Cantor Confusion
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From: Eckard Blumschein on 8 Dec 2006 11:14 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.
From: Eckard Blumschein on 8 Dec 2006 11:21 On 12/7/2006 4:09 PM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/5/2006 8:52 PM, Virgil wrote: >> > In article <1165322064.705072.182240(a)80g2000cwy.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > >> >> The number of your contributions has increased by 1 with your post I >> >> just answer. The same holds for the set of your contributions. >> > >> > Such time dependent "sets" are not the same as sets under the rubric of >> > set theories, as they do not, for example, obey the axiom of >> > extensionality. >> >> Fraenkel 1923, p.190: >> "Dieses Axiom besagt, dass eine Menge m als vollst�ndig festgelegt gilt, >> sobald bestimmt ist, welche Elemente in ihr enthalten sind." > > How is that relevant? I appreciate that you seemingly understand German. Elsewhere I gave the even less stringent mathematical axiom of extensionality.
From: stephen on 8 Dec 2006 12:15 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>stephen(a)nomail.com wrote: >> >>>>But everything can be modelled as a set. >> >>>Define "everything" and prove that claim. >> >> By "everything", I meant everything mathematical. Of course that is not 100% precise. >> And no, I cannot prove it. But so far all the various objects of mathematics can be >> modelled using set theory. That is what is meant by set theory being a foundation >> for mathematics. If someone were to invent something "mathematical" (whatever that may >> mean exactly) that could not be described in terms of set theory, then set theory would >> no longer serve as a foundation. But given that the basics such as the real numbers, >> functions, limits, calculus, etc. all can be founded in set theory, it would have to >> be something strange indeed. Not that there is anything wrong with strange, but you >> probably would like it less than set theory. > Correction. By "everything" you probably mean "everything according to > nowadays mainstream mathematics", which _is_, of course, "mathematics", > according to your probably rather limited view. But since you can not > really prove anything of the kind, I will rest my case. > Han de Bruijn It's not much of a case. You have not presented any evidence that there exists any sort of mathematics not describable by set theory. Until such evidence exists, the hypothesis that mathematics can be modelled with set theory has not been falsified. And don't bother presenting something that uses limits, functions, etc. as all of those things can be modelled with set theory. Nobody can prove the Church-Turing thesis, but that does not prevent people from being confident that our notion of computability is accurate. Nobody can prove anything in science, but that does not prevent people from placing a lot of confidence in it. For example, there is no proof that gravity acts on all masses. I am surprised that you do not understand something as basic as that. Stephen
From: Eckard Blumschein on 8 Dec 2006 12:41 This time even Bob Kolker is correct. However, Dummkopf is one more unacceptable indication for mean personality. On 12/7/2006 3:47 PM, Bob Kolker wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> You cannot imagine the integer [pi*10^10^100]. > > That is not an integer, dummkopf. It is an irrational real number. > > Bob Kolker
From: Eckard Blumschein on 8 Dec 2006 13:19
On 12/7/2006 8:38 AM, Virgil wrote: > In article <MPG.1fe18bc534a955549899ed(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> Eckard Blumschein wrote: >> >> > I didn't find a single counter-example. >> >> That doesn't prove anything. In mathematics, we prove things. Who proved Cantor's interpretation of his second diagonal argument? Well, many attepts failed to disprove uncountablity of the reals. However, the fallacy was in the neglect of the 4th logic option: There is no possibility to quantitatively compare with each other infinite objects. Already Galilei came to this conclusion. Who proved that Dedekind's cut really created irrational numbers after he himself confessed that he did not have a proof of a basic assumption? Who proved that Cantor's interpretation concerning the power set was correct? Who cared for the open secret that there is no valid definition of a set. Who proved the putative identity of the reals used in the 2nd diagonal argument with mandatory definitions of reals? Who provided a well-ordering of the reals? Who showed that one really needs alephs and not just the notions countably infinite and uncountable? Who gave at least one positive example for the necessity to have the misleading intermediate solutions in integral tables? .... |