An uncountable countable set
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From: Tony Orlow on 1 Oct 2006 15:54 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. Yes. I was flabbergasted by this example of "logic". The amazing thing is, set theory is supposed to apply where we know nothing except for the membership status of each element in a set, and yet, here is applied this property of labels that set theorists claim is crucial to answering the question. Set theory in the finite sense is a fine thing, but when it comes to the infinite case, set theorists don't even know anymore what they're TRYING to do. > > 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). :) That's precisely the same problem, and my guess is that Virgil thinks his autobiography will be a blank page, as long as he lives forever. That's pretty sensible, isn't it? Ha ha. > > When he is writing down always only the first on January, this > assertion of Fraenkel and Levy is certainly false. 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. > > Regards, WM > I suggest this Tristam Shandy start writing ahead if he's ever going to catch up. :) Have a nice day. Tony
From: Tony Orlow on 1 Oct 2006 16:02 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, > 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. (There are exactly twice so much > natural numbers than even natural numbers.) Therefore your assumption > of a uniquely defined number 0.111... is wrong. Indeed, where the "proper superset is larger" rule becomes violated, it's not a good theory, even if the subset relation does not apply to all set size comparisons. >> 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. Well, Wolfgang, that sets us apart, though I agree it's not a "specific" number. It's still some kind of quantitative expression, even if it's unbounded. Would you agree that ...333>...111, given a digital number system where 3>1? >> > then >> > you must be able to find a position which is different from those of >> > the numbers of my list. >> >> 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. It's at the "end", you know, the aleph_0th place. :) > > Take into account that also Cantor's diagonal argument cannot be > executed in unary 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. And it is in clear contradiction with your > requirement of completely indexing but not covering 0.111... by the > list numbers. That's because Cantor's diagonal argument concerning the reals is not about the reals, but about symbolic language, as he stated originally. I was later applied to the reals. > >> > > > 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)? >> > >> > 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. Cardinality is a weak measure of size for infinite sets, the operative word here being "measure". > >> >> > > > Which index distinguishes 0.111... from all the numbers >> > > > of the list? You cannot answer? >> > > >> > > I can. None. >> > >> > 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. >> > > So the number can be indexed. >> > >> > It is curious. You could also assert something like "I can shout louder >> > than anybody else but there is nobody who cannot shout louder than me". >> > But it is impossible to try to exorcise your burnt-in anti-logicism. >> >> 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 indicated >> by a natural number, which *by your* definition makes the number >> indexable. > > You have not *shown* that, but defined it, erroneously. But if you had > shown it, then 0.111... was in the list, which also would have been > wrong. Subtract 1 from 0 in binary. >> > > > 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, an
From: Virgil on 1 Oct 2006 16:04 In article <1159727459.165196.109230(a)b28g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Randy Poe wrote: > > > > Hence my continued statement that the vase does not > > "become empty". It is non-empty at certain times and > > empty at others. There is no transitional moment. > > Nature does not jump, Leibnitz said. Who said anything about this being "nature"? "Nature" does not have an endless supply of balls in the first place, > > > Noon is the first moment at which the vase is empty. > > > But noon is not the transitional moment. There's no > > time just before noon where the transition happened. > > Wow ! And _that_ calls himself a physicist ... Leave it to a physicalist to insist that a nonphysical problem is physics.
From: Virgil on 1 Oct 2006 16:05 In article <1159728969.000444.313320(a)i42g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil wrote: > > > In article <1159725088.430659.72630(a)m7g2000cwm.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > stephen(a)nomail.com wrote: > > > > > You have fundamentally changed the "experiment". > > > > > > Worse. I have fundamentally changed the mathematics. > > > > > > Han de Bruijn > > > > HdB only has the power to dictate his own version of mathematics. He has > > not the power to control anyone else's mathematics. > > How do you know I'm not teaching mathematics to anyone? Even then, the poison is localized.
From: Tony Orlow on 1 Oct 2006 16:13
Virgil wrote: > 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. That doesn't make it any more sound. That's the point. >> 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. So, you are a devout counter-intuitionist. If pigs could fly the moon would be a balloon. That's a valid logical statement? > > 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. That's a matter of contention, as you well know. > Similarly when the "then" clause is true, no matter what the "if" clause > says, the implication is true. > The "if" in that case is meaningless. Where the "if" is false, that is, 0, what is 0^x when x is 0? If the "then" clause is true, what is x^1, when x is 0? This is what the intuitionistic perspective boils down to, as I will show in a paper, hopefully soon. > >> 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. Vigilogic at its best (meaning worst). |