Cantor Confusion
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From: Eckard Blumschein on 8 Dec 2006 09:05 On 12/7/2006 9:23 PM, Virgil wrote: >> >> Just try and refute: >> >> Reals according to DA2 are fictitious > > Reference to any statement, other than by EB or his ilk, to the effect > that "Reals according to DA2 are fictitious" Cantors, Virgils and their ilk call this refutation. >> Reals according to DA2 are fictitious > > Repeating a falsehood does not make it any less false. You may try and furnish a complete numerical address of any irrational number. No chance! > >> With fictitious I meant: They must not have a directly approachable >> numerical address. > > What is this stuff about "addresses"? > > What rubric of mathematics demands that every "number" have a directly > approachable address? Logic. > That may be so in engineering, but is nowhere writ in mathematics. > > And what is so special about direct approaches. Lots of mathematical > theorems have to be approached quite indirectly, but are still valid and > important, despite that. Since you are obviously unable to grasp abstract reasoning, imagine a wooden folding rule. Because there is no place for pi and the like on it, it is equipped with at least an estimated trillion of filaments connected to only the most important additional notes without order. Good luck.
From: stephen on 8 Dec 2006 09:29 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Virgil wrote: >> In article <68588$45791ff3$82a1e228$8581(a)news2.tudelft.nl>, >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>>It's quite simple. Set Theory can not be the foundation for mathematics, >>>because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics >> >> A calculation is an application of mathematics, but may actually be >> physics, or chemistry, or merely commerce. > Geez! Can mathematics be separated from its "applications" in this way? > Han de Bruijn Sure it can, just as computer science can. Do you think the fact that computer programs exist to model chemical reactions makes computer science part of chemistry, or vice versa? Stephen
From: Eckard Blumschein on 8 Dec 2006 09:30 On 12/7/2006 9:15 PM, Virgil wrote: >> > The countability of a set requires no more than it have countably MANY >> > members, but is totally independent of the nature of those members. >> >> No. The numbers also have to have an approachable address. > > Which axiom says that? The customer is king. >> BTW: The meaning of "many" already includes countably. > > Where is that written in stone? German: how many = wieviele (considered a number) how much = wieviel (considered a single entity) >> >> >> > >> > For example: >> > >> > The set of square roots of prime naturals is as countable as the set of >> > prime naturals itself by an obvious bijection, >> >> In this case you are using addresses approachable via bijection. You are >> not really counting the square roots but primarily the prime naturals. > > In set theory, counting is done by bijection. So those square roots are > /really/ counted as much as the primes are counted. Bijection is not yet counting. You can also biject {Virgil, Cantor, Dedekind} with {nice, naive, clever}. This is no counting, not even a ranking. I explained what I meant with "not really counting". What did you mean with "really counted"? Does it differ from just "counted"?
From: Han de Bruijn on 8 Dec 2006 09:55 stephen(a)nomail.com wrote: > But everything can be modelled as a set. Define "everything" and prove that claim. Han de Bruijn
From: stephen on 8 Dec 2006 10:17
Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> But everything can be modelled as a set. > Define "everything" and prove that claim. > Han de Bruijn By "everything", I meant everything mathematical. Of course that is not 100% precise. And no, I cannot prove it. But so far all the various objects of mathematics can be modelled using set theory. That is what is meant by set theory being a foundation for mathematics. If someone were to invent something "mathematical" (whatever that may mean exactly) that could not be described in terms of set theory, then set theory would no longer serve as a foundation. But given that the basics such as the real numbers, functions, limits, calculus, etc. all can be founded in set theory, it would have to be something strange indeed. Not that there is anything wrong with strange, but you probably would like it less than set theory. Stephen |