Obections to Cantor's Theory (Wikipedia article)
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From: Martin Shobe on 29 Jul 2005 08:03 On Fri, 29 Jul 2005 10:29:05 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Patrick wrote: > >> The notion of subset, is distinct from the notion of member. >> >> {} is a subset of {}. >> >> {} is not a member of {}. > >I know that. But I'm changing the rules of the game in accordance with >the mantra: > > A little bit of Physics would be NO Idleness in Mathematics > >Then {} is a member of {} . > >> The principal of extentionality says that two sets >> are equal iff they contain exactly the same members. >> You can't, under ordinary rules, claim that {{}} = {}, >> since the LHS has 1 member, and the RHS has none. > >Under the new rules {{}} = {} . If by the new rules you mean Ax x = {x}. Then {} doesn't exist at all. (It's existence leads to a contradiction, namely that {} is a member of {} and that {} is not a member of {}.) That probably isn't a problem for you, as nothing is nothing after all. But, I'm not sure you'll care for all of the consequence of that axiom. For example, if you define a forest as a set of trees (some of your comment indicate that you find this acceptable), you can conclude that all forest have only one tree in them. Martin
From: Han de Bruijn on 29 Jul 2005 08:15 Martin Shobe wrote: > On Fri, 29 Jul 2005 11:14:56 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >>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. >> >>But nevertheless: *IN* your soup. > > > A "tiny pinch of salt"? From the axioms I've seen you proposing that > would be more like a ten pound bag of rock salt added to single > serving. No, no. Nobody would like the soup anymore. Just a pinch. But, everybody knows that a tiny pinch of salt in otherwise tasteless soup makes a _huge_ difference. Han de Bruijn
From: Robert Low on 29 Jul 2005 08:19 Han de Bruijn wrote: > But, everybody knows that a tiny pinch of salt in otherwise tasteless > soup makes a _huge_ difference. So does a small drop of Prussic acid in an otherwise nourishing meal.
From: Han de Bruijn on 29 Jul 2005 08:19 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? Han de Bruijn
From: David Kastrup on 29 Jul 2005 08:20
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. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |