An uncountable countable set
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From: mueckenh on 9 Sep 2006 17:14 Dik T. Winter schrieb: > In article <virgil-411702.16371406092006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > In article <1157574529.452614.131100(a)d34g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > ... > > > > Counting precedes adding, in that one may count when unable to add. > > > > > > Counting is addition of 1 to n. If one is unable to add 1, one cannot > > > count. > > > > Counting can take place without any numbers at all, much less any > > arithmetic. A set of n elements is *a number*. It need no necessarily be a latin or arabic symbol. > > I count with my fingers. Does that count? The positions of your fingers are numbers. And you know what to do in order to get from 32 to 33. > And I can go to 35 with my > fingers, and I use my feet (not toes) to extend to 143. I never really > managed to do it base 2, but 4 and 128 are interesting numbers in that > case. Every position of your fingers and feet is an expression of a number. Counting without numbers would be nonsense. A boy like Virgil may count 1, 2, 3, 3, 3, ... Regards, WM
From: Virgil on 9 Sep 2006 19:24 In article <1157836319.591764.139810(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1157574332.235885.113030(a)d34g2000cwd.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > Talking about 0.111... defined as: for all natural p digit p is 1, there > > are no other digtits. And the list is the list of natural numbers. > > > > > > > Your "each" means in symbols of logic: "A = (for) all". > > > > > The number is nothing than all of its digit positions. > > > > > Therefore your statement is a self contradiction. > > > > > > > > Where? In logical terms (A meaning "for all" and E meaning "there > > > > is"): > > > > (1) A{p = digit position} E{q = list item} {such that q indexes p} > > > > > > That is your definition. But what we can safely say is only: > > > (1') A{p = digit position of list item} E{q = list item} {such that > > > q indexes p} > > > > Why can we only say that? The definition of 0.111... is such that (1) > > holds. > > Then it would be in the list. > > > 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. > > > > > > (2) A{p = digit position} E{q = list item} {such that q covers p} > > > > > > If your definition could be satisfied, the construction of the list > > > would imply this, yes. What we can safely say, however, is only: > > > (2') A{p = digit position of list item} E{q = list item} {such that > > > q covers p} > > > > The same here. But apparently you think my definition of 0.111... can not > > be satisfied. Why not? > > 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. > > > > > 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. > > 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. > 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.
From: Virgil on 9 Sep 2006 19:37 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. > > > > I count with my fingers. Does that count? > > The positions of your fingers are numbers. And you know what to do in > order to get from 32 to 33. > > > And I can go to 35 with my > > fingers, and I use my feet (not toes) to extend to 143. I never really > > managed to do it base 2, but 4 and 128 are interesting numbers in that > > case. > > Every position of your fingers and feet is an expression of a number. > Counting without numbers would be nonsense. A boy like Virgil may count > 1, 2, 3, 3, 3, ... Listen, child, if you ever learn what numbers really are you might begin to count. Until then you don't count.
From: David R Tribble on 9 Sep 2006 21:17 mueckenh wrote: >> All we can attach to it is the number of elements known >> or existing. Disregarding physical constraints ... > David R Tribble schrieb: >> I was not aware that abstract mathematical concepts (e.g., sets) >> had any physical constraints. > mueckenh wrote: >> Then you should learn it. It you are unable to physically (i.e. in >> written form or in your mind) distinguish all the elements of a set, >> then the set does not exist. > David R Tribble schrieb: >> Nope, that still does not explain why abstract mental concepts >> are limited by physical constraints. I don't believe it. > 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. Those same pieces can, in fact, represent multiple abstract concepts 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.
From: David R Tribble on 9 Sep 2006 21:26
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. Supposing that I can in fact determine all 10^100 digits of this number. What then? 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? Likewise, until I actually enumerate the digits of some specific number, are you saying it does not (yet) exist? If that's the case, does pi or e exist? |