Cantor Confusion
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From: Dik T. Winter on 7 Dec 2006 07:13 In article <1165492756.322548.255040(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > The set in the quantifiers you are using is not finite either. The > > quantifier are not over a single line, but over the set of natural > > numbers. > > For finite natural numbers, we have finite lines only. It is not the > question of a single line. Every line is finite. Therefore there is no > line where quantifier reversal could not be applied. But the quantifier reversal is *not* applied to individual lines, it is applied to the set of natural numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 7 Dec 2006 07:26 Virgil schrieb: > It appears that the negation of "the set of all even numbers has > cardinal greater than any even number" would have to be " there is an > even natural number as great as the cardinality of the set of all even > natural numbers." Of course. Even greater even numbers must exist. And therefore, there is no set of all even numbers. > > > > By their fruits ye shall know them. Mathematics based on infinite sets > > > has produced useful and even indispensible results. Try getting modern > > > physics without it. > > > > For applications the usual way to interpret things is sufficient. > > Absent accepting the infinitenesses of analysis, a lot of analysis goes > missing. The infinity of standard analysis is only potential and will remain after ZF will have perished. Regards, WM
From: Han de Bruijn on 7 Dec 2006 07:26 Bob Kolker wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> >> Those who pretend to be able of knowing every integer and, therefore, >> to imagine the whole actually infinite set. > > If there were an integer I could not imagine, there would have to be a > least such integer. Well, it isn't 0 or 1. So the least integer I could > not image has a predecessor. But I can imagine it. I could also imagine > adding one to it. Despite your manifest lack of inspiration (since a "spirit" or soul does not exist according to you), you nevertheless have a lot of imagination. > Contradiction. With the abovementioned mathematical induction you have only established the potential infinite, not the actual one. The former is not denied by any of us. Han de Bruijn
From: Eckard Blumschein on 7 Dec 2006 07:36 On 12/7/2006 4:32 AM, Virgil wrote: > In article <4576DCFC.6040901(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> Every single integer is a countable element. >> > >> > Since the set of integers is a subset of the set of reals, every integer >> > is a real in any real set theory, whatever EB may claim. >> >> Just this is fallacious. The reals are continuous like glue. > > Being like glue is not a mathematically relevant quality. The Dedekind > cut for a real, e.g., for sqrt(2), is as actual as is the rational for > 1/2. Dedekind dealt with numbers ordered like points along a line. He assumed to consider _all_ points and _all_ rationals. This was the elusive basis of several fallacies. See my reply this morning. >> Embedded genuine numbers lost their numerical address >> >> > Countability >> >> > /Uncountability are properties of -sets-, not individuals. >> >> >> >> Do not reiterate what I know but deny. >> > >> > Then do not reiterate your falsehoods >> >> Cantor's proof (DA2) of uncountability required numbers of perfectly >> infinite length. > > Cantor's first proof Would you please clarify what proof or paper you are referring to! >did not require anything but one of the extant > models of the reals within set theory. No decimal or other basal > representation was required. > > Note that the Dedekind cut is only a mere refinement of Eudoxus > treatment of incommeasurables which has been acceptable since the time > of Euclid. Perhaps you mean Eudoxos of Knidos (408-355). Dedekind uttered that he did not understand what Euclid wrote (325-275). Perhaps you are calling this refinement. Eudoxos, Parnenides, and Aristotele long before Spinoza denied the idea that the continuum has been composed of atoms. So they maintained the opposite of Dedekind's illusion. > You understand: Such numbers contradict common sense. > > Common sense is irrelevant in mathematics. It is only irrelevant if it contradicts to well-founded theory and has proven wrong. In case of Cantor's transfinite numbers all putative proofs turned out to be not correctly founded. >> Writing very fasr and without proofreading, I cannot guarantee correct >> command of my English. Sometimes words or letters may be missing or >> confused. Nevertheless, all these arguments of mine are most likely >> flawless. > > Your flaws in English are excusable, your flaws in logic are not. I will hopefully understand at least one flaw of mine in logic, provided you are correct. Do not hesitate pointing me to what you consider wrong and why.
From: mueckenh on 7 Dec 2006 07:37
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Franziska Neugebauer schrieb: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > Bob Kolker schrieb: > >> >> [...] > >> >> >> Rational numbers are non-genuine. Nowhere in the physical world > >> >> >> outside of our nervouse systems do they exist. > >> >> > > >> >> > The nervous system is a part of the physical world. > >> >> > >> >> Hence this issue is to be disussed in a physics or neuro sciences > >> >> newsgroup. > >> >> > >> >> > The integers belong to the nervous system. > >> >> > >> >> The nervous system is generally not an issue in mathematics. > >> > > >> > That is why many mathematicians overestimate their capabilities so > >> > grossly. > >> > >> Which mathematician overestimates her or his capabilities? > > > > Those who pretend to be able of knowing every integer and, therefore, > > to imagine the whole actually infinite set. > > I don't know any mathematician who "pretends" an ability of > "knowing" (what does this mean?) every integer. Who "pretends" so? > Virgil can imagine all natural numbers, he said. Regards, WM |