Date: Apr 5, 2013 6:46 PM
Author: fom
Subject: Re: Matheology § 224

On 4/5/2013 4:51 PM, Sam Sung wrote:
> fom wrote:
>> On 4/5/2013 11:04 AM, WM wrote:
>>> On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote:
>>>> On Apr 5, 11:09 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>>>>

>>>>> On 4 Apr., 23:08, William Hughes <wpihug...@gmail.com> wrote:
>>>>
>>>>>> Nope. Any single element can be removed. This does not
>>>>>> mean the collection of all elements can be removed.

>>>>
>>>>> You conceded that any finite set of lines could be removed. What is
>>>>> the set of lines that contains any finite set? Can it be finite? No.

>>>> correct
>>>>> So the set of lines that can be removed form an infinite set.
>>>>
>>>> More precisely. There is an infinite set of lines D
>>>> such that any finite subset of D can be removed.

>>>
>>> What has to remain?

>>>>
>>>> This does not imply that D can be removed.

>>>
>>>> It does however imply that there is no single element
>>>> of D that cannot be removed. That this does not
>>>> imply that D can be removed is a result that
>>>> you do not like, but it is not a contradiction.

>>>
>>> It is simple mathological blathering to insist that |N contains only
>>> numbers that can be removed from |N but that not all natural numbers
>>> can be removed from |N.
>>>
>>> It is a contradiction with mathematics, namely with the fact that
>>> every non-empty set of natural numbers has a smallest element.

>>
>> There is no mathematical predicate "can be removed"
>> in the axioms by which the structure of natural
>> numbers are given.
>>
>> Since the natural numbers are not given by the axioms
>> as the union of subsets of the natural numbers taking
>> subsets away from the union of subsets of the natural
>> numbers has no effect on the definition-in-use given
>> by the axioms.

>
> (Ok, one may, however, build sets from other sets e.g. by
> doing unions, complements, etc., which can give modified
> copies of the original sets, resulting in copies that
> "are like" "modified sets".)
>


Sure. But it is hard to ever know what applies
in WM's uses.

Elsewhere, he asked about inductive sets. In
set theory, the "natural numbers" are derived
from the intersection over the class of inductive
sets containing the empty set. Some inductive
sequence satisfying the definition of ordinals
as transitive sets well-ordered by membership
is contained in that intersection. The domain
of "natural numbers" will be the intersection
of all of the inductive sets of ordinals in
in the intersection of all inductive sets
satisfying the axiom of infinity (as given
in Jech).

An arithmetic can be defined on this set
that corresponds with the Peano-Dedekind
axioms.

Now, what one makes of the transfinite sequence
of ordinals and its uses in the construction
of models in set theory is a different question
not unrelated to what WM does.

That one can investigate a transfinite arithmetic
is different from what constitutes the domain
of such an arithmetic. With respect to hierarchical
construction, transfinite recursions depend on
sequences (functions in relation to the replacement
schema) whose domains are limit ordinals. As
every cardinal number in the von Neumann representation
is a limit ordinal, and as the sequence of cardinals
arises in relation to the power set axiom, the
strength of set theory depends on the impredicative
nature of the power set operation.

Models of set theory have the appearance of being
constructed "from below" through cumulative
hierarchies.

But the domain upon which those hierarchies are
built is not a "from below" construction. The
Cantor diagonal argument presented a problem of
reference in regard to "infinity" as a singular
concept. The logico-mathematical approach to
an investigation of this fact is a system with
an arithmetical calculus. The existence of
limit ordinals and cardinals within that system
is not obtained by a process of construction.
They are the subject that is introduced by
the axiom of infinity and the power set operation
(and the others, of course, as needed for the
main purpose of investigation).

The reason forcing models work is because
they presuppose the partiality of the
ground model over which the forcing theorem
is applied. What is this partiality other
than the logical objection of Brouwer that
classical logic is effective when applied
to finite sets but not effective when applied
to infinite sets? When one introduces a
transfinite hierarchy, the problem of partiality
simply occurs with respect to "absolute infinity".
If one denies the existence of infinite sets,
then the issue occurs with the natural numbers as
in the constructive mathematics of the Russian
school.

WM has never introduced his assertions as
constructive mathematics. When he has been
asked about it he rejects that formalism
as he rejects all formalisms. He claims
that he "knows" what mathematics is by
virtue of "knowing reality".

Oddly, it is he that "knows" what no one
else does: namely, how to construct an
infinite set from finite sets. That
"knowledge" is the foundation of his
arguments in defense of his belief that
no such set exists.