Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 16 Jun 2010 16:46 In article <7a3e3a5a-2202-4cbb-9556-e3766ee5a745(a)k39g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 17:55, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > WM <mueck...(a)rz.fh-augsburg.de> writes: > > > It is very probable that every line has many different meaning, but no > > > line has uncounatbly many meanings. > > > > Every line has an indefinite and indeterminate number of possible > > meanings. It makes no sense to speak of the cardinality of the totality > > of all possible meanings of a string of symbols. > > It is fact that all possible meanings must be defined by finite > definitions. Therefore the meanings are countable. Therefore it makes > sense to call the set of all meanings countable. In proper set theory, one must be able to determine of an object whether or not it is a member of a set, but since those vague "meanings" of a string of symbols are so indeterminate, one cannot determine whether a "meaning" does belong to a "symbol" so WM's "sets" are not well enough defined to be sets in any proper set theory. > > In mathematics we can calculate or estimate things even if not all can > be named. But set membership must be less ambiguous than WM's. > > > > > Therefore the list contains only countably many finite definitions. Except that it is not at all clear what has been listed. > > > > The list contains just random, meaningless strings. Whenever we instill, > > with mathematical precision, these strings with some definite meaning, > > so that they become definitions of reals, we find there are definitions > > not included in the list. There is an absolute notion of computability > > in logic. > > If this notion yields numbers only that are in trichotomy with each > other, then the notion is acceptable. If this notion yields numbers > like you gave examples for (IIRC) like n = (1 if Obama gets a second > term and 0 otherwise) or so, then this notion together with your logic > should be put into the trash can. Only for a while. > > > There is no absolute notion of definability. > > We need not an absolute notion if we know that all possible > definitions of the notion of definability belong to a countable set. We cannot possibly know that. > > It is enough to prove by estimation that set theory is wrong. Compare > the famous irrationality proofs and transcendence proofs of number > theory. We need not calculate its deviation from truth to the fifth > digit. It is enough to see that ZFC is wrong unless there is a natural > between 0 and 1. But what everyone else sees is that ZFC is right unless there is a natural between 0 and 1.
From: Virgil on 16 Jun 2010 16:48 In article <12736859-7b42-437c-ab8c-2bb3882ef585(a)i28g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 18:07, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > Herman Jurjus <hjm...(a)hetnet.nl> writes: > > > Also many classical mathematicians appreciate this as an example > > > showing that the extensional notion 'Turing computable' is a slight > > > distortion of the intuitive notion 'computable'. > > > > Possibly, but I don't think this is quite the right diagnosis. The issue > > is more subtle. > > The issue is very simple. A real number is computable, if its place on > the real axis can be established, i.e., trichotomy. Otherwise it does > not deserve the name number but at most number form or interval (like > 0.1x means 0.10 to 0.19 in decimal). WM is free to restrict such meanings for his own use, but only for his own use, since when others use the same terms, they cannot be constrained to use them in ways foreign to themselves.
From: MoeBlee on 16 Jun 2010 17:43 On Jun 16, 3:34 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 21:40, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 16, 2:21 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > I don't have a thousand lifetimes to wait for you to show a > > > formula P such that both P and ~P are derivable in ZFC from the above > > > definition. > > > CORRECTION: I should have said: > > > [...] to show, for some formula P, a proof in ZFC (with said > > definition included) of P and a proof in ZFC (with said definition > > included)of ~P. > > 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". > 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. > 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"). If in some other sense you find ZFC not compatible with your own mathematical or other notions, is another matter. A technical note: It is conceivable that one could prove in ZFC that ZFC is inconsistent but as one does not show a particular derivation of a formula "P & ~P" (I think),i.e., showing "non-constructively" that ZFC is inconsistent. But still if ZFC is inconsistent then there does exist a constructive proof that ZFC is inconsistent, which could be picked out from an enumeration of the proofs (though, of course, it is recognized that this may not be feasible within the presumable lifetime of any human being). A note on the discourse: If you persist to present me with misunderstandings of ZFC and arguments and claims using terminology not defined to primitives and based in axioms in some stated formal language and logic (even if only informal arguments that nevertheless suggest some background formal langauge, logic, and axioms) then I may leave many, perhaps all, such arguments without my response, since I do need to start better valuing my time than to fruitlessly try to bring you to reason and understanding. This is not to say that I reject philosophical discussion or even yet-to-be formalized mathematical discussion; but rather that I don't wish to be mired in conversation where such as just mentioned are conflated with mathematical argumentation that at least in principle admits of resolution per some formal theory or another. MoeBlee
From: Virgil on 16 Jun 2010 18:09 In article <fe10d9c1-b497-43f7-91fa-53415e0b7b85(a)c10g2000yqi.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 03:05, "Mike Terry" > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > "Virgil" <Vir...(a)home.esc> wrote in message > > > > news:Virgil-3B2F0B.16291815062010(a)bignews.usenetmonster.com... > > > > > In article <87sk4ohwbt....(a)dialatheia.truth.invalid>, > > > �Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > > Virgil <Vir...(a)home.esc> writes: > > > > > > > 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. > > > > > > You need infinitely many unknown places. > > > > > If the value of some decimal digit of a number depends on the truth of > > > an undecidable proposition, can such a number be computable? > > > > Yes - e.g. imagine just the first digit of the following number depends on > > an undecidable proposition: > > > > � � 0.x000000000... > > This is not a real number. It restricts the set of numbers to 10 > differents numbers (in decimal). If x is unambiguously defined as one of the decimal digits then 0.x000000000... IS a number, even if we do not know which one. > > > > There are only 10 possibilities for the number, and in each case it is > > obviously computable... > > But it is not clear which case will be chosen. Two real numbers a and > b satisfy a < b or a = b or a > b. > 0.y and 0.x do not. That you cannot readily determine that trichotomy holds does not mean that it does not hold. These symbols are number forms like x and y in > 3x + 5y = 0 > or > n in "let n be an even number". > Obviously n need not be an even number. > It is no number at all. That n is still a number, even if which one is not known. According to WM's philosophy, one cannot speak of a number in the abstract, but only in the concrete.
From: Virgil on 16 Jun 2010 18:20
In article <c3c22786-a664-4fb8-bc90-66b29116484b(a)w12g2000yqj.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 15 Jun., 21:11, "Mike Terry" > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > > > 1) What you write is sloppy (mathematically) and so it's not > > clear what you intend to say with your individual statements. > > But actually everybody can easily imagine what is meant. Quite often not. > You prove it > for your person below. > > > > > > 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) > > > > Literally this just doesn't make sense. > > Obviously he means that this list contains every digit of pi with the > correct index enumerating its position. In discussions like the > present one it is not usual to state everything explicitly that can be > understood by an experienced reader. > > > Perhaps you mean to say: > > > > this list contains every finite prefix of the infinite > > digit sequence for pi (1)? > > > > I would agree with that... > > > > (What you actually said is gibberish because the list is not a list of > > digits. If we try to treat it as such, then the only digit in the list is > > 3). > > A list of people can contain and usually does contain letters. A list of people should contain only people. A list of people's names can contain and usually does contain letters, other than in, say, Chinese or Japanese. > Thiks list is not a set in orthodox set theory. > > > > > [My suspicion at this point is that your gibberish wording is actually the > > *key* in some way to how you want to introduce some incorrect conclusion, > > but time will tell on that. If I'm right, you won't like my clarification, > > because it will make it harder for you to express your mistake...] > > This suspicion is wrong. > > Every initial segment of the decimal expansion of pi is in at least > one line of your list > 3. > 3.1 > 3.14 > 3.141 > ... Actually, in all but finitely many of those lines. > What we can find in the diagonal, namely 3.141 and so on, exactly that > can be found in one line. This is obvious by construction of the > list. That is FALSE, since the "diagonal" is necessarily endless and each line ends. > So consider this: > > Every part of the diagonal is in at least one line. Each finite initial part, but not any non-empty terminal part. > That means That means WM is wrong. me. > > The latter proposition can be excluded. If there are more than one > lines that contain parts of pi, then it can be proved, be induction, > that two of them contain the same as one of them. This can be extended > to three lines and four lines and so on for every initial segment of n > lines. Every FINITE initial segment, but each of those omits more than it inlcudes. > > Hence we prove that all of pi, that is contained in at least one of > the finite lines of this list, is contained in one single line. Only if there is a last line, which there is not. > > Conclusion: Either the complete diagonal pi does not exist, or it > exists also in one and the same single line (because every line that > can contain a digit sequence, is a finite line). Conclusion false, as it requires the existence of a last line, which does not occur. If one has a (finite) last line, one does not have pi. And if one does not have a last line, WM's argument fails. |