Date: Mar 13, 2013 2:19 PM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 3/13/2013 12:33 PM, WM wrote:
> On 13 Mrz., 17:59, William Hughes <> wrote:
>> On Mar 13, 5:37 pm, WM <> wrote:

>>> On 13 Mrz., 13:19, William Hughes <> wrote:
>> <snip>

>>>> If you wish to contest this, use my words not
>>>> yours (e.g. I have never said "The list contains more
>>>> numbers than fit into a single line", I have said
>>>> "There is no line in the list which contains every
>>>> number in the list".)

>>> Correct. The list has more numbers than a single line has. Since every
>>> number that is in the list, must be in at least one line, this implies
>>> that the numbers are in more than one line.

>> To be precise, a set of lines, say K, that contains all the numbers
>> contains at least two lines.

> In actual infinity, this is not avoidable.
> We note: At least two lines belong to the set that contains all
> numbers. We call these lines necessary lines.
> So the set of necessary lines is not empty.

>> However, this does *not* imply that
>> there are two numbers that are not in a single line.

> Why then should two lines be necessary?
> One being the substitute in case the other falls ill?

>> Nor does it imply that there is a necessary line in K.
> If there is not one necessary line, then there are two or more
> required.
> Proof: If you remove all lines from the list, then there remains no
> line and no number.

>> Note that a sufficient set does not imply a necessary line
>> even in potential infinity. There is no line that is needed
>> to make L have an unfindable last line.

> So you believe that there can remain all numbers in the list after
> removing all lines? That is a remarkable claim. I would not accept it
> in mathematics.
> Note in actual infinity it makes sense to talk about all lines and to
> remove all lines.

I think this is just a difference of interpretation
concerning "necessary".

To borrow from linear algebra, you are describing
something that might be more along the lines of
a "spanning set". Any particular lines are not
necessary, but whenever one speaks of the possibility
of given lines containing all the numbers, the
count of those lines would necessarily have a
non-zero value greater than one because of