An uncountable countable set
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From: mueckenh on 18 Sep 2006 11:22 David R Tribble schrieb: > > Yes, I can see now that these are all finite sets. > > And which are proper subsets of infinite sets. The set of all naturals > that have been written now, for example. Obviously it's an ever > growing set as time goes on, and will never contain the entire set > of naturals that are possible. So it's simply a finite subset of N, > and always will be. > > Somehow you are using this fact to "prove" that N can't exist, perhaps > employing some marvelous mathematical logic that has not been > tainted by mainstream teachings. You show several finite sets. > How do they prove anything about infinite sets? "a reasonable way to make this conform to a platonistic point of view is to look at the universe of all sets not as a fixed entity but as an entity capable of 'growing', i.e. we are able to 'produce' bigger and bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. Why should simple infinite sets exist in another way? Just because there is an axiom which cannot be satisfied like the axiom that there be a straight bent line? Regards, WM
From: mueckenh on 18 Sep 2006 13:33 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Mike Kelly schrieb: > > > > > > > The mathematics of > > > > the infinite can only be derived from the mathematics of the finite > > > > (because nobody has an idea what "the infinite" is). > > > > > > Don't extrapolate from yourself so harshly! > > > > I talked to a lot of first-rate mathematicians. This in parentheses is > > a qoute from many of them. > > Is that your idea of a citation? It is my idea of trust and honour. I do not publish names of my private correspondents. Regards, WM
From: mueckenh on 18 Sep 2006 13:36 stephen(a)nomail.com schrieb: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > stephen(a)nomail.com wrote: > > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >> > >>>mueckenh(a)rz.fh-augsburg.de wrote: > >> > >>>>Representation is number. There is no difference. Numerals have no > >>>>"soul". > >> > >>>Whew! I've never heard someone expressing this fact so lucidly! > >> > >> So 3/2 is a different number than 6/4? The representations > >> differ, and if there is no difference between representation > >> and number, then different representations must imply different > >> numbers. For that matter 6 birds is a different representation > >> of 6 than 6 cars. And 6 birds is a different representation > >> of 6 than 6 different birds? How many 6's are there? > > > Try to understand what equality means. > > > Han de Bruijn > > That does not seem to be an answer. Your position is > that "represenation is number". According to you there > is no difference between a number and its representation. > "3/2" is a representation. It is different than "6/4". > Or does your definition of "equality" somehow allow for > different strings to be equal? *The number* is each of its representations. The number 3 1) either is realized by a fundamental set like the following fundamental set of 3: {III, Dik, {a,b,c}, {father, mother, child}, {sun, moon, earth}, ....} 2) or is completely determined by a series of digits like 3.000000 or 3.00 or 3 or 3/1 or 6/2. (6/2 can also be interprtede as an exercise.) Each of them and they all (together with what you think if you think of 3) *is the number 3 (because all these representations are equal to each other and to 3, i.e., to themselves). Regards, WM
From: mueckenh on 18 Sep 2006 13:41 Mike Kelly schrieb: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > Mike Kelly wrote: > > > > > [ ... snip ... ] It's not clear to me that providing finite examples then > > > saying "obviously this holds for infinite cases too" without any > > > justification whatsoever should be at all convincing to anyone. > > > > It may be not clear to any mathematician, but it is clear to any > > scientist. The reason is that infinities do not really exist. > > They only exist as an attempt to make the "very large" rigorous > > in some sense. The moment you forget this, you get into trouble. > > But we are discussing whether there exists a uniform distribution over > the naturals. Please make just the experiment. Choose at random 30 natural numbers from the whole set N. What is the result? How many of these 30 numbers are in fact divisible by 3? (In case you have problems with large numbers: It is easy to check the divisibility of the number by checking the divisibility of the sum of its decimal digits.) Now, it there a distribution lacking, or is the complete set of natural numbers lacking? > If you don't think this claim means anything at all then > why do you dispute it? If you reject the existence of the set of > natural numbers then you reject the set theory probability is based on. In order to calculate probability we do not need set theory. Pascal and Fermat, for instance, did it without set theory very well. Your result shows only that set theory is not useful in any branch of useful mathematics. > So why bother to argue against individual theorems? You don't accept > *any* of probability theory. We have a better probability theory. > > It seem your argument is based on the idea that infinites do not exist > in physical reality. But mathematics is abstract, so this seems an > absurd objection. > > If you refuse the idea of infinite sets, what does it mean to you to > say a function has domain and range R? As an argument you can choose any real which you really can choose. See the experiment above. You don't really believe that you can choose a natural from the whole set N, do you? But if so, what then is N god for in probability theory (and elsewhere)? Regards, WM
From: MoeBlee on 18 Sep 2006 14:07
Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> David R Tribble wrote: > >>> Tony Orlow wrote: > >>>>> Yes, I am including infinite values on the number line, since it's > >>>>> "infinitely long". > >>> David R Tribble wrote: > >>>>> Yet another thing you have to define or prove. Where do these > >>>>> "infinite values" appear on your real number line? > >>> Tony Orlow wrote: > >>>> Further from 0 than any finite number. > >>> Then how are they "on" the same "line"? > >>> > >> By trichotomy. For all x ad y on the line, either x>y, x=y or x<y. > >> That's what makes a line in concept. > > > > That's question begging. Indeed, trichotomy is necessary for an > > ordering to be linear. But you have not proven the existence of such an > > ordering with the values you claim to be in its field. Just saying that > > your claim follows from trichotomy is just assuming what you are being > > asked to show, viz. that there is such a linear ordering with the > > values you claim to be in its field. But more basically, it doesn't > > matter, since you have no axiomatization nor rules of inference upon > > which to prove anything whatsoever in a mathematical system. > Well, MoeBlee, I have been trying to get that axiom together so that it > properly ties together count with measure, and trichotomy is one of the > axioms that defines the values along the line. If such a rule is > declared for all pairs of members of a set, then that can be considered > the definition of a linear set, I'm trying to get an axiom that construes identity with position, and transitivity is one of the axioms that defines iterations in a plane, and if this rule is posited for all singletons of subsets, then that can be a definition of a rectangular set. Now, the question is why think that anyone would care more about your gibberish than about the gibberish in the previous sentence? > be it what you would call a sequence, or > a dense linear set like a real interval. 'linear ordering' has a set theoretic definition that does not require sequence, density, nor interval. > I guess when I finally publish a set of axioms, then the subject will > suddenly "matter". Funny how that works in transfinitology. My axiom set > must be some sort of limit ordinal or something. Sure, Tony. MoeBlee |