Why can no one in sci.math understand my simple point?
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From: MoeBlee on 17 Jun 2010 13:23 On Jun 17, 2:49 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 23:43, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > Look simply at the results. > > > If a theory says that there is an uncountable set of real numbers such > > > each number can be identified as a computable or definable or > > > constructable one, or in other ways, then this theory is provably > > > wrong. > > > Please tell me which EXACT, SPECIFIC theorem of ZFC "says" that "each > > real number can be identified as a computable or definable or > > constructable one, or in other ways". > > If you start with Cantor's list, then the diagonal can be identified. > Do you mean that in ZFC Cantor's argument is wrong? In this particular conversation, my interest is in Cantor's argument as it is formalized in Z set theory. And of course it's not "wrong". It is simply first order logic applied to axioms. You may wish to have any feeling or conviction you like about first order logic and the axioms, but that Cantor's argument can be formalized as proof in Z set theory is not controvertible. (As I alluded, a proof being a certain kind of finite sequence of finite sequences of symbols.) > Further: A number that cannot be put in trichotomy with others is not > a number. Is such an unknown entity called a number in ZFC? I make no use of a predicate 'is a number' where 'number' is STAND- ALONE nor does an ordinary formalization of Cantor's argument. And I don't know what failure of trichotomy (in ZFC) you might have in mind. > > > Reason: Cantor either proves that a countable set is uncountable or > > > that a constructible/computable/definable number is not constructible/ > > > computable/definable. > > > Whatever Cantor proves, my question is specifically as to ZFC. > > I consider results only. If ZFC allows the conclusion that Cantor's > diagonal argument is wrong, then ZFC may be right. Is it right? "right", "wrong" are your terminology. I've stated already now my most basic view of Cantor's argument as formalized in Z set theory. Whether this or that is, in some other sense, "right" or "wrong" is another discussion, but not needed to address the mere question of the theoremhood of a certain formula in the language of ZFC. > > > What the theory internally may be able to prove or not to prove is, at > > > least for my person, completely uninteresting. > > > What interests you is entirely up to you, of course. But you've not > > shown that ZFC is inconsistent (where 'inconsistent' is defined as > > "having as a theorem some formula and its negation"). > > I have been told that ZFC yields this result: There are uncountable > many real numbers. And by number I understand something that can be > put in trichotomy with other numbers. ZFC doesn't make use in this context of a predicate "is a number" where 'number' is STAND-ALONE. As to 'real number" we may regard that as single predicate. In light of the formal itself itself, we could as well write 'realnumber' or 'realschmumber' or 'schmeal schmumber' so that 'number' has no component meaning onto itself. You should understand that names of predicates in formal mathematics are not always compositional, particular with adjectives. I can define "is a slooper spockle" without committing that 'slooper' and 'spockle' have independent meaning. This is because English is only an INFORMAL rendering of certain formal predicates. So, as to the FORMAL theory, for example, where we say "x is a real number", the actual formula would be "Rx" where 'R' (or whatever other typographic shape) has been chosen as the formal predicate symbol ordinarily rendered in English as 'is a real number'. Please tell me whether you understand this. If you find fault in it, then so be it. I'm not interested in arguing with you about it. But at least you know now how I regard such matters in terminology. > If this is not the case, ZFC may > be right. > Others have told me that ZFC does not prove any uncountability at all > and that the "standard model" is not a model of ZFC at all. If this is > the case, ZFC may be right. ZFC proves a FORMULA that we READ or RENDER in ENGLISH as "there exists an x such that x is uncountable". As to models, there are different senses of the word 'model' depending on context of discussion. MoeBlee
From: MoeBlee on 17 Jun 2010 13:36 Re: Jun 17, 12:23 pm, MoeBlee <jazzm...(a)hotmail.com>e: P.S. If the discussion about the formal theory ZFC, then I don't know what lack of trichotomy you're referring to. Of course, many relations fail to satisfy trichotomy. But the standard strictly less than ordering on the reals satisfies trichotomy: If x and y are reals, then exactly one of these holds: x <_r y y <_r x x=y where '<_r' stands for the standard stricly less than relation on the set of real numbers. Or to be pedantic: <x y> e <_r <y x> e <_r x=y And there is no ordering on the cardinals, since there is no set that has all cardinals in it, but still, we have the theorem (rendered here in English and symbols; as I'll not make that note usually now): If X and Y are cardinals, then exactly one of these three: X <_c Y Y <_c X X = Y where '<c' stands for the cardinal strictly less-than predicate. Thus for any sets S and T, exactly one holds: card(S) <_c card(T) card(T) <_c card(S) card(S) = card(T) MoeBlee
From: WM on 17 Jun 2010 14:21 On 17 Jun., 12:16, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Virgil <Vir...(a)home.esc> writes: > > In article > > <995d761a-f70f-4bca-b961-8db8e1663...(a)d37g2000yqm.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > >> On 15 Jun., 22:24, Virgil <Vir...(a)home.esc> wrote: > > >> > Note that it is possible to have an uncomputable number whose > >> > decimal expansion has infinitely many known places, so long as it > >> > has at least one unknown place. > > >> That is mathematically wrong. > > > It may not match every definition of 'uncomputable', but otherwise it is > > right. > > It doesn't really match any definition of 'computable'. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus- Zitierten Text ausblenden - As you seem to be able to understand German: Here is a paragraph from Fraenkel celebrated book (3rd ed. 1928, Springer). The annotations in double curly brackets are mine. Die Ordnungsregel kann, wenn M eine endliche Menge ist, durch Aufzählung aller Paare von Elementen und Angabe der jeweils gültigen Ordnungsbeziehung ausgedrückt werden (etwa tabellarisch). Ein derartiges Verfahren ist natürlich bei einer unendlichen Menge M unmöglich. {{Natürlich! Wie konnte diese Erkenntnis so in Vergessenheit geraten?}} Hier muß an die Stelle einer Aufzählung oder Tabelle das Hilfsmittel treten, dessen sich die Mathematik allgemein bedient, um unendlichviele Einzelsachverhalte in einen endlichen Ausdruck zu kleiden: das Gesetz (zu dessen Aussprache man meist Formeln benutzt). Es verhält sich damit also ebenso wie mit der Abbildung zweier äquivalenter Mengen (d. h. der umkehrbar eindeutigen Zuordnung ihrer Elemente) die zwar bei endlichen Mengen durch eine Summe von Einzelvorschriften, bei unendlichen aber nur durch Angabe eines Zuordnungsgesetzes zu vollziehen ist. {{Kann man es noch klarer sagen? Diese Aussage gilt aber nicht nur für äquivalente Mengen, sondern auch für die Angabe einer unendlichen Ziffernfolge. Die Zuordnung einer Ziffer zu jeder natürlichen Zahl ist ohne Gesetz nicht möglich! Deshalb gibt es für unendliche Folgen auch kein Diagonalverfahren. Und daher ist das hier zitierte Buch fast gänzlich falsch und überflüssig - mit einer einzigen Ausnahme, nämlich der hier vorgestellten. Allein dieser Erkenntnis wegen lohnt die aufmerksame Lektüre.}} [Adolf Fraenkel: "Einleitung in die Mengenlehre" 3. Aufl., Springer, Berlin (1928) p. 125f] Gruß, WM
From: WM on 17 Jun 2010 14:27 On 17 Jun., 15:56, Sylvia Else <syl...(a)not.here.invalid> wrote: > On 15/06/2010 2:13 PM, |-|ercules wrote: > > > > > > > Consider the list of increasing lengths of finite prefixes of pi > > > 3 > > 31 > > 314 > > 3141 > > .... > > > Everyone agrees that: > > this list contains every digit of pi (1) > > > as pi is an infinite digit sequence, this means > > > this list contains every digit of an infinite digit sequence (2) > > > similarly, as computable digit sequences contain increasing lengths of > > ALL possible finite prefixes > > > the list of computable reals contain every digit of ALL possible > > infinite sequences (3) > > Obviously not - the diagonal argument shows that it doesn't. > There is no diagonal element for a list of finite lines. And there is no infinite representation of an irrational number. The function f : N --> {0, 1, 2, ..., 9} can be defined by a finite word but not by an infinite sequence. The list of all computable reals is a sub-list of the following list of all words: 0 1 00 01 10 11 000 .... in binary representation. This list has no diagonal. There is no word missing. Regards, WM
From: WM on 17 Jun 2010 14:54
On 17 Jun., 19:23, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 17, 2:49 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 16 Jun., 23:43, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > Look simply at the results. > > > > If a theory says that there is an uncountable set of real numbers such > > > > each number can be identified as a computable or definable or > > > > constructable one, or in other ways, then this theory is provably > > > > wrong. > > > > Please tell me which EXACT, SPECIFIC theorem of ZFC "says" that "each > > > real number can be identified as a computable or definable or > > > constructable one, or in other ways". > > > If you start with Cantor's list, then the diagonal can be identified. > > Do you mean that in ZFC Cantor's argument is wrong? > > In this particular conversation, my interest is in Cantor's argument > as it is formalized in Z set theory. And of course it's not "wrong". > It is simply first order logic applied to axioms. No, this application is erroneus. A real number cannot be defined by an infinite sequence. A real number can be defined only by a finite word. But there is no diagonalization over finite words. > You may wish to have > any feeling or conviction you like about first order logic and the > axioms, but that Cantor's argument can be formalized as proof in Z > set theory is not controvertible. (As I alluded, a proof being a > certain kind of finite sequence of finite sequences of symbols.) But you start with a wrong assumption. This is proved by the contradiction: Either Cantor proved the uncountability of a countable set or Cantor proves the inconstructibility of a constructible number. The translation of these notions into your "incontrovertible" theory is the weak point. > > > Further: A number that cannot be put in trichotomy with others is not > > a number. Is such an unknown entity called a number in ZFC? > > I make no use of a predicate 'is a number' where 'number' is STAND- > ALONE nor does an ordinary formalization of Cantor's argument. > > And I don't know what failure of trichotomy (in ZFC) you might have in > mind. Then forget it. It is not important. Imporrtant is: Either Cantor proved the uncountability of a countable set (namely the constructible numbers) or Cantor proves the inconstructibility of a constructible number. > > > > > Reason: Cantor either proves that a countable set is uncountable or > > > > that a constructible/computable/definable number is not constructible/ > > > > computable/definable. > > > > Whatever Cantor proves, my question is specifically as to ZFC. > > > I consider results only. If ZFC allows the conclusion that Cantor's > > diagonal argument is wrong, then ZFC may be right. Is it right? > > "right", "wrong" are your terminology. I've stated already now my most > basic view of Cantor's argument as formalized in Z set theory. Whether > this or that is, in some other sense, "right" or "wrong" is another > discussion, but not needed to address the mere question of the > theoremhood of a certain formula in the language of ZFC. Does ZFC not prove that all constructible numbers are countable? Is Cantor's diagonal not a constructible number? > > > > > What the theory internally may be able to prove or not to prove is, at > > > > least for my person, completely uninteresting. > > > > What interests you is entirely up to you, of course. But you've not > > > shown that ZFC is inconsistent (where 'inconsistent' is defined as > > > "having as a theorem some formula and its negation"). > > > I have been told that ZFC yields this result: There are uncountable > > many real numbers. And by number I understand something that can be > > put in trichotomy with other numbers. > > ZFC doesn't make use in this context of a predicate "is a number" > where 'number' is STAND-ALONE. As to 'real number" we may regard that > as single predicate. In light of the formal itself itself, we could as > well write 'realnumber' or 'realschmumber' or 'schmeal schmumber' so > that 'number' has no component meaning onto itself. You should > understand that names of predicates in formal mathematics are not > always compositional, particular with adjectives. I can define "is a > slooper spockle" without committing that 'slooper' and 'spockle' have > independent meaning. This is because English is only an INFORMAL > rendering of certain formal predicates. So, as to the FORMAL theory, > for example, where we say "x is a real number", the actual formula > would be "Rx" where 'R' (or whatever other typographic shape) has been > chosen as the formal predicate symbol ordinarily rendered in English > as 'is a real number'. > > Please tell me whether you understand this. If you find fault in it, > then so be it. I'm not interested in arguing with you about it. Of course, this is not difficult to understand. You may call these objects as you like. But why not use the common abbreviations? Cantor uses some x of R and shows that R is an uncountable set by constructing an x_0 of R. We can show, probably also in ZFC, that the constructible elements of R are not an uncountable set. > > > If this is not the case, ZFC may > > be right. > > Others have told me that ZFC does not prove any uncountability at all > > and that the "standard model" is not a model of ZFC at all. If this is > > the case, ZFC may be right. > > ZFC proves a FORMULA that we READ or RENDER in ENGLISH as "there > exists an x such that x is uncountable". This proof is probably done by constructing another element called the anti-diagonal? So, concerning the constructible elements alone, ZFC cannot able to prove their countability? Or how else could you avoid a contradiction? Regards, WM |