An uncountable countable set
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From: imaginatorium on 15 Sep 2006 00:22 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <1158102358.446630.158500(a)i3g2000cwc.googlegroups.com> "David R Tribble" <david(a)tribble.com> writes: > > ... > > > [http://en.wikipedia.org/wiki/Number] > > > A number is an abstract entity that represents a count or > > > measurement. In mathematics, the definition of a number > > > has extended to include abstractions such as fractions, > > > negative, irrational, transcendental, and complex numbers. > > > > > > I'd also add algebraic numbers, ordinals, cardinals, quaternions, > > > octonions, matrices, tensors, p-adics, hyperreals, and > > > surreals to that list. > > > > Somthing like that. In my opinion, when you have a set of objects, two > > operations (+ and *) of which one is distributive over the other, > > you can talk about numbers. This seems an odd definition, since it means that anything that forms a ring [Tony: I'm not talking to you here, since you don't understand what "ring" means] would count as numbers: polynomials, matrices, and goodness knows what. > > ... But perhaps some more properties are > > required? I have no idea. But that is only my opinion. In mathematics, > > unless the context is clear, the word "numbers" is always qualified. > > There is *no* mathematical definition of number. > > We can also include exponentiation and tetration, in which case we close > or complete the circle of operations, in a sense. What is the "circle of operations"? In what sense? Or is this the Finlayson "in a sense", as in: <<... paragraph of incoherent babble ... >> in a sense. ... which somehow makes it all ok? Brian Chandler http://imaginatorium.org
From: mueckenh on 15 Sep 2006 05:02 David R Tribble schrieb: > Tony Orlow wrote: > > Wolfgang's and my position is that N is unbounded but finite, > > "Unbounded but finite" is a contradiction, meaning "not finite but > finite". I'm sure you and Wolfgang think this double-think makes > sense, but the rest of us don't. Your position only reflects the miseducation in mathematics during the last decades. Actual or finished infinity is a contradicton. Surpassed infinity is a contradiction. Unbounded but finite is mathematical reality. Think of the set of all natural numbers which have been realized by writing down these numbers. Think of the set of known prime numbers. Think of the set of written novels. Think of the set of postings. These sets are unbounded because they can be extended without end. Nevertheless they are always finite. Regards, WM
From: mueckenh on 15 Sep 2006 05:05 Virgil schrieb: > > III is a representation of 3. > > "III" and "3" are both numerals representing the same number, but > neither is anything more that a representation, neither is the number > itself. What is the number denoted by 3? Some cloudy idea in the platonic universe or heaven? The number 3 is present in this line. It is "3" and it is "III" and it is "{a,b,c}". Everything else is purest matheologial rubbish. Regards, WM
From: mueckenh on 15 Sep 2006 05:07 Virgil schrieb: > > III is a representation of 3. > > "III" and "3" are both numerals representing the same number, but > neither is anything more than a representation, neither is the number > itself. What is the number denoted by 3? Some cloudy idea in the platonic universe or heaven? The number 3 is present in this very line. It is "3" and it is "III" and it is "{a,b,c}". Everything else is purest matheological rubbish. Regards, WM
From: mueckenh on 15 Sep 2006 05:09
Virgil schrieb: > Counting can be done by making tally marks or moving pebbles, for > example, entirely without numbers, though we have become so > sophisticated that we may have trouble realizing it. > > It is whether the tally marks or collected pebbles biject with the > objects counted which is the issue. > > And no number need ever be mentioned or used. Your tally marks and moving pebbles are numbers. Get more sophisticated in order to see it. Regards, WM |