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:

<snip>

>... consider the list of finite initial segments of natural numbers

>

> 1

> 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.

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.

]