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 <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).

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 http://mathe-online.fernuni-hagen.de/MIB/HTML/node80.html

> 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...