> 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".
> 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...