An uncountable countable set
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From: Dik T. Winter on 30 Sep 2006 22:23 In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. "set number of numbers of sets"? What does *that* mean? > The error of you is to believe that the possibility of a > bijection rules the number o numbers. Your error is still that you believe that a cardinal number should act as a natural number. You have some presupposed ingrated definition for the word number that I do not understand. Would you be happy if we used the terms "ordinal fluffs" and "cardinal fluffs". Bijection rules the "cardinal fluffs". Are you happy now? > (There are exactly twice so much > natural numbers than even natural numbers.) By what definitions? You never state definitions. > Therefore your assumption > of a uniquely defined number 0.111... is wrong. Well, I almost always gave in this context the word number in quotes. So your assumption that I assumed a uniquely defined number 0.111... is wrong. Read back some time ago where I said (explicitly) that the only way we could interprete it was a string of digits. Note especially the article were I wrote about three possible interpretations. In my latest articles I simply interprete it as a string of symbolss, that you wish to see it otherwise is your responsibility. > > > 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. But it is the representation of some numbers, but that entirely depends on how you interprete that string of symbols. > > No. It is sufficient to prove that for each number on your list there is > > a position where it is different from that number. That does *not* > > imply that there is a position where it is different from each number > > of the list. > > 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. Now you are interpreting it as *number* with some additional connotations. Well, yes, if you wish to interprete it as number, I would say it is omega in unary notation. And so it is larger than any number on the list. What you mean with that I should give it a position is unclear to me. > Take into account that also Cantor's diagonal argument cannot be > executed in unary representation. Two red herrings in a single sentence. Can you get more? (1) Cantor's diagonal argument was about countable sequences of two symbols. There is only one countable sequence of one symbol. (2) Cantor's argument as augmented by Zorn and later by somebody who I do not know can not be executed for reals represented in base 3 or smaller. But reals are not tied to their representation. > The unary representation is capable > of modelling rational numbers like 0.111 / 0.11111 and even some > irrational numbers like sqrt(0.11). But it is not capable of modelling > Cantor's diagonal argument. What is the problem? The diagonal argument is not about representation, but about numbers (when you look at the modified argument). > And it is in clear contradiction with your > requirement of completely indexing but not covering 0.111... by the > list numbers. I think there was a proof, not a requirement. > > > k + omega is omega. And -k + omega is omega too. There is no well > > > defined set. > > > > In what way is that an answer to my question? Do you understand that > > 1 + omega = omega != omega + 1? > > And (as far as I know) -k + omega is not defined for positive k. (With > > the ordinals addition is defined only between ordinals.) > > Of course you can set up a bijection beween the sets > k + omega = {-k, -k+1, -k+2, ... , 0, 1,2,3,...,} and > -k + omega = {k+1, k+2, k+3, ...}. > But that does not mean that both sets have the same number of elements. You are utterly confused. Ordinals are concerned with order preserving bijections. A set with ordinal "k + omega" is, by definition of the operator, the set that is formed from the elements of a set with ordinal "k" followed by the elements of a set with ordinal "omega". Now try to apply that definition with -k, where k is positive. What is a set with ordinal "-k"? Do you know how ordinal addition works? > > > The axiom extensionality tells us that two sets are different, if they > > > differ in at least one element. If 0.111... differs from number n, then > > > it differs from all numbers m < n. As 0.111... is different from each > > > number of the list, it also differs from each one which is smaller than > > > another one. As every number of the list is smaller than another one, > > > 0.111... cannot be covered by all numbers. Hence, it cannot be indexed > > > by all list numbers. > > > > Again, that last conclusion is not justified. > > On the contrary, your assertions that indexing is possible but covering > is impossible, is completely unjustified and obviously wrong, as is > easily seen by the unary representation. I did give proofs for a specific sequenve. Your only response was that my definition was wrong. > > Apparently not with your burnt-in anti-logicism. I have *proven* that > > it can be indexed, by the simplest of all possible proofs. Namely by > > showing that there is no digit 1 at any position other than indic
From: David R Tribble on 30 Sep 2006 23:49 Tony Orlow wrote: >> For the sake of this argument, we can talk about infinite reals, of >> which infinite whole numbers are a subset. > David R Tribble wrote: >> What are these "infinite reals" and "infinite whole numbers" that you >> speak of so much? >> >> If you've got a set containing the finite naturals and the "infinite >> naturals", how do you define it? N is the set containing 0 and all >> of its successors, so what is your set? > Tony Orlow wrote: > The very same, with no restriction of finiteness. Any T-riffic number > has successor. :) Well, 0 is finite, and the successor of 0 is finite, and the successor of any finite in N is just another finite in N. Therefore N must contain only finite naturals. It's sporting of you to drop the requirement that all the naturals in N have to be finite, but since all of them are, it's meaningless to say "with no restriction of finiteness". That's kind of like saying N contains all naturals "with no restriction of non-integer values". I can say that, but it does not change the fact that all the members of N are integers. So I ask again, where are those infinite naturals and reals you keep talking about? It's obvious they are not in N.
From: mueckenh on 1 Oct 2006 06:23 Virgil schrieb: > 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. So we will keep only the numbers. The vase will be empty at noon if it emits 1,2,3,... but not empty if it emits 1, 11, 21, ...? Even for pure numbers that point of split-brains view is lot too silly. > > > > 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. Sorry, but that is a poor performance of yours. According to that we could also say: "but if he lived forever then large parts of his biography would remain unwritten"? Or even: If there is a complete set of natural numbers, then [insert anything you like]. That is not what Levy meant. Concerning Tristram Shandy, let his son and his grandson etc. continue the diary forever. Or would you think "forever" is in principle a false premise? Have you become a realistic thinker? Regards, WM
From: mueckenh on 1 Oct 2006 06:25 Virgil schrieb: > In article <1159648683.273921.24350(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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.) > > "Mueckenh" has it backwards, "+1" is merely a veiled form of "another", > which predates counting by millennia. And both Zermelo's and von > Nuemann's successor echo "another" faithfully without any requirement > for "+1". Another one. Yes, that is +1. Regards, WM
From: mueckenh on 1 Oct 2006 10:00
Dik T. Winter schrieb: > In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > 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. > > "set number of numbers of sets"? What does *that* mean? Meant is "numbers of elements2, alas everything is a set. > > > The error of you is to believe that the possibility of a > > bijection rules the number o numbers. > > Your error is still that you believe that a cardinal number should act > as a natural number. You have some presupposed ingrated definition for > the word number that I do not understand. Would you be happy if we used > the terms "ordinal fluffs" and "cardinal fluffs". Bijection rules the > "cardinal fluffs". Are you happy now? > Here I am interested in the possible indexes and the number of 1's in 0,111... . What else you my do, derive define, and prove with your fluffs interests me less. > > (There are exactly twice so much > > natural numbers than even natural numbers.) > > By what definitions? You never state definitions. By the only meaningful and consistent definition: A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. Do you challenge its truth? > > > Therefore your assumption > > of a uniquely defined number 0.111... is wrong. > > Well, I almost always gave in this context the word number in quotes. > So your assumption that I assumed a uniquely defined number 0.111... > is wrong. Read back some time ago where I said (explicitly) that the > only way we could interprete it was a string of digits. Note especially > the article were I wrote about three possible interpretations. In my > latest articles I simply interprete it as a string of symbols, that > you wish to see it otherwise is your responsibility. > Your main argument and my main target is the full presence of all digits and their indexibility by numbers all of which do not cover 0.111... . > > > > 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. > > But it is the representation of some numbers, but that entirely depends > on how you interprete that string of symbols. "..." is the representation of potential infinity, nothing more and nothing less. 0.111... cannot be interpreted as you try to do. > > > > No. It is sufficient to prove that for each number on your list there is > > > a position where it is different from that number. That does *not* > > > imply that there is a position where it is different from each number > > > of the list. > > > > 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. > > Now you are interpreting it as *number* with some additional connotations. I use your language if I discuss with you. I would never use such names like aleph in a monolog. > Well, yes, if you wish to interprete it as number, I would say it is > omega in unary notation. And so it is larger than any number on the list. > What you mean with that I should give it a position is unclear to me. > I mean: Indexing of all digit position of 0.111... by the unary numbers 0.1, 0.11, 0.111, ... is impossible unless all digi positions f 0.111... are also covered by these unary numbers. Instead of "to index position" we can also say "to cover up to position n". Hence you assert that it is possible to cover 0.111... up to every position but it is impossible to cover every position. > > Take into account that also Cantor's diagonal argument cannot be > > executed in unary representation. > > Two red herrings in a single sentence. Can you get more? > (1) Cantor's diagonal argument was about countable sequences of two > symbols. There is only one countable sequence of one symbol. Cantor's argument was about reals. He strived for generality but did not see that two symbols are not enough. > (2) Cantor's argument as augmented by Zorn and later by somebody who > I do not know can not be executed for reals represented in base > 3 or smaller. But reals are not tied to their representation. Therefore a general truth should not depend on the base 4 or larger. > > > The unary representation is capable > > of modelling rational numbers like 0.111 / 0.11111 and even some > > irrational numbers like sqrt(0.11). But it is not capable of modelling > > Cantor's diagonal argument. > > What is the problem? The diagonal argument is not about representation, > but about numbers (when you look at the modified argument). > > > And it is in clear contradiction with your > > requirement of completely indexing but not covering 0.111... by the > > list numbers. > > I think there was a proof, not a requirement. > > > > > k + omega is omega. And -k + omega is omega too. There is no well > > > > defined set. > > > > > > In what way is that an answer to my question? Do you understand that > > > 1 + omega = omega != omega + 1? > > > And (as far as I know) -k + omega is not defined for positive k. (With > > > the ordinals addition is defined only between ordinals.) > > > > Of course you can set up a bijection beween the sets > > k + omega = {-k, -k+1, -k+2, ... , 0, 1,2,3,...,} and > > -k + omega = {k+1, k+2, k+3, ...}. > > But that does not mean that both sets h |