Cantor Confusion
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From: Han.deBruijn on 7 Oct 2006 16:38 Virgil wrote: > In article <1160223586.604282.269450(a)h48g2000cwc.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > Tonico wrotef: > > > > > Maths, just like the other sciences, isn't grounded on dogma, and > > > people forwarding REASONABLE, well-based objections, opinions or ideas > > > on whatever are always welcome. > > > > _There_ is your problem! Ask Virgil, and the other mathematicians here: > > > > MATHEMATICS IS _NOT_ A SCIENCE > > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > HdB's version of mathematics certainly seems to be grounded in dogma. > > My mathematics is merely grounded on determining what can be derived > from a given set of axioms. Yes. And these axioms are your dogmas. And I don't trust your "derived" either, but that's another story ... Han de Bruijn
From: Tonico on 7 Oct 2006 17:15 Han.deBruijn(a)DTO.TUDelft.NL wrote: > mueckenh(a)rz.fh-augsburg.de schreef: > > > Tonico schrieb: > > > > > It is nonsense, but it is the truth if infinity can be calculated. I > > said it merely to show that actual infinity is not only a huge nonsens > > but rather an infinite nonsense. > > And when you're debating with me (HdB). I'm on Mueckenheim's side > in these matters. ************************************************************* No wonder I confused between you two. You both use a lot of very weird, non-mathematical mumbo-jumbo, just like JSH uses to do though in a slightly different manner, and you can't seem to agree even with yourselves in things widely accepted names. It's useless to continue this discussion. It was fun for me for a while, though. You don't want to accept ZF's Axiom of Infinity? Well, then don't: see if I care! You won't be playing some of the most thrilling and outstanding of all games ever invented by the human mind called maths, or at least some of its best parts. To me this is a huge lost just to have our rather deceiving intuition calmed and in peace. Your choice. Just as Brouwer chose his way and then lost, imfho., a lot of fun. **Shrugs** Regards Tonio > Han de Bruijn
From: Virgil on 7 Oct 2006 17:44 In article <1160252933.461908.146830(a)m7g2000cwm.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Tonico wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > Tonico wrotef: > > > > > > > Maths, just like the other sciences, isn't grounded on dogma, and > > > > people forwarding REASONABLE, well-based objections, opinions or ideas > > > > on whatever are always welcome. > > > > > > _There_ is your problem! Ask Virgil, and the other mathematicians here: > > > > > > MATHEMATICS IS _NOT_ A SCIENCE > > > > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > > > > ****************************** > > Wrong: whether maths is a science or not, it is NOT grounded on dogma > > There are several forms of mathematics. Some of them are NOT grounded > on dogma (presumably the forms you are aware of), but others certainly > are. They call it not dogmas but axioms. There is nothing wrong with > axioms as such. But there CAN BE something wrong with axioms that > come out of thin air and are contradictory to physical experience. > Such as axioms which embody the idea that completed infinities could > possibly exist. HdB has as an axiom that the truth about reality can be determined, which axiom many mathematicians reject for sound reasons. > > > (or better: mention one dogma of mathematics). > > About maths being or not a (natural) science: I think that may depend > > on what definition we use for "natural science". Maths doesn't require > > labs, observation of physical phenomena or testing of results again and > > again, just as chemistry, physics or biology may need, but maybe not > > only that is what makes a science according to other definitions. > > How about computers? Aren't they the labs of modern mathematics? No more than pencil and paper are. They are just a lot faster than working things out with pencil and paper. > > > Anyway, I really don't care whether someone thinks maths is or not a > > science. > > Because in your view, mathematics IS a science. No? According to my > personal opinion, mathematics _should be_ a science. Which opinion plus a dime won't buy a cup of coffee. > > Han de Bruijn
From: Virgil on 7 Oct 2006 17:49 In article <1160253510.579062.129230(a)k70g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil wrote: > > > In article <1160223586.604282.269450(a)h48g2000cwc.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > Tonico wrotef: > > > > > > > Maths, just like the other sciences, isn't grounded on dogma, and > > > > people forwarding REASONABLE, well-based objections, opinions or ideas > > > > on whatever are always welcome. > > > > > > _There_ is your problem! Ask Virgil, and the other mathematicians here: > > > > > > MATHEMATICS IS _NOT_ A SCIENCE > > > > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > > > HdB's version of mathematics certainly seems to be grounded in dogma. > > > > My mathematics is merely grounded on determining what can be derived > > from a given set of axioms. > > Yes. And these axioms are your dogmas. Except that our axioms are assumed true only for the purposes of seeing what can be derived from them, not declared necessarily true in the way HdB declares his axioms to be, not declared false the way HdB declares ours to be. > > And I don't trust your "derived" either, but that's another story ... Fair enough, we do not trust any of your declared truths either.
From: David Marcus on 7 Oct 2006 18:30
Tonico wrote: > Omega has NEVER, ever being a cardinal, as far as I know (perhaps I > missed something, true), but an ORDINAL!! > And since when cardinals are...sets?!?! Talking of mixing up terms...!! Actually, this is standard. For example, the book "Set Theory" by Kunen has the following definition. Definition. If A can be well-ordered, then the cardinality of A (denoted |A|) is the least ordinal alpha such that there is a bijection from alpha to A. Kunen then remarks that with AC, |A| is defined for every A. -- David Marcus |