Cantor Confusion
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From: Lester Zick on 8 Dec 2006 14:27 On Fri, 08 Dec 2006 17:14:39 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 12/7/2006 7:06 PM, Lester Zick wrote: >> On Thu, 7 Dec 2006 02:04:34 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >>>Bob Kolker wrote: >>>> Eckard Blumschein wrote: >>>> >>>> > Roughly speaking, it just claims that a set is unambiguously determined >>>> > by its elements. If i recall correctly A=B<-->(A in B and B in A) >>>> > >>>> > Perhaps the Delphi oracle provided less possibilities of tweaked >>>> > interpretation betwixed and between potential and actual infinity. >>>> >>>> What is "potential" infinity. Can you define it rigorously? >>> >>>Even a non-rigorous defintion would be a start. >> >> Well since according to David a definition is "only an abbreviation" >> how about "X"? >> >> ~v~~ > >Strictly speaking there is no potential infinity. Infinity is a >fictitious quality. The series of natural numbers is potentially >infinite. Aristotele wrote: Infinity exists potentially. There is no >actual infinity. > >Marcus is quite right. We should better explain such basic terminology. Of course. I was just mocking David's idiotic definition of definition as only an abbreviation which he doesn't seem particularly anxious to defend. By the way I hope the "we" you refer to is the editorial "we". I myself take no position with respect to such issues. I'm opposed to set "theory" and it's nonsensical use of points, definitions, infinity and such. But alternative doctrines are another matter. I don't claim any affinity with anyone else on these matters. I've posted my opinion of Aristotle as history's first formal empiric but beyond that I have no more use for mathematical empiricism than scientific empiricism and none at all for dialectical philosophy regardless of subject. ~v~~
From: Virgil on 8 Dec 2006 14:27 In article <45794E19.5050608(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 1:05 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 12/5/2006 9:23 PM, Virgil wrote: > >> > >> >> Do not confuse Cantor's virtue of belief in god given sets with my power > >> >> of abstraction. > >> > > >> > Cantor's religious beliefs are as irrelevant as EB's beliefs in his own > >> > infallibility. > >> > >> I am not infallible. Show me my errors, and I will express my gratitude. > > > > Show you your errors or convince you that they really are errors? The > > former is simple, but the latter appears to be impossible. We can't > > force you to learn, if you don't wish to. > > > I can force you to either refute e.g. my hint that Cantor's definition > of a set has been declared untenable or tacitly accept this fact. How? By sending storm troopers to visit him?
From: Virgil on 8 Dec 2006 14:31 In article <457954B2.5000907(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > I think that to find that 4th one must use most of a 5th. > > Sorry, an insider has to know: There are exactily 4 options, cf. e.g. > Fraenkel 1923 or a good book on logic. EB has not only opened the fifth, he has finished it and is part way through another.
From: Eckard Blumschein on 8 Dec 2006 14:39 On 12/8/2006 8:04 PM, Virgil wrote: > In article <45793B7C.4050105(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: with higher abstraction. >> While the imaginary numbers are obviously different from the ordinary >> numbers, the corresponding distinction between rationals and reals is >> more subtle. > > Rationals are ratios of integers, non-rational reals are not ratios of > integers. That should not be too subtle even for EB. Do not try cheating. I did not refer to non-rational reals but to reals in general. DA2 is valid for reals in general. >> Perhaps it is most helpful to declare the reals just >> fictions, while the rationals, including naturals and integers, are >> genuine numbers. > > It is most unhelpful to misuse words whose common meanings tend to > mislead one about their technical meanings. In this case the common meaning is quite appropriate. Or can you suggest better words? >> The difference between rationals and reals corresponds to the difference >> between potentially infinite and perfectly infinite. > > Since in such set theories as ZF or NBG or NF there do not exist any > such things as potentially infinite sets but there do exist infinite > sets, the distinction is irrelevant in those set theories. This is perhaps the most important achievement of these systems of axioms, maybe it has even positive aspects too. > And in those > theories each real is a set just as each rational is a set and each > natural is a set. How to get this information out from the axioms? > If EB wishes to produce an axiomatic system which distinguishes between > potential and actual, let's see him do it. Admittedly, I would not be in position for doing so, I do not intend it, and I do not even criticise the effective neglect of the difference in mathematical pratice. Nonetheless I suggest to clarify that 1) There is a categorial difference between rational and real numbers 2) Corresponding reasoning of mine is not wrong (some consequences) 3) It is overdue to delete nonsensical basics and terminology from books and lectures (in particular the distinction between ordinals and cardinals, the notion cardinality, the German word ueberabzaehlbar, transfinte numbers, all alephs, etc.). Mathematics will not suffer from this lost but benefit from it. >> Infinity is in some sense the opposite of being infinite. > > Sanity is clearly the opposite of being EB. Just try and understand what I meant.
From: Virgil on 8 Dec 2006 14:41
In article <45796266.4030309(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 9:58 PM, Virgil wrote: > > The only fallacies here are by those who proclaim that ZF or NBG must > > have internal contradictions because they do not conform to their > > critic's intuitions. > > Forget ZF, NBG, and intuitions altogether. The original topic is Cantor > and confusion, your confusion. I am not confused in stating that within ZFC and NBG and NF, Cantor is exactly right. So it is EB who confuses himself. > >> > Cantor's first proof > >> > >> Would you please clarify what proof or paper you are referring to! > > > > The one before what you mislabel "DA2" > > If you refer to Cantor's so called 1st diagonal argument stolen from > Cauchy, then you may reiterate what you intended to say. DA1 just > illustrates that discrete points can be arranged along a line. So > rational numbers are countable. Those who so misread that theorem to mean what EB states are incompetent. > > > > >> > 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. > > > > According to EB, who confesses that he is not a mathematician. > > Nonetheless I can read and reason. We have considerable disproofs of both these claims. Just above EB claims that Cantor's first theorem on the uncountability of the reals says things it does not say. > > >> >> 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. > > > > Been there! Done that! Have the t-shirt! > > A dirty kind of surrender. When EB calims that things which have happened have not happened, why need I reiterate? I am satisfied that EB is wrong and has been shown to be wrong by several posters in ways satisfactory to convince mathematicians. That EB's ego an d ignorance of mathematics will force him to deny it is his problem, not mine. |