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.