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.