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.