Cantor Confusion
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From: Han de Bruijn on 16 Oct 2006 03:47 Virgil wrote: > In article <1160933229.072292.316580(a)e3g2000cwe.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Dik T. Winter schreef: >> >>>In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com> >>>Han.deBruijn(a)DTO.TUDelft.NL writes: >>> > stephen(a)nomail.com schreef: >>> > > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: >>>... >>> > > > Pray warn me when 2 has changed sufficiently to be the square of a >>> > > > rational. >>> > > > I would not like to miss that moment. >>> > > >>> > > The day the circle is squared cannot be to far behind. >>> > >>> > Come on, guys! You all know that, in the world of approximations, >>> > 2 _is_ the square of a rational and the circle _is_ squared. >>> >>>I thought you were talking mathematics? >> >>I thought approximations were a part of mathematics? > > But approximations are not all of mathematics in the way that HdB > preaches, and "approximately equal" is still mathematically > distinguishable from "exactly equal". How? Han de Bruijn
From: Han de Bruijn on 16 Oct 2006 03:54 Virgil wrote: > In article <1160933968.829143.108720(a)f16g2000cwb.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>imaginatorium(a)despammed.com schreef: >> >>>You have been told, probably hundreds of times now: mathematics is the >>>study of consistent abstract systems. If you don't like that, push off >>>to sci.physics, or alt.crystal-gazing. >> >>The funny thing is that you are not in the position to DICTATE what >>mathematics is, especially not in a FREE forum like 'sci.math'. If you >>don't like that, push off to a censored forum like 'sci.math.research'. > > He does not have to dictate what is common knowledge among > mathematicians. Cantorians/Hilbertian mathematicians, as distinguished from intuitionist and constructivist mathematicians. Not to speak of the many more people who find their employment in Applied Mathematics (e.g. Computer Science) Han de Bruijn
From: Han de Bruijn on 16 Oct 2006 04:01 David Marcus wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>David Marcus schreef: >> >>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> >>>>David Marcus schreef: >>>> >>>>>Han de Bruijn wrote: >>>>> >>>>>>I'm not interested in the question whether set theory is mathematically >>>>>>inconsistent. What bothers me is whether it is _physically_ inconsistent >>>>>>and I think - worse: I know - that it is. >>>>> >>>>>What does "physically inconsistent" mean? Wouldn't your comments be >>>>>better posted to sci.physics? Most people in sci.math are (or at least >>>>>think they are) discussing mathematics. >>>> >>>>ONE world or NO world. >>> >>>Sorry. No idea what you mean. Do you have anything to say about >>>mathematics or would you prefer we all just ignore you? >> >>It would save me a lot of time if some people here start to ignore me. > > Does that mean you have nothing to say about mathematics? Sigh! Start digging into my website. I've said more about mathematics than anybody else in 'sci.math'. Han de Bruijn
From: Han de Bruijn on 16 Oct 2006 04:08 Dik T. Winter wrote: > In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > Han.deBruijn(a)DTO.TUDelft.NL writes: > > Virgil schreef: > ... > > > I do not object to the constraints of the mathematics of physics when > > > doing physics, but why should I be so constrained when not doing physics? > > > > Because (empirical) physics is an absolute guarantee for consistency? > > Can you prove that? Is it possible to live in a (physical) world that is inconsistent? Han de Bruijn
From: mueckenh on 16 Oct 2006 08:25
Virgil schrieb: > In article <1160814799.951385.291960(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > Consider the binary tree which has (no finite paths but only) infinite > > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > > edges is countable, because we can enumerate them > > > > > > > > 0. > > > > /a\ > > > > 0 1 > > > > /b\c /\ > > > > 0 1 0 1 > > > > ............. > > > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > > a is inherited by all paths which begin with 0.00, the other half of > > > > edge a is inherited by all paths which begin with 0.01. > > > > > > One can relate one of the 'a' edges, say the left one, to all paths > > > beginning with 0.0 and the other 'a' edge, the right one with. the > > > string beginning 0.1 > > > > You misunderstood the notation (even after so many postings!). There is > > only one edge a taken as an example. The edge on the right hand side is > > not labelled. > > Every edge must have its own label if they are to be referred to by > label. Of course, but what I did is only to give an example how the edges and their parts are inherited. For my proof I do not need to refer to every single edge but only to count the shares. > > How about L and R a labels for the left and right branches at the root > node, LL and LR for the left and right branches at the left node and RL > RR for the rightmost pair at the right node , > then LLL, LLR; LRL, LRR; RLL, RLR; RRL, RRR for edges at the next level, > and so on ad infinitum. > > That gives every edge in the entire tree a unique label by which it may > be referenced. Yes, if necessary, one could do so. Regards, WM |