Date: Feb 22, 2013 4:57 PM
Author: Virgil
Subject: Re: Matheology � 222 Back to the roots

In article 
<90cbfe69-ca5b-454c-8126-6543716186ed@w7g2000yqo.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 22 Feb., 02:08, William Hughes <wpihug...@gmail.com> wrote:
> > So
> >
> >      A) For every natural number n, P(n) is true.
> >
> >         implies
> >
> >      B)  For any n: There does not exist a natural number
> >          between 1 and n such that P(n) is false
> >
> > However, we cannot conclude
> >
> >      B')  There does not exist a natural number
> >           m such that P(m) is false

>
> In potential infinity there is no actually infinite set



But in standard set theories there is no potentially infinite set that
is not actually infinite.

At least WM has yet to provide us with any axiom system or comprehesive
set of rules for a coherent set theory which even allows for potential
infiniteness without requiring actual infiniteness into the bargain.

We have ZF and NBG, among others, but WM has nothing.





>
> Compare my example:
>
> A B
> --> 1 -->{ }
> --> 2,1 -->{ }
> --> 2 -->1
> --> 3, 2 -->1
> --> 3 -->1, 2
> --> 4, 3 -->1, 2
> --> 4 -->1, 2, 3
> ...
> --> n -->1, 2, 3, ..., n-1
> --> n+1, n -->1, 2, 3, ..., n-1
> --> n+1 -->1, 2, 3, ..., n-1, n
> ...
>
> There is no last element in B.


Another evidence of WM's habitual sloppy thinking, confusing a sequence
of subsets of |N with a subset of |N.

A and B are not subsets of |N, at least in any standard set theory, as
if they were, their memberships would be fixed and unchanging.

One can have a sequence of sets in which different members of the
sequence can have different membership, but one cannot have a single set
whose membership is not fixed. At least not outside WMytheology.

If A and B are to be understood as sequences of sets, then WM must
define what he means by the limit of a sequence of sets before claiming
any such limit exists or does not exist. or claiming a value for such a
limit.



> Every x from 1 to every n you desire
> can be in B.


Perhaps x as a natural can be in a term or member of B, but the members
of B are subsets of |N rather than members of |N.







>
> That is all we can know about infinity. (Not my fault.)


It is certainly your own fault that you can know no more, as there are
myriads of sources all around you from which, once you open your eyes,
you can learn about infinity.
--