Cantor Confusion
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From: Eckard Blumschein on 4 Dec 2006 12:46 On 12/1/2006 9:42 PM, Virgil wrote: > In article <45706268.1020005(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 1:37 AM, Dik T. Winter wrote: >> > In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein > >> > Oh, for once, try to talk mathematics. By the axiom of infinity the >> > set of all naturals is neither hypothetical nor fictitious. >> >> This axiom combines flawless Archimedean reasoning with an at least >> questionable replacement of the notion number by the notion set. > > When EB presents a completed axiom system from which he can generate > mathematics, or at least arithmetic, he may join the lists, but until > then he is merely a spectator at mathematics, and not competent to be a > judge. At least, a spectator is not blind.
From: Eckard Blumschein on 4 Dec 2006 13:06 On 12/1/2006 9:35 PM, Virgil wrote: > In article <1164982199.959381.134510(a)j72g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Eckard Blumschein schrieb: >> >> >> > I recall being a little boy wondering when I was told that while there >> > is no evidence proving the existence of god there is also no evidence >> > showing his non-existence. Are those crippled who don't believer in CH? >> > I consider the background of CH given in the difference between number >> > and continuum. This might be crippled down to the truth? Do you agree? >> > >> No, I am sorry, I do not. The continuum is nothing but our failure to >> look closely enough. In physics it lasted 2000 years to settle the idea >> of the atom and to supplement and complete it by the uncertainty >> relations. The majority of matematicians is not yet far sighted enough >> to recognize the same situation in their realm. > > > We do not yet /know/ that the physical universe is not continuous, so > why should we reject a mathematically continuous real number system? I do not just agree. I would even like to stress once again that the mathematical concepts of continuum is independent from its application. What about continuous real numbers, I only vote for a little bit more honesty and more consequent reasoning. If irrational numbers are thought to complete the rationals which sounds quite logical, then the constituted entity of the reals has to be as fictitious as the irrationals. The name real "numbers" is misleading due to lacking trichotomy in this case. Nonetheless, one may largely operate with the real "numbers" AS IF they were actually numbers. Likewise oo ist often treated like a number while oo + a = oo obviously violates the ordinary rules.
From: Bob Kolker on 4 Dec 2006 13:08 Eckard Blumschein wrote: > > > At least, a spectator is not blind. Neither are the players. And in addition the players know something, and you don't. Bob Kolker >
From: Eckard Blumschein on 4 Dec 2006 13:24 On 12/1/2006 8:55 PM, Virgil wrote: > In article <45700723.3060406(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/30/2006 1:39 PM, Bob Kolker wrote: >> > Division by zero in a field yeilds a contradiction. >> >> Just this contradiction resides already in the notion of (actual) >> infinity. > > Division by zero in standard sets of numbers is not defined because > there is never a unique x in such sets of numbers for which a = 0*x. > Either no x works or more than one works. > > Infinity has nothing to do with it. > > A finite example: > > The residues of the integers modulus a prime is always a finite field > under the usual addition and multiplication, so there is no > "infinity" involved, but division by zero in those fields is still > barred for the reason above, a = 0*x can never have a unique solution. I do not feel limited in thinking to the indefinitely large. I likewise consider the indefinitely small (infinitesimal). >> Isn't it better to understand why it is incorrect than simply to learn >> it is forbidden? >> >> Eckard Blumschein > > It is better to understand the real reason (see above), but Eckard > doesn't seem to understand the real reason. It has nothing to do with > "infinity". Not directly with the indefinitely large, yes.
From: MoeBlee on 4 Dec 2006 13:30
mueckenh(a)rz.fh-augsburg.de wrote: > No. Compare Fraenkel et all. They talk about to look at the universe of > all sets not as a fixed entity but as an entity capable of "growing". > What they understand and how this growing can take place has lead to > many misunderstandings by underinformed mathematicians. But however one > may interpret their sentence. The universe of all sets can change, to > put it cautiously. That is not at all ridiculous. What they convey is that the universe if different upon adopting different axioms. You just want to skip all of what they wrote in that section. MoeBlee |