Cantor Confusion
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From: David Marcus on 29 Oct 2006 15:06 Lester Zick wrote: > On Sun, 29 Oct 2006 13:03:00 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> David Marcus schrieb: > >> > >> > mueckenh(a)rz.fh-augsburg.de wrote: > >> > > William Hughes schrieb: > >> > > > >> > > > And once again WM deliberately confuses, "all positions are > >> > > > finite", with "there are a finite number of positions". > >> > > > >> > > There is no confusing! Every finite position belongs to a finite > >> > > segment of positions (indexes). If you don't believe that and assert > >> > > the contrary, then try to find a finite position which dos not belong > >> > > to a finite segment (of indexes). > >> > > > >> > > It is simply purest nonsense, to believe that "all positions are > >> > > finite" if "there are an infinite number of positions". > >> > > >> > Before any of us wastes more time on this, please pick one: > >>> > >> > 1. You are making a statement that you say is provable within some > >> > standard mathematical system, e.g., ZFC. > >> > > >> > 2. You are making a statement that is true within your own system. > >> > >> My above statement is true within any system which is free of self > >> contradictions. > > > >That's nice, but it isn't what I asked. I'll try again with a more > >specific question: > > > >1. Is your statement provable within ZFC, i.e., using the axioms and > >rules of inference of ZFC? > > Is ZFC free of self contradictions? In W. Muckenheim's opinion? I couldn't say. Some days he seems to say no, other days he seems to say yes. He is impossible to pin down to a clear statement about anything. Anyway, my #1 asked about "provability" and W. Muckenheim replied about "truth". So, it didn't seem worthwhile to worry about the rest of his sentence. Best to go one step at a time. -- David Marcus
From: Han.deBruijn on 29 Oct 2006 15:10 mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > Can you describe a continuous version of the problem where each > > "unit" of water has a well defined exit time? A key part of > > the original problem is that the time at which each ball is > > removed is defined and reached. This is crucial to the problem. It is > > not just a matter of rates. If you added balls 1-10, then 2-20, > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > not empty at noon, even though the rates of insertions and removals > > are the same as in the original problem. So you cannot just > > say the rate is 10 in and 1 out and base an answer on that. > > > The answer for any time t *before noon* is independent of the chosen > enumeration of the balls. Doesn't that fact make you think a bit > deeper? And add this to the fact that noon and beyond cannot exist in this problem. Han de Bruijn
From: Virgil on 29 Oct 2006 14:19 In article <MPG.1faed1fae99d87889897b2(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > Real numbers have two standard definitions generally accepted: > > (1) as Dedekind "cuts" > > (2) as equivalence classes of Cauchy sequences of rationalsmodulo the > > null sequences. > > You can also construct the real numbers as infinite decimals. WM argues that lack of an explicit and completely determiniable decimal expansion is equivalent to non-existence for real numbers. And as that is directly counter to the infinite decimals model, and has certain other problems, I didn't choose to consider it.
From: Virgil on 29 Oct 2006 14:22 In article <1162152643.939045.128030(a)i42g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > stephen(a)nomail.com schrieb: > > > > > Can you describe a continuous version of the problem where each > > > "unit" of water has a well defined exit time? A key part of > > > the original problem is that the time at which each ball is > > > removed is defined and reached. This is crucial to the problem. It is > > > not just a matter of rates. If you added balls 1-10, then 2-20, > > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > > not empty at noon, even though the rates of insertions and removals > > > are the same as in the original problem. So you cannot just > > > say the rate is 10 in and 1 out and base an answer on that. > > > > > The answer for any time t *before noon* is independent of the chosen > > enumeration of the balls. Doesn't that fact make you think a bit > > deeper? > > And add this to the fact that noon and beyond cannot exist > in this problem. > > Han de Bruijn Perhaps not in HdB's philosophy, but that only limits him.
From: David Marcus on 29 Oct 2006 15:31
Han.deBruijn(a)DTO.TUDelft.NL wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > stephen(a)nomail.com schrieb: > > > > > Can you describe a continuous version of the problem where each > > > "unit" of water has a well defined exit time? A key part of > > > the original problem is that the time at which each ball is > > > removed is defined and reached. This is crucial to the problem. It is > > > not just a matter of rates. If you added balls 1-10, then 2-20, > > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > > not empty at noon, even though the rates of insertions and removals > > > are the same as in the original problem. So you cannot just > > > say the rate is 10 in and 1 out and base an answer on that. > > > > The answer for any time t *before noon* is independent of the chosen > > enumeration of the balls. Doesn't that fact make you think a bit > > deeper? > > And add this to the fact that noon and beyond cannot exist > in this problem. Are you still doing physics, water, and a finite number of molecules? Let us know when you switch to mathematics. -- David Marcus |