Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Robert Low on 29 Jul 2005 08:30 Han de Bruijn wrote: > Robert Low wrote: >> 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? I think that's a question for you to answer: you're the one telling us to throw away set theory. Because adapting it by including the new rule a={a} really does break it at a very fundamental level. Apart from anything else, it means that = {} (by taking a to be nothing at all) and hence that everything in the cumulative hierarcy then also collapses down to absolutely nothing at all. So presumably you need a bunch of urelements. And then the whole universe is going to collapse down just to the set of urelements, because, for example, the ordered pair (a,b) is {a,{a,b}} = {a,a,b} = {a,b}, and similarly the power set of any set is just the same set back again. Oh, and the ordered pair (a,b) is the same as the ordered pair (b,a), which is a little unfortunate. So, why should we adopt your physical principle that a={a}? > And BTW, did General Relativity "throw away" Newtonian Mechanics? Conceptually, yes, but in computational practice it refines it. 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.
From: Han de Bruijn on 29 Jul 2005 08:36 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": http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#bo Han de Bruijn
From: Robert Kolker on 29 Jul 2005 09:02 Han de Bruijn wrote: > > > That comprises a difference, because we are still designing cars with > Newtonian mechanics, not with GR. True. Auto drivers (among others) use GR to find out where their classically designed car is on the face of the earth. You realize that your computer depends to some extent on quantum field theory which is based in part on special relativity. If we stuck with newton we would still be using semaphores. Even Samuel F. B. Morse did not use classical mechanics to make his telegraph to work. Bob Kolker
From: Robert Kolker on 29 Jul 2005 09:03 Martin Shobe wrote: > > Not entirely accurate. For one thing, a number of cars include GPS > systems nowdays. And those don't work properly if designed with > Newtonian mechanics. Particularly the solid state chips that control the ignition and firing. Bob Kolker
From: Robert Kolker on 29 Jul 2005 09:06
Han de Bruijn wrote: > > > I'm not going to succeed in "throwing it away". So why bother? > And BTW, did General Relativity "throw away" Newtonian Mechanics? As far as gravotation is concernt, GTR -replaced- Newtonian Gravity. The underlying mathematics of GTR is quite different from that of Newtonian gravitation. Overall, classical physics has been replaced by quantum physics, properly relativized and Lorentz invariant. Bob Kolker |