Date: Feb 16, 2013 5:01 PM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 2/16/2013 3:37 PM, Virgil wrote:
> In article
> <e827583d-6246-4dd1-a860-bc7da80bfbcd@r3g2000yqd.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> On 15 Feb., 23:27, Virgil <vir...@ligriv.com> wrote:
>>> In article
>>> <8847efa6-6663-40e9-a61e-76bba7f34...@dp10g2000vbb.googlegroups.com>,
>>>
>>>
>>>
>>>
>>>
>>> WM <mueck...@rz.fh-augsburg.de> wrote:

>>>> On 15 Feb., 00:53, Virgil <vir...@ligriv.com> wrote:
>>>
>>>>>>> And just this criterion is satisfied for the system
>>>
>>>>>>> 1
>>>>>>> 12
>>>>>>> 123
>>>>>>> ...

>>>
>>>>>>> For every n all FISs of d are identical with all FISs of line n.
>>>
>>>>> For every n there is an (n+1)st fison of d not identical to any FIS of
>>>>> line n.

>>>
>>>> That does not prove d is not in the list, but only that d is not in
>>>> the first n lines of the list.

>>>
>>> For every n.

>>
>> For every n there are infinitely many lines following.
>> You never can conclude having all n.
>>

>
> Then WM must be willing to give up all induction arguments and proofs by
> induction, as they are all have the same basis as Cantor diagonal
> arguments: if something is true for the first natural, and whenever true
> for a natural 'n' is also true for the natural 'n + 1', then it is true
> of ALL n.
>
> Thus true for the first natural and if true for n then true for n+1
> DOES allow one to conclude having all n.
>
> At least outside of Wolkenmuekenheim.
>
> Does WM really want to give up the proof by induction?
>


Yes.

Once again, as I have watched these discussions,
it is clear that what is involved here is the
structure of the natural numbers as a directed
set.

And, as I consider how WM never tires of the same
meager statements rather than substantive discussion,
it occurs to me that his issue with the reversal
of quantifiers in relation to directed set structure
is the key.

Could it be that the underlying sense of proof by
induction is the same as the construction of a
forcing language?

That is, it is Euclid's proof that there is no
greatest prime number which establishes the
directed set structure. Euclid's proof involves
an application of a successor.

In forcing, the order relation of a directed
set is inverted. From this, one constructs a
forcing language through which a forcing model
is obtained. What proves that the forcing model
is not the ground model? It is the fact that
one can produce a name not in the ground model.
The forcing model has names for every element
of the ground model plus at least one name
different from those names.

This is not the diagonal proof.

So, in relation to your question, is mathematical
induction an arithmetical expression of what
the algebraic methods associated with forcing
do. Does mathematical induction rely on the
reversal of order relative to the directed set
structure?

If this is reasonable, then WM's assertions
concerning the reversal of quantifiers do imply
that WM rejects induction.

Apparently, WM is a paragon ultrafinitist.

But, then which set of prime numbers are *the*
*finite* set of prime numbers?

This is where his problem of not understanding
the meaning of "singular term" reappears.