinct from 0; so... > > > > B*(0) = 1/B > > 0 = 1/B > > > > So 1/B is identical to 0. Where is my error? > > > > Cheers - Chas > > > > Hi Chas - > > I haven't been online lately, so please excuse the delay. You haven't > made any error, except in interpretation of my position. I said that I > got an interesting result from applying a segment-sequence definition of > your staircase-and-diagonal comparison, which gave "weight" to the > notion of an infinitesimal being defined in such a way. Well, why doesn't the logic I gave above give obvious "weight" to the notion that your nilpotent infinitesimals imply that 1/B = 0? Or, since then B*1/B = B*0, that 1 = 0? Doesn't that tend to contradict your "intuition" regarding T-numbers? Cheers - Chas
From: Dik T. Winter on 25 Sep 2006 22:09 In article <1159183624.817772.5540(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > No. There are "numbers" (exactly one) that can be indexed by that list > > > > but that is not in that list. > > > > > > That is a false belief. And why should there be only one of those > > > numbers? Why couldn't it be two at least or even ten? If you answer > > > this question, you have a good chance to recognize that not even one > > > exists that is not in that list but can be indexed by the numbers of > > > that list. > > > > Because there is only one set that contains *all* natural numbers. > > Why? Your assertion is without proof. I should have stated: "there is only one set that contains *all* natural numbers and no other numbers". But that is easily proven once you understand set theory. > Why should there be only one number > 0.111... ? By what property is this 0.111... different from all the > numbers in the list? Because it has infinitely many digits? > And why can't there be more than one number with > infinitely many digits? You cannot answer these questions because > already one infinite set is a contradiction. No, I can not answer this question because I have no idea what you mean with a number with more than omega digits. Consider K = 0.111... . What is K+1? Can you provide a definition (as I did for K)? > > There is only *one* set of natural numbers, so there is only *one* > > sequence 0.111... that can be indexed using *all* natural numbers. > > Who said so? If, in fact, one set of all natural numbers could exist, > then also many of them could exist. Proof? > Think of the axiom of > comprehension. Which one? > Which index distinguishes 0.111... from all the numbers > of the list? You cannot answer? I can. None. So the number can be indexed. > So we cannot answer which index > distinguishes the many different infinite digit sequences 0.111... from > each other. What different infinite digit sequences? I note that digit sequences are countable, and so there is only one infinite digit sequence. > > > > Do you agree that the non-terminating list can index non-terminating > > > > numbers? If the answer is no, why? > > > > > > Because it should index at least two different of those infinite > > > numbers if it could index one of them. > > > > Why? > > Either: There is an index which distinguishes 0.111... from any number > of the list. > Or: There is a number which cannot be distinguished by indexes. > But if there was one such number admitted, how could the existence of > many of them be excluded? Because there is a single minimal countable but infinite set? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Sep 2006 22:22 In article <1159184068.040882.61000(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In modern mathematics there are no self-evident truths. There are the > > axioms that are the basic material to work with and the theorems that > > follow from them. In that sense, Cantor's Grundsatz is completely > > equivalent with an axiom as used in modern mathematics. And so, > > translating it with "axiom" is neither wrong, nor misleading. > > It is wrong and misleading because Cantor was not a modern > mathematician and did not accept "axiom" in the modern meaning of this > notion. His "Grundsatz" is not subject to arbitrary choice. So be it. In that case target Cantor, not modern mathematics. The translation I gave was the best I could do with modern mathematics. I have no idea how to translate it to a term of early 20-th century mathematics. > > > > Educate me. What are they? Pray provide sources. But your rubbish > > > > is not my rubbish... > > > > > > Cantor's truths are self-evident truths which cannot be changed > > > arbitrarily in contrast to the axioms of modern set theory. > > For instance: I + I = II. In some cases, yes. In other cases, no. Depends entirely on how you define "+" and the other symbols. In Greek mathematics I would expect I + I = K (as in 10 + 10 = 20). > > > > Yeah, whatever. Is this a reply to my question? > > That was my intention, yes. If I failed, you should improve the > precision of your question. I ask for self-evident truths. Upto now you have not provided any. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: cbrown on 25 Sep 2006 22:34 Tony Orlow wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Mike Kelly schrieb: > > > >>>>> So lim [n-->oo] n/aleph_0 < 1 > >>>> Division is not defined for infinite cardinal numbers. > >>> Is that your only escape? If you dare to say that aleph_0 > n for any > >>> n e N, then we can conclude the above inequality. > >> No, because division is not defined on infinite cardinal numbers. The > >> above inequality is meaningless. > > > > For cardinals we have well defined n*aleph_0 = aleph_0 for any natural > > n (see any book on set theory). Multiply with 1/n for any natural n, > > then you get well defined > > > > A n e N: aleph_0 / n = aleph_0 > 1. Hence, my followin statement is > > correct. > >>> But remedy is easy. > >>> Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions > >>> analogously. > > [...] > > > >> Your position seems very inconsistent. You claim that numbers have no > >> existence outside their representation. And now you are claiming there > >> exists a "true" arithmetic. > > > > It is obvious. You can verify it by experiment: II + III = IIIII. > > > > [...] > >> It is not a proof. Division is not defined where either operand is an > >> infinite cardinal number. > > > > But you can conclude n / aleph_0 < 1 by inserting aleph_0 > n which is > > definied *if aleph_0 is a number in trichotomy with natural numbers*. > > > > You cannot have both, assert that aleph_0 is a number larger than any n > > but on the other hand prohibit that the inequality n < aleph_0 be > > utilized. > > > > Regards, WM > > > > I agree. If x is infinite, and that means greater than any finite, and > trichotomy holds, then 1/x is in [0,1], and is real, though > infinitesimal, of course. But, I am getting the feeling that set > theorists now don't want to claim that infinite is greater than finite. I think it would be more correct to say that set theorists don't now want, and have never wanted, to claim that some mathematical object is greater than another mathematical object without an /explicit definition/ of what is meant by "greater than" in the context of the discussion. > Hmmm... Indeed. Cheers - Chas
From: Dik T. Winter on 25 Sep 2006 22:36
In article <1159185473.917531.157040(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Representation is number. There is no difference. Numerals have no > > > "soul". > > > > Makes no sense to me. You may represent a natural number as you wish, > > that does not change anything. > > Correct. 3 or III or {D,i,k} is a means which can be used to calculate > with. Or a or whatever. > > You stated that you needed counting to determine the successor. That is > > false. The successor is defined without any reference to counting. > > The successor function *is* counting (+1). Wrong. > The successors are defined > without counting only over a very restricted domain. In the usual > decimal systems only from 1 to 12 and then repeating again and again > from X to X + 9. You think so because you again focus on the decimal system. I wonder how you get at 12. German influence? > > > > Sometimes from 0. > > > > > > Who did so before Cantor? > > > > Some people from India. There are quite a few where you can read about > > the zeroth year of the reign of somebody. Have a look at the vast number > > of different calendars and year countings that have been in use in India. > > That is curious, but consequent. Our calendar consists of years BC and > AC but does not contain a year zero. That frequently leads to errors. You really do not understand. > Of course it is correct, to count divisible objects (like years or > electric current) beginning with 0, 0.1, 0.333 or so. But that is not > applicable to counting of indvisible elements of sets. In the different way the people from India count the era of the reign of a sovereign they either start at 0 or at 1. They switch to the next number (1 or 2) either at the virthday of the sovereign or when he reigns one year. This has nothing to do with the divisibility or whatever. Most curious is a way of counting in some sovereignty where they count the years of the reign as 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, etc. A truly octal counting based on 0, using the digits 1, 2, 3, 4, 5, 7, 8 and 9. The successor function? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |