Cantor Confusion
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From: Eckard Blumschein on 27 Nov 2006 06:39 On 11/25/2006 3:58 AM, Dik T. Winter wrote: > In article <456692F9.7070103(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > > On 11/24/2006 2:25 AM, Dik T. Winter wrote: > > > > > > Mathematicians like you are hopefully aware of the trifle that the > > > > relation >= cannot be applied to the really real numbers. > > > > > > Oh. What are "really real numbers"? > > > > Those, like the imagined numerical solution to the task pi, which are > > just fictions. Those which are assumed as basis for DA2, > > In that case what are the "not-really real numbers"? Those putatively real numbers as defined by Dedekind's cut, nested intervals, etc. without the step into a different quality. I learned the word "Grenzuebergang". Is "limit" the mathematically correct translation? In my understanding, a limit ia a border while Grenzuebergang means border crossing. When Eudoxus coined the term, what did he mean? > > > What you are missing is that mathematics provides an idealisation of > > > arithmetic (amongst others), and from that background provides > > > processes to do "real" calculations to get results in (amongst others) > > > the physical world. The whole field of numerical mathematics could > > > barely have been build without the idealised mathematical background. > > > > No no. I do not deny that numbers can come as close as you like to the > > ideal concept of infinity and continuum. Rational numbers already do > > that job. > > You completely misunderstood what I wrote. Cholesky decomposition While I did not understand you wrong, I nonetheless very appreciate your hint to Cholesky decomposition. This way I found http://sepwww.stanford.edu/sep/prof/gem/hlx/paper_html/node11.html Notice, I am mainly interested in causality and spectral analysis, cf. http;//iesk.et.uni-magdeburg.de/~blumsche/M283.html
From: Eckard Blumschein on 27 Nov 2006 07:21 On 11/25/2006 7:59 PM, Virgil wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > The form of induction I use is based on having, as one has within ZF and > NBG, an infinite set of finite naturals. If one has the first natural > and one has the successor to every natural, then one has all of the > members of that infinite set of finite naturals. Isn't this Cantor's childish naive alias selfcontradictory assumption? What makes any number finite? Obviously it is the break of the potentially never-ending process of counting. Do not confuse any with all. The expression "all natural numbers" does not have a conceivable meaning. Since the natural numbers count themselves, it would be identical with the largest possible number, clearly at odds with Archimedes. Since there is no such number, there is no "all natural numbers", there is no actual infinity except for the realm of fiction. You may have any natural you like. You may never get them all just by counting. In order to have them all, you have to move onto the opposite quasi devine point of view on the expense of loosing the ability to see them like single elements.
From: Eckard Blumschein on 27 Nov 2006 07:22 On 11/25/2006 8:03 PM, Virgil wrote: > In article <1164446180.867559.37730(a)l39g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Philosophy of mathematics belongs to mathematics. > > > If that is so then those like WM and EB who try to impose the philosophy > of physics on math are out of bounds. Geometry is not outside mathematics.
From: Bob Kolker on 27 Nov 2006 07:25 Eckard Blumschein wrote: > > > Geometry is not outside mathematics. True. But Geometry is not all of mathematics either. The two historical sources of mathematics has been land measurement and counting objects. After which comes tracking the cycles in the heavens. Bob Kolker
From: Eckard Blumschein on 27 Nov 2006 07:41
On 11/25/2006 9:50 PM, Ross A. Finlayson wrote: > McGill's proof checker found the rationals uncountable. Can you point us to an available source, please? I guess, the mathematical object under test was not the rationals but the set of rationals understood like a whole entity. If so, did the checker also test the naturals? I see the naturals countable while the entity of all naturals an uncountable fiction. My notion of the reals is different from Dedekind's. Already each single out of these reals is uncountable. |