Date: Feb 13, 2013 3:03 AM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 2/12/2013 3:47 PM, Virgil wrote:
> In article
> <13ead2a9-0acc-452f-99aa-c2cd81d02ae7@m4g2000vbo.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> On 12 Feb., 18:13, William Hughes <wpihug...@gmail.com> wrote:
>>> On Feb 12, 2:12 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>>>

>>>> There is no line of the list that contains all FISs of d (because
>>>> there are not all).

>>>
>>> Your claim is that there is a line of the list that contains
>>> every FIS of d (there is no mention of all)

>>
>> But you seem to interpret some completeness into "every".
>> Remember, beyond *every* FIS there are infinitely many FISs.

>
> But no FISs beyond *every* FIS.

>>>
>>> you also claim
>>>
>>> the potentially infinite sequence d is not equal to the
>>> potentially infinite sequence given by a line of the list.

>>
>> The lines of the list are finite.
>> 1
>> 12
>> 123
>> ...
>> Or does that property disturb you?I use this model only because it is
>> simpler to treat.
>>

>>>
>>> Note that a potentially infinite sequence x is
>>> equal to a potentially infinite sequence y iff
>>> every FIS of x is a FIS of y and every FIS of
>>> y is a FIS of x. No mention or need of completion.

>
> That does not hold when comparing an infinite sequences of finite
> sequences of digits with an infinite sequence of digits.
>
> A FIS of a sequence of sequences is not the same as a FIS of a sequence
> of individuals. at least not for sequences of more than one object.
>
> For WM's argument to be calid here one would have to have sequences of 2
> or more digits identical to single digits.
>
>
> You must also ASSUME the existence of your allegedly potentially
> infinite sets which behave as you claim.


Right. So, he is manipulating the structure of
the naturals as a directed set.

His notion of successor treats each successor
as a choice function.

|({|})=|
||({|,||})=||
|||({|,||,|||})=|||

and so on

His diagonal is

|({|})=|
|({|})=|, ||({|,||})=||
|({|})=|, ||({|,||})=|| ,|||({|,||,|||})=|||

and so on

So, given some initial segment, any pair of choice
function names corresponding to any initial segment
of the given initial segment are in relation to the
name of the choice function for the given initial
segment so as to satisfy the axioms of a directed
set.

But, when a choice function name corresponding to any
initial segment of the given initial segment is
paired with the choice function name of the given
initial segment, the selected pair of choice function
names do not satisfy the axioms of a directed set
as the choice function name of the given initial
segment is a greatest element.

Without the equal signs, his choice functions form
the collection,

{|({|}),||({|,||}),|||({|,||,|||}),...}

So, what application of a choice function
orders his choice functions so that a new
choice function can be selected to resolve
the problem?

One could apply the same strategy.

Given the new choice functions to account
for a well-order of the collection above

|({|})({|({|})})=|({|})
||({|,||})({|({|}),||({|,||})})=||({|,||})
|||({|,||,|||})({|({|}),||({|,||}),|||({|,||,|||})})=|||({|,||,|||})

and so on

one has the collection

{
|({|})({|({|})}),
||({|,||})({|({|}),||({|,||})}),
|||({|,||,|||})({|({|}),||({|,||}),|||({|,||,|||})}),
...
}

Now, of course, what do I need?

You guessed it.... ad infinitum.