An uncountable countable set
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From: mueckenh on 13 Sep 2006 03:51 Dik T. Winter schrieb: > In article <1157743126.475610.189460(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > A law is derived from the natural properties of arithmetics. > > > > > > > > > > Oh. What law derived from the natural properties of arithmetics is he > > > > > talking about when he gets the completely determined set of all integral > > > > > finite numbers? > > > > > > > > That is but his conviction. > > > > > > I ask you what law, but you are not willing to answer? Again, what law is > > > he using (note that in English law generally refers to Theorem). > > > > Cantor held the opinion that there are some natural "laws" or "rules" , > > not theorems, but Grundwahrheiten, so say truths, which are valid for > > the natural numbers and which cannot be changed without leading to > > rubbish. > > Yes, in English such things are called axioms. An axiom can be chosen arbitrarily, either the axiom or its negation. Cantor's truths cannot be chosen but exist prior to any mathematics. > Although I disagree that > changing them leads to rubbish. because you have not yet understood what Cantor's truths are. > Changing them leads to at most a > different kind of mathematics. > Therefore, the truths are *not* axioms. Changing these truths leads to rubbish. > > Something like axioms but not arbitrarily stated but derived > > from "nature" or "reality". > > Ah, like the parallel postulate of Euclides. > > > One of them is 2 + 2 = 4 (in decimal > > notation). Another one, according to his opinion, is the existence of > > infinitely many finite numbers. > > Well, if you want to use only Cantor's view you should not attack current > set theory. I have only tried to explain the difference between Cantor's view and what you think what was Cantor's view. > In current set theory that is an axiom, and so I think my > translation with the word "axiom" is quite correct. Then you should try to improve your thinking. You wanted to translate Cantor and you have completely neglected his words and his view. A worse translation is impossible. Regards, WM
From: mueckenh on 13 Sep 2006 03:55 Virgil schrieb: > > > If it does not hold you should be able to give an index position > > > such that it is false. > > > > It is not in the list, although we cannot give an index position which > > is responsible for that fact. All possible indexes are in the list. > > But that list is capable of indexing as many more objects as it already > has. And it has many more objects as it already has. > > Because it is not in the list which, by definition, contains all > > numbers which can be indexed and which can index. > > Not so. Even if one says it contains all the numbers which are indices > that list still can index as many again as already indexed. Yes, because they are all in the list. But 0.111... is not there. > > > > > > > Now we may ask: Is it possible that a list item indexes or covers other > > > > digits than those which are indexed or covered by list items? The > > > > answer is: no. > > If one insists that a number may only be used to index itself, perhaps, > but that is not necessary, and when no necessary the answer becomes yes. The axiom of infinty applies to my list. All natural numbers which do exist (and can be indexed) are there. 0.111... is not there. > > > > > Because you assert that all digits of 0.111... can be indexed, which is > > wrong. Indexing and covering "by all list numbers" is equivalent. Both > > is true or both is false. > > > It is false that indexing and 'covering "by all list numbers" ' are > equivalent. Indexing merely means tagging each object with some index > value, and may be done in many ways for any but the smallest of sets. If you deny that indexing and 'covering "by all list numbers" ' are equivalent, then give me a number of the list (no other naturals do exist) which can be indexed by all list numbers but which cannot be covered by all list numbers (or the reverse). > > > It has been proven by showing that 0.111... does not belong to the set > > of numbes which can be indexed completely. > > Except that an equally valid proof has shown the opposite. No proof has shown this. Regards, WM
From: mueckenh on 13 Sep 2006 03:57 Virgil schrieb: > In article <1157836470.071804.254270(a)i3g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > A set of n elements is *a number*. It need no necessarily be a latin or > > arabic symbol. > > In that case "Mueckenh" has problems because this requires that some > numbers, as ordinals, are members of themselves which ordinals cannot be. The set III is the number 3. If that is impossible according to your mathematics, then your mathematics is deplorable. Regards, WM
From: mueckenh on 13 Sep 2006 03:59 David R Tribble schrieb: > mueckenh wrote: > > All your mental concepts are nothing but electric loads and currents in > > your brain. > > That's true, but the mental thoughts they embody are not. > > It's like saying the pieces on a chessboard are constrained by physical > laws, but the abstract game they represent have no such constraints. It's like saying the ship is carried by the water, but the persons on board do not require the water to be carried. > Those same pieces can, in fact, represent multiple abstract concepts How do they manage that? Who decides? What is a decision? > while still being embodied by the same physical objects. Same thing > goes for poker cards, which can be used to represent player hands > in many different games (which might be a better analogy). > > Thus it is with atoms and photons, constrained as they are by physical > laws but used to embody many different objects and even mental ideas > inside brain cells. The atoms, yes. And they are physics. Therefore it would be short sighted to believe that mental ideas were not physics. Regards, WM
From: mueckenh on 13 Sep 2006 04:01
David R Tribble schrieb: > David R Tribble schrieb: > >> I really don't see where the physical limitation is for visualizing the > >> elements or the set. More to the point, I don't see how the phyics > >> of the real world have any limiting effect on abstract concepts. > > > > mueckenh wrote: > > What you are arguing is only the beginning of infinity, because you are > > unable to see more than this little realm. Try to determine the natural > > number which consists of the 10^100 digits following the first 10^1000 > > digits of pi. Your unability is not due to lack of time. This is only > > one of the infinitely many numbers which you cannot deal with (because > > it is not a number). > > Why is it not a number? It has all the mathematical properties > required of a number. Having "all its digits enumerated" is not a > required property. It is the only important property of a number. > > Supposing that I can in fact determine all 10^100 digits of this > number. What then? Then it would be a number, but you cannot determine these digits. > Does that number now magically possess > an existence that countless others do not, i.e., has it suddenly > become a "more real" number than, say, the next 10^100 digits > of pi? Not necessarily suddenly. It has become existence for you. You know it. > > Likewise, until I actually enumerate the digits of some specific > number, are you saying it does not (yet) exist? Would you say that a poem I am going to make tomorrow is already existing today? > If that's the > case, does pi or e exist? pi and e do exist as ideas or as problems or as irrational proportios but they do not exist as numbers. And they will never exist as numbers, in contrast to numbers with 10^20 digits. Regards, WM |