```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...
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