fom
Posts:
1,026
Registered:
12/4/12
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Re: Matheology � 203
Posted:
Feb 6, 2013 6:07 AM
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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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