Date: Apr 5, 2013 7:24 PM
Author: Tanu R.
Subject: Re: Matheology § 224

fom schrieb:

> 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 <> wrote:
>>>>> On Apr 5, 11:09 am, WM <> wrote:

>>>>>> On 4 Apr., 23:08, William Hughes <> 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).

Otherwise they use to call it "unmenge" - not a set
which is assured to be containing well discernable
element - which is why we like linearity over any
more dimension in "our" phase space (graph space).

> 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.

Which contains that each node is unique in the same way
as each path is well discernible even in trees.

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

Thats part of these kind of partitions - they like
to treat it globally...

> 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.

WM does not "accept" that ;)

> 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".

Where that symbol stands for uneigentlicher Grenzwert,
see and understand

> 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.

WM is incapable of introducing himself - thats the way it is an asshole.

> 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".

It also tells the biggest number ever "is" some estimated
number of fermions in every proper horizon aspect about 10^80.

> 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.

WM is even incapable of recognizing our arguments...