Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 12 Dec 2006 09:24 On 12/6/2006 8:18 PM, David Marcus wrote: > Eckard Blumschein wrote: >> He assumes that his list of all reals is complete and shows that this is >> not the case. From this contradiction he was forced to conclude that the >> reals are uncountable >> but he intentionally misinterpreted the outcome by >> claiming there are more real than rational numbers. > > Please define the word "more" in this context. What exactly are you > saying is not correct (or misinterpreted)? The word more is nearly synonymous to the > relation. There is no number x of all real "numbers". There is also no number y of all rational numbers. There is not even a number z of all natural numbers. So it may be intuitively obvious but it is not justified to guess x>y>z.
From: Eckard Blumschein on 12 Dec 2006 09:57 On 12/6/2006 8:29 PM, Virgil wrote: > In article <4576E233.7030802(a)et.uni-magdeburg.de>, >> >> You all know that DA2 shows by contradiction that real numbers are >> >> uncountable. >> > >> > WRONG! It is not, at least in Cantor's version, a proof by contradiction. >> >> He assumes that his list of all reals is complete and shows that this is >> not the case. > > > Not so. He assumes one has an arbitrary list, with no other > qualifications than that it is a list of reals, and then shows it is not > complete. You wrote not so. Why? Did you read his original text? > There is no necessity to assume it complete and convert a > straightforward direct proof into a proof by contradiction. Do you refer to literature or is this just your own comment? > In is only those who misread the proof that themselves supply that > additional assumption that Cantor himself does not require. I do not intend to question historical sources. >> From this contradiction he was forced to conclude that the >> reals are uncountable but he intentionally misinterpreted the outcome by >> claiming there are more real than rational numbers. > > In the sense in which Cantor defined "more", there are more. I am not aware of Cantor's definition of "more". He was so daring to allegedly count the uncountable by just ascribing Maechtigkeit (later renamed into cardinality) to it. Ending up at the quality infinity (oo), he misunderstood oo like a quantum. Then he defined countable a bit different from ordinary use like the possiblity to order single elements by bijection to the naturals. If EB > wishes to use his own meanings for words in place of the clearly defined > mathematical ones, he will, as here, repeatedly come a cropper. You are not in position to grade me. >> > was not in any way the same and IS valid for >> > Dedekind cuts, any flaws in his second are irrelevant. >> >> I do not understand what you mean in this contex regarding Dedekind >> cuts. I do not refer to any flaw in DA2 itself, just the interpretation >> is wrong. > > It is the interpreter making that interpretation who is in error because > he insists on using his own non-mathematical meanings for in place of > mathematically standard definitions. Standard definitions may be inappropriate. In contrast to the bizarre theory of more than infinitely many numbers, my interpretation is logical and makes not just naive set theory, CH, AC and many other worries unnecessary. It has only advantages.
From: Eckard Blumschein on 12 Dec 2006 09:58 EoD
From: Eckard Blumschein on 12 Dec 2006 10:00 On 12/11/2006 10:35 PM, Virgil wrote: > Given any rational, it is easy to construct the set of all larger > rationals, Really? Who gives me all rationals?
From: Eckard Blumschein on 12 Dec 2006 10:25
On 12/11/2006 9:56 PM, Virgil wrote: > Cantor was able to show >> himself that all natural and rational numbers do not have a different >> "size". Infinity is not a number. > > No mathmatician says "infinity" in that context. They will call some > sets "infinite", but that is not the same thing. Yes. Infinity is a fiction. It cannot be achieved by conting. Infinite is almost the opposite. >> It has been understood like something >> which cannot be enlarged and not exhausted either: > > Infinite sets can be enlarged in the sense that for each > of them there are proper supersets, at least in ZFC and NBG. So infinite sets are potentially infinite. Quite reasonble. > They can be exhausted by taking away all their members. One more indication for my insight that ZFC does not really cover pi. Unfortunately, I will be hindered for a while to continue our discussion. Who could grasp my insights and suggestions? Robert Kolker is out of anything. TO does not even understand Cantor. Virgil will not be able to surrender. Lester Zick? ??? Maybe someone else. EoD. I regret it. |