fom
Posts:
1,033
Registered:
12/4/12
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Re: Matheology § 210
Posted:
Feb 5, 2013 8:21 PM
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On 2/5/2013 5:49 PM, Virgil wrote:
> In article
> <5ba22415-f653-437c-9aef-cdd140fd92c9@fw24g2000vbb.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>
>> On 5 Feb., 21:55, William Hughes <wpihug...@gmail.com> wrote:
>>> On Feb 5, 6:02 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>>>
>>> <snip>
>>>
>>>> ... one cannot even put all natural numbers in
>>>> correspondence with all natural numbers.
>>>
>>> However, one can put every natural number in
>>> correspondence with every natural number.
>>
>> One can have that by belief or decreed by the powers that be.
>> n <--> n is the finite expression of this belief. But its validity is
>> dubious because one cannot have possibly in a possible list that
>> contains every finite initial sequence of the sequence of natural
>> numbers the sequence s of every finite initial segment of the sequence
>> of natural numbers.
>
> If one cannot even justify an n to n correspondence for naturals n, then
> one has thrown out a large baby with a very little bathwater, as one has
> automatically thrown out induction as well.
>
> Induction requires that one be able to conclude that something holds for
> all n in |N.
>
> But if one cannot anything for all n in |N, no induction!
>
Even Wittgenstein's finitism admitted correspondence
by finitely specified rules (as opposed to the Dirichlet
abstraction of any collection of ordered pairs satisfying
the usual constraint of well-definition).
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