Obections to Cantor's Theory (Wikipedia article)
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From: Robert Low on 29 Jul 2005 09:20 Han de Bruijn wrote: > Robert Low wrote: >> What you're proposing doesn't refine set theory, it >> breaks it into tiny little pieces, and then jumps >> up and down on the tiny little pieces until they're >> broken too. > I wonder what you think of my "Boolean Objects": I think you're trying to have your cake and eat it. You simply ignore the problems raised by your additional axiom when they're inconvenient. From the ZFC axioms, there is a set with no elements, {}. But your axiom says that {} is an element of {}, which is an immediate contradiction. So it 'blows up' a lot more than the 'non-finitistic' part of ZFC. This does mean that you can prove lots of theorems: however, it also means that there is no collection of objects that is described by your axioms, which most of us regard as a drawback.
From: Han de Bruijn on 29 Jul 2005 09:28 Robert Kolker wrote: > Overall, classical physics has been replaced by quantum physics, > properly relativized and Lorentz invariant. Only too many nerds in the sci.* groups think so. People here don't even have a clue about what's involved in making a working model of something real & continuous. And that's simply because they have never _done_ it. It's impossible to explain to them that GR and QM only have _refined the limits_ wherein classical physics has to be applied. GR and QM _didn't_ "replace" classical physics. Not at all. Ask any expert in Computational Fluid Dynamics (CFD) and maybe you'll gain some understanding. Han de Bruijn
From: malbrain on 29 Jul 2005 12:02 David Kastrup wrote: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > Martin Shobe wrote: > > > >> On Thu, 28 Jul 2005 15:56:32 +0200, Han de Bruijn > >> <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> > >>>But without the claim that it > >>>is the one and only foundation possible. Why not have _several_ pillars > >>>that provide a foundation, instead of just one? > >> Actually, I don't have a problem with that. In a sense, we have > >> that > >> now with set theory and category theory (And lets not forget logic). > >> But while physics will continue to provide inspirition to mathematics, > >> it will not qualify as a foundation for methematics. > > > > And I don't want that either. Read my lips: > > > > A little bit of Physics would be NO Idleness in Mathematics > > > > See? Just that tiny pinch of salt in your otherwise tasteless soup. > > Axioms don't come in tiny pinches. Anyway, you are free to do with > your own soup whatever you like. Yes they form SYSTEMS, but NO, each one stands/falls on its own as a PART/PINCH. karl m
From: Robert Kolker on 29 Jul 2005 12:02 Han de Bruijn wrote: > It's impossible to explain to them that GR and QM only have _refined the > limits_ wherein classical physics has to be applied. GR and QM _didn't_ > "replace" classical physics. Not at all. Ask any expert in Computational > Fluid Dynamics (CFD) and maybe you'll gain some understanding. Classical electrodynamics cannot account for the properties of semi-condictors. The anomalous precession of planets falsify Newtonian gravitation. Galilean Invarant physics is just plain wrong. It is empirically falsifiable. Bob Kolker
From: malbrain on 29 Jul 2005 12:11
Han de Bruijn wrote: > Robert Low wrote: > > > Han de Bruijn wrote: > > > >> Robert Low wrote: > >> > >>> OK, Han. Let's try your game. > >>> What are the axioms you use in place of ZF(C)? > >>> Once you've listed them, tell us what your > >>> standard model is (an equivalent for the > >>> cumulative hierarchy), and what the natural > >>> numbers are in that standard model. > >> > >> Whoa! Not so fast! Even Einstein didn't finish GR in one day! > > > > > > Right. So you want to throw away something that > > works and replace it with something you haven't > > got yet. You understand our reluctance to > > quit our game and try to play one where the > > rules haven't even been given? > > I'm not going to succeed in "throwing it away". So why bother? > And BTW, did General Relativity "throw away" Newtonian Mechanics? If you want to change a SYSTEM you do have to break-it-down into smaller pieces. With his transform, Fourier showed that you don't have to go to the AXIOMS to do that. karl m |