An uncountable countable set
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From: Virgil on 1 Oct 2006 17:11 In article <45202171(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. It does away with all that ambiguity that TO was trying to introduce. > > >> 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? If that is equivalent to "If pigs could fly then the moon would be a balloon. ", yes. > > > > > 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. Whom does TO allege contends it? > > > 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. > > Vigilogic at its best (meaning worst). It is none of my doing. It is entirely material implication at its best/worst. http://en.wikipedia.org/wiki/Material_implication q.v. A close approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences.
From: Virgil on 1 Oct 2006 17:24 In article <4520236c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > 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. > > No, the successor to an infinite, after an infinite number of > successions from 0, is infinite. But that does not occur in N, but after N. N is defined as minimal, so it cannot contain those infinites. > > > > > 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. > > Is the successor to ...11110000 not equal to ...11110001? That is no more relevant than asking whether the successor of -9 is -8. > > > > > So I ask again, where are those infinite naturals and reals you keep > > talking about? It's obvious they are not in N. > > > > Not it's not. It is to anyone who knows how N is defined. There are inductive sets. Every limit ordinal is one. But there are limit ordinals which are not minimal among limit ordinals. There is a unique minimal limit ordinal, and that is the set we call N. Every member of that set is a finite ordinal, and every non-empty finite ordinal has an immediate predecessor. Since ordinals are, by definition, well ordered, they cannot contain and endlessly decreasing sequences, which TO's models require.
From: Virgil on 1 Oct 2006 17:26 In article <4520254e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > You do realize that my statements involve a considerable amount of > personal reflection, don't you? There is more to the number circle than > "proven". In the binary number circle, "100...000" is both positive and > negative infinity. What about "100...0001"?
From: Tony Orlow on 1 Oct 2006 17:26 Virgil wrote: > In article <1159711218.812268.276490(a)c28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Dik T. Winter schrieb: >> >>> In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> >>> mueckenh(a)rz.fh-augsburg.de writes: > >>> > (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? > > I challenge the "truth" of its being the ONLY meaningful and consistent > definition. > > "Mueckenh"'s claim is like that of a blind man claiming that colors are > imaginary. They are, even for the seeing. http://www.sciam.com/article.cfm?chanID=sa022&articleID=000DA6AC-F10C-1492-A7CE83414B7F0000 > > I, and many others, find both meaning and consistency in the definition > of cardinality. That "Mueckenh" does not is more a measure of his > incapacity than of any lack of meaning and consistency. WM, HdB, Finlayson, others and I see that the definition is lacking. There's also Zuhair and Petry, and a slew more. This is the most contentious of issues here. Perhaps you and I touched on the root of it, The nature of logical implication. Can you appreciate out striving for more. Are you so complacent in your position? Mine is out here. I'm not complacent. Would that I could be. > >>> > 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. > > A specific proof of a general truth can be based on whatever it is based > on. To the detriment of its generality. > > There are other proofs , including Cantor's first proof, which do not > depend on any sort of representations of the reals. > Cantor's first is an interesting proof of the uncountability of the continuum, and I consider it valid. It demonstrates that the notion that, for numerical strings a, b, and c in set S containing representation of all r in R, ((aeS ^ ceS ^ a<c) -> Eb ^ beS ^ a<b<c) -> (Es ^ seS ^ A neN length(s)>n) > >>> > 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. >>> >>> Stated without proof at all. What is erroneous about my definition? >>> Do you assert that definitions can be erroneous? If so, why? Do you >>> think the definition >>> Let a be the number such that a = 4 and a = 5 >>> is erroneous? I think not. It is a proper definition, but there is just >>> no 'a' that satisfies the definition. >> It is erroneous, because you say let a *be* which is false, if a cannot >> *be*. > > There is no such thing as an "erroneous" definition, except possibly in > the sense of a grammatically incorrect one. A definition may lack any > instantiation, such as a 4 sided triangle, but as a definition is valid. yada blabba flob. Sure Virgule.
From: Tony Orlow on 1 Oct 2006 17:27
Virgil wrote: > In article <45201554(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451df41c(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <451dd1f2(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Virgil wrote: >>>>>>>> Are you saying that aleph_0 naturals only require ln(aleph_0+1)/ln(2) >>>>>>>> bit positions? >>>>>>> Not at all. I am talking about indvidual natural numbers as members of >>>>>>> N, ,not N itself, which is not a member of N. >>>>>> And also for every set of contiguous naturals starting at 0 EXCEPT for >>>>>> N. Why EXCEPT for N? >>>>>> >>>>> For the same reason that a paper sack holding oranges is not an orange. >>>>> >>>> A set is a sack? It is nothing besides the elements it includes. >>> A set is a container, and is not one of the objects that it contains. >> It is nothing more or less than its contents. > > It is determined uniquely and entirely by its contents, as stated in the > axiom of extentionality. So we agree. There is nothing besides the members. |