Cantor Confusion
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From: Virgil on 8 Dec 2006 15:47 In article <457987C1.5090209(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:47 PM, Virgil wrote: > > In article <4577F0AD.7070802(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > >> Moreover, rational numbers loose > >> their property of being countable if they are embedded into the > >> continuum. > > > > Then according to EB, finite subsets of the reals are uncountable. > > Eb's misuse of mathematical terms is.unaccountable > > Well, I have to stress that I am referring to reals as they were > homogenously considered in DA2 (all fictitious in that they were assumed > to have actually infitely much of decimals). Actually no more than two decimals apiece, and for most of them only one. But in Dedekind cut form, decimals are irrelevant. > The presently mandatory reals are an inconsistent mix of irrationals and > rationals. It is perfectly consistent in every mathematical sense. > >> At least there is no possiblity to > >> decide inside the genuine continuum whether a fictitious "element" > > > > How can a "genuine" continuum be made up entirely of "fictitious" > > elements? > > What looks fictitious from one point of view looks genuine from the > opposite one and vice versa. Then let us by all means look at each real from a point of view that allows it to be genuine, and all our problems vanish. > > > > >> The primary continuum is strictly speaking amorph. > >> There is no structure available inside this continuum. > > > > Then it is not a mathematical object at all, > > Geometry and nonlinear functions are mathematics too. But not when amorphical. > > > as every mathematical continuum has a good deal of internal structure. > > You are confusing the original continuum according to Peirce with > Hausdorff's pseudo-continuum. As Pierce's does not seem to have any reason for existing or justification for existing, at least in mathematics, it has no part in mathematics. > > > Such non-mathematical > > notions are of no interest within mathematics. Let us not hear about > > them further! > > > > EB's anti-mathematical rants create more smoke than light. > > Anti-illusive is not anti-mathematical, on the contrary. If there is no model for a Pierce continuum in mathematics, then it is no part of or concern of mathematics.
From: MoeBlee on 8 Dec 2006 16:26 Han de Bruijn wrote: > It's quite simple. Set Theory can not be the foundation for mathematics, > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > it's not a set. What pure mathematical calculation cannot be represented as a proof in set theory? (And a proof, being a sequence of formulas, is, in Z set theory as a meta-theory, a set.) MoeBlee
From: Virgil on 8 Dec 2006 16:35 In article <4579ACBC.3090709(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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? Cantor? > 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. If there is an injection from set A to set B, one can quantitatively conclude that Card(A) <= Card(B). > > 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? That's how axiom systems work. > > Who proved that Cantor's interpretation concerning the power set was > correct? Cantor, among others. > > Who cared for the open secret that there is no valid definition of a set. No one. In most axiom system for set theories, sets are primitives, and so need no definition and are exempt from definition. > > Who proved the putative identity of the reals used in the 2nd diagonal > argument with mandatory definitions of reals? That is a bunch of nonsense unrelated to any actuality. > > Who provided a well-ordering of the reals? No one. In ZF the reals need not even be well-orderable. > > Who showed that one really needs alephs and not just the notions > countably infinite and uncountable? One never /needs/ a definition. Definitions merely abbreviate things. One can be prolix instead. > Who gave at least one positive example for the necessity to have the > misleading intermediate solutions in integral tables? I am not convinced that any 'misleading' was necessary.
From: Virgil on 8 Dec 2006 16:42 In article <4579AE2A.4080001(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:24 AM, David Marcus wrote: > > Eckard Blumschein wrote: > > >> Why do you not admit the possibility that countability of > >> a set requires countable numbers. Doesn't it make sense? > > > > Of course, it doesn't make sense. The reasons are: > > > > 1. You haven't defined/explained what a "countable number" is. > > Countability is self explaining. Except that different people seem to use it with different meaning, so it must "self-explain" differently to different people. How does it "self-explain" to you? > I argue: The current deviating use in > set theory is not necessary if set theory turns out to be a nice fancy. Until you can /prove/ all current versions of set theory to be "nice fancies", whatever they may be, the current use stands. > > > > > 2. There is already a standard definition of "countability of a set" and > > it seems unlikely that you can define "countable number" in such a way > > to make your statement true. > > I know it and consider it pretty narrow minded. Being narrow minded is a virtue in mathematics. The narrower the better.
From: Virgil on 8 Dec 2006 16:50
In article <4579B47C.8030204(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:18 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 12/5/2006 2:13 PM, Bob Kolker wrote: > >> > For the latest time. Uncountability is a property of sets, not > >> > individual numbers. > >> > >> I know this widespread view. > > > > So you claim. However, last time I asked you to give the standard > > definitions, you failed. Care to try again? Define "countable" and > > "uncountable". > > Do you believe someone who is urged to say the words of pater noster ... > will immediately become a believer? Why is someone who refuses to say what he means by "countable" and "uncountable" bringing his religion into this discussion. > > > >> > There is no such thing as an uncountable real > >> > number. > >> > >> Real numbers according to DA2 are uncountable altogether. If DA2 says "real number are uncountable all together", and why would that imply that they are uncountable individually? > > At the moment I am reading Fraekel et al. Foundations of Set Theory. 1958. > It does not reveal any solid fundamental, just endless attempts to > repair an illusion. The illusions may not be in the book but in yourself. > > > > >> > Countability > >> > /Uncountability are properties of -sets-, not individuals. > >> > >> Do not reiterate what I know but deny. > > > > How come you get to deny things, but we don't? Doesn't seem fair. > > Well, I am unbiasedly looking for strong arguments. Everyone thinks themselves unbiased, but everyone is biased. So EB is biased even though he denies it. > Dedekind and Cantor made a big mistake EB would be making a bigger one if he ever got anyone to take him seriously. |