Obections to Cantor's Theory (Wikipedia article)
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From: Robert Kolker on 21 Jul 2005 20:19 Han de Bruijn wrote: > > It's the standard definition of the "actual infinite", but it is not > "perfectly good". Worse. It's not good at all. What is wrong with it? No one has shown it leads to a contradiction. Bob Kolker
From: Martin Shobe on 21 Jul 2005 20:20 On Thu, 21 Jul 2005 14:39:50 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Martin Shobe wrote: > >> On Thu, 21 Jul 2005 12:21:38 +0200, Han de Bruijn >> <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>an "all-encompassing definition of 'infinite' must be provided". >> >> Does a "Christmas tree" in certain engineering contexts actually have >> be a conifer? > >No. But it seems that you have deleted an essential add-on: >>> For the simple reason that 'infinity' is not a concept that is limited >>> to mathematics alone. It spreads out i.e. into physics, and gives rise >>> there to singularities that exist but one can never perceive them, due >>> to a Cosmic Censorship that prevents us to take a look into the inside >>> of a Black Hole. Thanks to Roger Penrose. Does somebody believe this ? It wasn't relevant. The colloquial definition of "Christmas Tree" spread out too, so why should it not have to have a single definition too? (And it's not like there is a single definition of "inifinite" within mathematics.) Martin
From: Robert Kolker on 21 Jul 2005 20:21 Han de Bruijn wrote: > > For the simple reason that 'infinity' is not a concept that is limited > to mathematics alone. It spreads out i.e. into physics, and gives rise > there to singularities that exist but one can never perceive them, due > to a Cosmic Censorship that prevents us to take a look into the inside > of a Black Hole. Thanks to Roger Penrose. Does somebody believe this ? You are beating a dead horse. Mathematics as such as no empirical content, whatsoever. It is purely abstract. It may be the case that a mapping or correspondence can be established bewteen some mathematics systems and measurable quantities, but that is purely happenstantial. Bob Kolker
From: Robert Kolker on 21 Jul 2005 20:23 Peter Webb wrote: > > > How do you disagree with an axiom? By assuming a contrary axiom, as is done in non-Euclidean geometry. The parallel postulate is denied in non-euclidean geometry. Axioms are posits, or assumptions. They are NOT self evidence truths. Bob Kolker
From: MoeBlee on 21 Jul 2005 22:31
Han de Bruijn wrote: > Daryl McCullough wrote: > > > Instead of using the term "size" to refer to sets, we could > > refer to the "bloppitude". > > > > Instead of using the words "infinite", we could use the term > > "mega-bloppity". > > > > Nothing of any importance about mathematics would change > > if we substituted different words for the basic concepts. > > In contrast, your arguments are about nothing *but* terminology. > > To me, that shows that there is no actual content to your > > arguments.An actual mathematical argument does not depend on word > > choice. > > > > As a challenge, see if you can express your claims about > > infinite sets, or infinite naturals, or set size, or whatever, > > *without* using the words "infinite", "larger", "size", etc. > > This clearly represents the formalist (Hilbertian) view on > mathematics as a "senseless game with symbols". > > Han de Bruijn Would you please say exactly where Hilbert said that mathematics is a "senseless game with symbols"? And the remarks of the poster to whom you responded do not "clearly represent" that he considers mathematics to be a "senseless game of symbols". On the contrary, I take him to be saying that mathematics is made senseless by appropriating mathematical terminology in vague, equivocal, and inconsistent ways. I understand the poster's point to be that one's mathematics shouldn't depend on the suggestiveness of metaphorical jargon, but rather on defintions and proof. Set theorists express ideas through the relations among primitive and defined terms, not by relying upon penumbral associations of the terms. You missed that point and made a strawman of the poster's view and Hilbert's views too. MoeBlee |