Date: Mar 6, 2013 1:44 PM
Subject: Re: Matheology § 222 Back to the roots

On 6 Mrz., 13:18, William Hughes <> wrote:
> On Mar 6, 12:48 pm, WM <> wrote:

> > On 6 Mrz., 12:05, William Hughes <> wrote:
> > > > L_m is a single line if m is a natural number.
> > > > Would you prefer to call L_m infinitely many lines?

> > > Nope, I would prefer to call L_m a function
> > > (of time and person). A function may have as
> > > value a "single line of the list"
> > > but calling something that changes a "single line of the
> > > list" is silly.-

> > I said always that L_m is a function (of several arguments) and that
> > this function takes as vaules lines of the list. As it takes single
> > lines, I don't see why we should not call them single lines.

> Because calling L_m a single line
> is certain to cause miscommunication
> and using language in a way certain
> to cause miscommunication is silly.

No. L_m is a single line. You have misunderstood as becomes clear from
the following.
> So the statement
> "there is no line which contains every
> FIS of d"
> becomes in the language of Wokenmuekenheim
> "there is no findable line which contains
> every FIS of d"
> Similarly, there is no statement about
> the behaviour of "actually infinite"
> sets that does not have an analogue
> in the language of Wolkenmuekenheim.
> For example:
> in Wolkenmuekenheim you would say
> (about potentially infinite sets)
> A subset K of the lines of L
> contains every FIS of d iff
> K has no findable last line.

No, it is exactly false to require an infinite subset K to contain
every subset of d. Every FIS of d is always in one single line. This
line is always the last line. Every other line is not necessary and
not sufficient to contain every FIS of d. Without a last line we can
prove that no line is sufficient and necessary to contain every FIS of

This again testifies that you have not understood yet the nature of
potential infinity.

> to mean the same thing as the
> statement (about "actually infinite sets")
> A set of lines K contains
> every FIS of the diagonal
> iff K has infinite cardinality

That is actual infinity and as such different from the former. And, by
the way, the claim is nonsense, since, even in actual infinity, for
every element of K we can prove, that it does not belong to the set of
lines necessary or sufficient to contain every element of d.
> This is what I mean when I say
> that "potential infinity" behaves
> like "actual infinity".

So you have not yet comprehenden the nature of potential infinity.

It is ridiculous to see how many matheologians claim that a set K that
contains every FIS of d although it is clear that none of its FIS can
satisfy this claim.

Everey line of the complete list

1, 2
1, 2, 3

is not containing the actually infinite set |N.
It is claimed that the union does.
But this list is constructed such that the union is the same as the
And the limit |N does not belong to the sequence.
Therefore it does not belong to the union.

Regards, WM