An uncountable countable set
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From: mueckenh on 30 Sep 2006 16:34 Dik T. Winter schrieb: > In article <1159437845.922031.117160(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > 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). > ... > > > I ask for self-evident truths. Upto now you have not provided any. > > > > For Cantor I + I = II is such a self-evident truth. (Of course with the > > usual meaning of "+" and "=".) > > How do you *define* "+"? I have not yet seen a definition from you. In > mathematics that operation can be defined in terms of the Peano axioms. In *mathematics* that operation has been known and been defined well enough 3000 years before Peano and before the cheeky set theorist. Regards, WM
From: mueckenh on 30 Sep 2006 16:38 Dik T. Winter schrieb: > In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > The successor function *is* counting (+1). > > > > > > Wrong. > > > > After a while you will have run out of the predefined successor, > > unavoidably. > > succ(x) = {x}. That is nothing else but a veiled form of +1. (This form of addition of 1 is due to Zermelo's, a little bit different from that of von Neumann's.) > > > Then you have no other choice but to add 1 each time you > > proceed. That is counting. > > Take the reals. succ(r) = sqrt(1 + r^2). Where is the counting? Didn't you really see it? It is twofold in this example. You count +, in the absolute value from 0 to 1 and in the exponent from 1 to 2: succ(0 + r^1) = srt(1 + r^2) > > > > > 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? > > > > Of course, but not only in German or English we see that phenomenon. I > > would be surprised if it were different in Dutch. > > Germanic languages in general. In Indian languages in general to 99. In > Welsh to 20. In Yoruba most similar to from -4 to 5. Thank you. Regards, WM
From: Virgil on 30 Sep 2006 16:39 In article <1159648032.835876.237760(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > > >Do I "misunderstand" that if you remove balls 1, then 11, then > > 21, etc, that the vase will NOT be empty? > > This is an extremely good example that shows that set theory is at > least for physics and, more generally, for any science, completely > meaningless. Because the numbers on the balls cannot play any role > except for set-theory-believers. Then by all means. less us do way with the balls and keep only the numbers. > > The same issue we have with Tristram Shandy "who writes his > autobiography so pedantically that the description of each day takes > him a year. If he is mortal he can never terminate; but if he lived > forever then no part of his biography would remain unwritten, for to > each day of his life a year devoted to that day's description would > correspond." (Fraenkel, Levy). > > When he is writing down always only the first on January, this > assertion of Fraenkel and Levy is certainly false. Actually, since its premise is false (no one lives forever) the implication itself ( if...then... statement) is quite true. That is the puzzling thing about material implications ( if...then... statements), when their "if" clauses are false, no matter what the "then" clauses say the entire "if...then..." statement is true. Similarly when the "then" clause is true, no matter what the "if" clause says, the implication is true. > And as absolutely > nothing is changed with respect to speed and quantity if Tristram > Shandy decides to follow another sequence, we see that this assertion > is always false. "Mueckenh"'s logic again fails as he has tripped over material implications.
From: mueckenh on 30 Sep 2006 16:43 Dik T. Winter schrieb: > > > So, again, no definition. Where did I not speak the truth? > > > > Here: "...because he never answered to questions about it". > > You gave on Usenet a definition of natural number in answer to my question > for it. I posted questions about that definition, and you never answered > them. So in what way is it a lie when I state that you never answered them? I did not see an unanswered question. Please repeat. By the way, please switch to the thread "Cantor Confusion" because this one has become too lengthy and, at home, I have only a slow internet access. So I am not able to follow this thread firmly. > > > Most > > questions on the representation of a number are answered in my paper. > > The questions were about the definition you gave in Usenet. And I do not > ask about "representations", I ask for a *definition* of the concept > "number". A proper, mathematical, definition. In my paper which you read I quoted Peano and v. Neumann on the first page. Then, on page 2, I wrote "Of course this realization of 2 presupposes some a priori knowledge about 2. But here we are concerned with the mere realization." That is my point! > > > The definition of an object does not provide its existence. > > Indeed. But when it is *defined* as the limit of a sequence, and if that > limit exists, that means that the object does exist. The limit omega does not exist. > > > The set of known prime numbers is bounded. Period. It is a specific set > > > that now and today consists of a fixed number of elements. It may be a > > > different set tomorrow, but that is something different, and again, > > > tomorrow it will be fixed and bounded. > > > > The cardinality of the set of prime numbers known today P(t) is as > > unbounded as the time variable t of today. > > If you talk like that you can not talk about "the set of known prime > numbers", because that is not a set, but a function of time. A set can be a function of time. > For > each 't', the outcome is a specific set. A properly defined set does > not change over time. That is your definition, not mine. We have the same with numbers. Of course, every number is a constant. Nevertheless, variables can exist for numbers (which can, but need not, be defined as sets) and for sets. "The letters X and Y in these expressions are variables; they stand for (denote) unspecified, arbitrary sets." Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel Dekker Inc., New York, 1984, 2nd edition., p. 4. > So when you talked about the set of known prime > numbers, I thought you were talking about the set of prime numbers known > at the time you wrote it, as I can give it no other interpretation. You fall back behind Cantor. He could. "ein in Veränderung Begriffenes Endliches, das also in jedem seiner actuellen Zustände eine endliche Größe hat." An example is the largest natural number you can imagine. Try it. There is always a larger one. Regards, WM
From: Virgil on 30 Sep 2006 16:53
In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > For instance: > > > Does the set o all natural numbers include 0? In old Greek it did not > > > even include 1. In future it may include even -1. > > > > Yes, indeed, that depends on your starting point with the natural numbers. > > That does not make it "all natural numbers and some more natural numbers". > > There is a bijection *possible*, but that does not mean that this > bijection is ruling the set number of numbers of sets. There is a > bijection possible between the sets {1,2,3,...} and {11, 12, 13, ...} > but the latter set obviously contains 10 elements more than the former "Mueckenh" has things a bit backwards here. , > even though a bijection between {1,2,3,...} and {101, 102, 103, ...} is > possible. The error of you is to believe that the possibility of a > bijection rules the number o numbers. It rules cardinality, because of how cardinality is defined. > (There are exactly twice so much > natural numbers than even natural numbers.) Therefore your assumption > of a uniquely defined number 0.111... is wrong. Unless some specific notation is specified, 0.11... is not even a numeral. > > > > Why not? Each and every number of the list terminates. That one is a > > number > > that does *not* terminate. > > > > > If you think that 0.111... is a number, but not in the list, > > > > It is *you* who insists it is a number. In most of my communications > > with you I put the word number in quotes, because it depends on how you > > interprete it on whether it is a number or not. > > It is me who insists that it is not a representation of a number. Once a notation is specified in which it is well formed as a numeral, then it represents a number. > > You could come up with that argument for arbitrary numbers, but not for > unary numbers. what you require is impossible. Either 0.111... is > larger than any number of the list, then you have to give a position > which is not covered by a list number or not. In binary to rational conversions, 0.0 = 0/1 0.1 = 1/2 0.11= 3/4 0.111 = 7/8 .... 0.111... = 1/1 |