Date: Mar 15, 2013 12:13 PM
Author: William Hughes
Subject: Re: Matheology § 224
On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>... consider the list of finite initial segments of natural numbers
> 1, 2
> 1, 2, 3
> According to set theory it contains all aleph_0 natural numbers in its
> lines. But is does not contain a line containing all natural numbers.
> Therefore it must be claimed that more than one line is required to
> contain all natural numbers. This means at least two line are
> necessary. There are no special lines necessary, but there must be at
> least two. In this case, however, we can prove, by the construction of
> the list, that every union of a pair of lines is contained in one of
> the lines. This contradicts the assertion that all natural numbers
> exist and are in lines of the list.
Nope, two lines are necessary but not sufficient.
Two lines can never do a better job than 1.
Any finite number of lines is necessary but not sufficient.
Any finite number of lines can never do a better job than 1.
An infinite number of lines is necessary and sufficient
An infinite number of lines can do a better job than 1
[In potential infinity things go
Any number of findable lines is not sufficient
An unfindable line is necessary and sufficient
An unfindable line can do a better job than
any number of findable lines.