Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 28 Jul 2005 05:08 MoeBlee wrote: > If you'd like to have a theory in which Ax x = {x}, then that's fine. > But your claim - pretty much bragging - on your web site, that this is > consistent with Z without infinity is false. Hmm, seems that I have to pay more attention to this issue. Thanks. > Meanwhile, I had posted to you a few days ago asking for substantiation > of your claim that Hilbert said that mathematics is a senseless game. > Unless I missed something, you did not respond. Geez, did you notice how _big_ this thread has become? Yes, it's easy to miss something, as I obviously did. (I wish T.O. stops monopolizing ...) > That is, of course, > your prerogative, but another poster picked up on your post, so that > the claim was iterated, thus contributing to the propogation of your > unsubstantiated assertion. There is spread over the Internet a supposed > quote of Hilbert saying that mathematics is a meaningless game of > symbols. But Martin Davis's question whether this quote is in Hilbert's > writings has so far not had an affirmative response. (Let alone that > whatever Hilbert's remarks were, they must be taken in a context that > includes Hilbert's explicitly stated concerns with intuition and > content.) You've made an unsubstantitated and misleading claim about an > important matter in the history of mathematics, and you haven't the > intellectual responsibility to address this. But, hey, you've got > mantras do... But hey, that Hilbert quote, *I* didn't make it up, honestly. I don't know precisely where it comes from, but I've always thought that it's from a reliable source. Reliable? Hmm, perhaps it's the Internet ... Han de Bruijn
From: Han de Bruijn on 28 Jul 2005 05:24 Tony Orlow (aeo6) wrote: > Robert Kolker said: > >>Sets are not physical. They are abstract. Search the world over and you >>will not find a set. In the real world trees exist but forests do not. >>Forests exist in our heads. > > Forests do not exist? Do people exist, or only cells? Maybe cells don't exist, > but only organelles. Do organelles exist? Nope, just molecules. Oh wait, > molecules don't exist, only atoms, which are, of course, unsplittable. They're > not? Okay they don't exist. Only quarks exist. Quarks are made of what? Okay > they don't exist either. Nothing exists, except Bob. I find this quite an intelligent answer. It adresses very well all the problems that some people in this thread have with using common speech (: Robert Kolker, Randy Poe, David Kastrup, to name a few). Han de Bruijn
From: David Kastrup on 28 Jul 2005 05:34 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Tony Orlow (aeo6) wrote: > >> Robert Kolker said: >> >>> Sets are not physical. They are abstract. Search the world over and >>> you will not find a set. In the real world trees exist but forests >>> do not. Forests exist in our heads. >> Forests do not exist? Do people exist, or only cells? Maybe cells >> don't exist, but only organelles. Do organelles exist? Nope, just >> molecules. Oh wait, molecules don't exist, only atoms, which are, of >> course, unsplittable. They're not? Okay they don't exist. Only >> quarks exist. Quarks are made of what? Okay they don't exist >> either. Nothing exists, except Bob. > > I find this quite an intelligent answer. It adresses very well all > the problems that some people in this thread have with using common > speech (: Robert Kolker, Randy Poe, David Kastrup, to name a few). But this is _exactly_ the reason why we have axioms in mathematics: they are the point where you don't need to look further. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 28 Jul 2005 05:48 Martin Shobe wrote: > Since sets aren't physical, it is not physically correct to assert > that a e A ==> a c A. From what I understand of the history of set > theory, physics wasn't the inspiration, and it certainly isn't its > purpose at the current time. Since sets aren't physical, since straight lines aren't physical, since numbers aren't physical. Of course they _are_ physical. Or better: they _were_ physical. Though some have been more physical than others. Those which have been more physical are the better ones, i.e. straight lines. > That doesn't mean that you can't explore theories where a e A ==> a c > A. Just don't confuse what you are doing with standard set theory. > (And it would be nice if you indicate that you aren't doing standard > set theory, if for no other reason than to avoid some confusion.) Heh, heh. That's precisely the first thing I want: throw _doubt_ upon everything that everybody finds so certain. And I thought that I have indicated that well enough, all over the place, ever since 1989. > If you are going to use that as an axiom, then the set theory you add > it to should include sets that aren't well founded. Make that very > few well-founded sets. As a consequence, mathematics can no longer be founded on set theory. That's precisely what I want: set theory as a relatively _unimportant_ branch of mathematics. Han de Bruijn
From: Robert Kolker on 28 Jul 2005 06:36
Poker Joker wrote: > > I disagree with grouping infinity in with the natural > numbers. That's what Cantorians do. They do not. There is a clear distinction between finite and infinite cardinality. Bob Kolker |