> On 2/3/2013 10:50 PM, Ralf Bader wrote: >> Virgil wrote: >> >>> In article >>> <firstname.lastname@example.org>, >>> WM <email@example.com> 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?