```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 appliesin WM's uses.Elsewhere, he asked about inductive sets.  Inset theory, the "natural numbers" are derivedfrom the intersection over the class of inductivesets containing the empty set.  Some inductivesequence satisfying the definition of ordinalsas transitive sets well-ordered by membershipis contained in that intersection.  The domainof "natural numbers" will be the intersectionof all of the inductive sets of ordinals inin the intersection of all inductive setssatisfying the axiom of infinity (as givenin Jech).An arithmetic can be defined on this setthat corresponds with the Peano-Dedekindaxioms.Now, what one makes of the transfinite sequenceof ordinals and its uses in the constructionof models in set theory is a different questionnot unrelated to what WM does.That one can investigate a transfinite arithmeticis different from what constitutes the domainof such an arithmetic.  With respect to hierarchicalconstruction, transfinite recursions depend onsequences (functions in relation to the replacementschema) whose domains are limit ordinals.  Asevery cardinal number in the von Neumann representationis a limit ordinal, and as the sequence of cardinalsarises in relation to the power set axiom, thestrength of set theory depends on the impredicativenature of the power set operation.Models of set theory have the appearance of beingconstructed "from below" through cumulativehierarchies.But the domain upon which those hierarchies arebuilt is not a "from below" construction.  TheCantor diagonal argument presented a problem ofreference in regard to "infinity" as a singularconcept.  The logico-mathematical approach toan investigation of this fact is a system withan arithmetical calculus.  The existence oflimit ordinals and cardinals within that systemis not obtained by a process of construction.They are the subject that is introduced bythe axiom of infinity and the power set operation(and the others, of course, as needed for themain purpose of investigation).The reason forcing models work is becausethey presuppose the partiality of theground model over which the forcing theoremis applied.  What is this partiality otherthan the logical objection of Brouwer thatclassical logic is effective when appliedto finite sets but not effective when appliedto infinite sets?  When one introduces atransfinite hierarchy, the problem of partialitysimply occurs with respect to "absolute infinity".If one denies the existence of infinite sets,then the issue occurs with the natural numbers asin the constructive mathematics of the Russianschool.WM has never introduced his assertions asconstructive mathematics.  When he has beenasked about it he rejects that formalismas he rejects all formalisms.  He claimsthat he "knows" what mathematics is byvirtue of "knowing reality".Oddly, it is he that "knows" what no oneelse does: namely, how to construct aninfinite set from finite sets. That"knowledge" is the foundation of hisarguments in defense of his belief thatno such set exists.
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