```Date: Feb 6, 2013 6:07 AM
Author: fom
Subject: Re: Matheology � 203

On 2/5/2013 9:47 PM, Ralf Bader wrote:> fom wrote:>>> On 2/3/2013 10:50 PM, Ralf Bader wrote:>>> Virgil wrote:>>>>>>> In article>>>> <bc3c4c0e-d017-49b3-a4f3-22aba84aa3c7@5g2000yqz.googlegroups.com>,>>>>    WM <mueckenh@rz.fh-augsburg.de> wrote:>>>>>>>>> On 3 Feb., 22:29, William Hughes <wpihug...@gmail.com> wrote:>>>>>>>> We can say ?"every line has the property that it>>>>>>>> does not contain every initial segment of s">>>>>>>> There is no need to use the concept "all".>>>>>>>>>>>>> Yes, and this is the only sensible way to treat infinity.>>>>>>>>>>>> So now we have a way of saying>>>>>>>>>>>> s is not a line of L>>>>>>>>>>>> e.g. ?0.111... ?is not a line of>>>>>>>>>>>> 0.1000...>>>>>> 0.11000...>>>>>> 0.111000....>>>>>> ...>>>>>>>>>>>> because every line, l(n), ?has the property that>>>>>> l(n) does not ?contain every ?initial>>>>>> segment of 0.111...>>>>>>>>>> But that does not exclude s from being in the list. What finite>>>>> initial segment (FIS) of 0.111... is missing? Up to every line there>>>>> is some FIS missing, but every FIS is with certainty in some trailing>>>>> line. And with FIS(n) all smaller FISs are present.>>>> But with no FIS are all present.>>>>>>>>>>> Is there a sensible way of saying>>>>>> s is a line of L ?>>>>>>>>>> There is no sensible way of saying that 0.111... is more than every>>>>> FIS.>>>>>>>> How about "For all f, (f is a FIS) -> (length(0.111...) > length(f))" .>>>>>>>> It makes perfect sense to those not permanently encapsulated in>>>> WMytheology.>>>>>> By the way, Mückenheim's crap is as idiotic from an intuitionistic point>>> of view as it is classically. Intuitionists do not have any problems>>> distinguishing the numbers 0,1...1 with finitely many digits and the>>> sequence formed by these numbers resp. the infinite decimal fraction>>> 0,11....>>>>>>> No.  His finitism seems to be more of a mix of Wittgenstein and>> Abraham Robinson.  Although it is not apparent without reading the>> original sources, it has a certain legitimacy.  Names complete>> Fregean incomplete symbols.  So names are the key to model theory.>> Robinson explains this exact relationship in "On Constrained>> Denotation".  It is, for the most part ignored by the model>> theory one obtains from textbooks.  The model theory that one>> learns in a textbook parametrizes the quantifier with sets.>> Thus, the question of definiteness associated with names is>> directed to the model theory of set theory.  In turn, this is>> questionable by virtue of the Russellian and Quinean arguments>> for eliminating names by description theory.  So, the model>> theory of sets consists of a somewhat unconvincing discussion>> of how parameters are constants that vary (see Cohen).  If one>> does not know the history of the subject, then one is simply>> reading Cohen to learn some wonderful insights and does not>> question his statements (after all, it is Paul Cohen, right?)>>>> In Jech, there is an observation that forcing seems to>> depend on the definiteness of "objects" in the ground>> model such as the definiteness of the objects in the>> constructible universe.>>>> If you read Goedel, there is a wonderful footnote explaining>> the assumption that every object can be given a name in>> his model of the constructible universe.>>>> If you read Tarski, there is an explicit statement that>> his notion of a formal language is not a purely formal>> language, but rather one that has formalized a meaningful>> language--by which one can assume that objects have>> meaningful names.  As for a "scientific" language generated>> by definition, Tarski has an explicit footnote stating>> that that is not the kind of language that he is>> considering.>>>> So, we have names being eliminated by Russell and Quine>> and descriptive names being specifically excluded by the>> correspondence theory intended to convey truth while the>> notion of truth in the foundational theory that everyone>> is using only presumes definiteness through parameters>> that vary.>>>> But, the completion of an incomplete symbol requires>> a name.>>>> Who wouldn't be a little confused?>> I am indeed slightly confused about what you wrote and what it has to do> with the previous discussion. This was centered around a "list" of decimal> fractions, namely:> To the natural number i, the fraction 0.1...100... with exactly i digits> equalling 1 is associated. And the assertion of Mückenheim was that> s=0.111... with infinitely many digits equalling 1 "is" somehow in this> list, because all its finite initial segments appear in the list.> And this I called idiotic crap, and I still do so; if I should have> overlooked something deeply profound, I still don't see it. These fractions> and the list are a pretty simple matter, and I really do not see why the> help of Wittgenstein, Russell, Quine, Goedel, Jech and Robinson is required> to find out what is "in" that list. I have just remarked that, whatever one> thinks about intuitionism, its representatives like Brouwer and, to some> extent, Weyl, on whose "sharp minds" Mückenheim called to support his> nonsense, did not commit such a blunder. Their reservations about classical> mathematics did not concern decimal representations of rational numbers or> simple sequences of rationals. According to Mückenheim, "There is no> sensible way of saying that 0.111... is more than every> FIS". Of the authorities you called upon, whom would you find capable of> regardng this as a sensible assertion?>On second thought, you are right.  My apologies.
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