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.