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From: albstorz on 28 Oct 2005 07:27 Robert J. Kolker wrote: > albstorz(a)gmx.de wrote: > > > > > > > I'm not shure if the reals build a set in spite of you and Cantor and > > others are shure. > > A set is defined by consisting of discrete, distinguishable, individual > > elements. Now tell me: what separates a point on a line from the very > > next point on the line to be discrete? > > There is no very next point under the ordinary ordering of reals. But > given a pair of reals they are either equal or not. Between any two > distinct real numbers there is always a third real different from the > two given (with respect to the standard ordering of the reals). > > Bob Kolker Do you have a test on objects to proof if they are sets or not? Or is a set just that what you want to have to be a set? What's your definition of sets? Is the water in a can a set of water? Regards AS
From: William Hughes on 28 Oct 2005 10:26 albstorz(a)gmx.de wrote: > You misinterpret totally when you say, I think there must be an > infinite natural number. I don't think so. I only argue that, if there > are infinite sets, there must be infinite natural numbers (since nat. > numbers are sets). > OK. Make the substitutions, natural numbers = Greeks, sets = mortals and infintite = German [1] Then the statment "if there are infinite sets, there must be infinite natural numbers (since nat. numbers are sets)" becomes "if there are German mortals, there must be German Greeks (since Greeks are mortals)". Aristotle must be rolling in his grave. -William Hughes [1] Deutchland, Deutchland, ueber alles
From: David R Tribble on 28 Oct 2005 20:01 David R Tribble said: >> [Alrecht] are claiming that one of the members of the infinite set is >> equal to the size of the set. This is not obvious to us because it >> contradicts proven set theory. It is your responsibility to prove >> that your claim is true, and that standard set theory is wrong. >> Saying that your claim is "obvious" is not a proof. > Tony Orlow wrote: >> His claim is obvious based on the diagram he offered, which graphically >> shows the element values along one side of a growing square, and the >> element count along the other side. They are obviously, graphically, >> inductively, always equal. One is not infinite while the other is finite. >> It's impossible. > > Here's his second diagram: > >> # O O O O O O O O O ... 1 >> # # O O O O O O O O ... 2 >> # # # O O O O O O O ... 3 >> # # # # O O O O O O ... 4 >> # # # # # O O O O O ... 5 >> # # # # # # O O O O ... 6 >> # # # # # # # O O O ... 7 >> # # # # # # # # O O ... 8 >> # # # # # # # # # O ... 9 >> . . . . . . . . . . . > > Every natural is represented by a horizontal string of #'s. The vertical > string of zeroes directly to the right of the last # in each natural denotes > the size of the set through that natural. With the addition of each natural, > both the horizontal number of #'s and the verticle number of 0's are > incremented in tandem, so this equality is preserved. The slope of -1 shows > that there is a 1-1 correspondence between element value and set size. > This is a good graphical illustration of what I have been trying to say > regarding the naturals. You only have an infinite number of them when you > allow infinite values for them. If that's the case, at what point do the naturals on the right side stop being finite and start being infinite? All I see is an infinite list of rows, where each row is labeled by a finite natural k and contains k leading # marks and an infinite number of trailing O marks. >From the other angle, I see an infinite list of columns, where each column contains a finite number of O marks on top and an infinite number of # marks on the bottom. But why can't there be an infinite number of finite naturals? Do you have some kind of proof of this? You have never offered one. You've offered plenty of statements about "unidentifiable largest naturals" and "set ranges" and "unit infinities", but you've never actually pinpointed how you get from the finites to the infinites.
From: David R Tribble on 28 Oct 2005 20:14 David R Tribble said: >> So your infinite natural that maps to the entire set *N is: >> s = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + ... >> s = 1 + 2 + 4 + 8 + 16 + ... >> >> Thus s is a sum of an infinite number of finite powers of two (2^p), >> each term being twice the previous term. So s is your 2^N-1, where >> N is your "unit infinity": >> N = 1 + 1 + 1 + 1 + 1 + ... > Tony Orlow wrote: > The dawn dawns on David Tribble.... > David R Tribble said: >> (We'll pretend for the moment that s and N are different, and ignore >> the fact that they are actually the same value.) > Tony Orlow wrote: > ....and the clouds enshroud the sun. > > Do you really think sum(n=0->N:1) is the same value as sum(n=0->N:2^n), when > every term of the second (except the first) is greater its corresponding term > in the first? Oy..... I know you can't comprehend this little tidbit dealing with infinite sums, but I'm glad you asked. Let: n = 1 + 1 + 1 + 1 + ... and: s = 1 + 2 + 4 + 8 + ... Now let's group the partial sums of n: n = 1 + (1 + 1) + (1 + 1 + 1 + 1) + (...) + ... n = 1 + 2 + 4 + 8 + ... Obviously, we can group each set of 2^p 1's in our infinite sum so they are identical to the terms in the series for s. So n = s. Looking at it from a TOmatics standpoint, s is an infinite binary number containing all 1 digits (one digit for each 2^p term). Likewise, n is the sum of an infinite number of 1's, so it must also be an infinite number composed of all 1 digits (what else could it be?). So n = s. Isn't this fun?
From: albstorz on 30 Oct 2005 16:34
William Hughes wrote: > albstorz(a)gmx.de wrote: > > > You misinterpret totally when you say, I think there must be an > > infinite natural number. I don't think so. I only argue that, if there > > are infinite sets, there must be infinite natural numbers (since nat. > > numbers are sets). > > > > OK. Make the substitutions, natural numbers = Greeks, sets = mortals > and > infintite = German [1] > > Then the statment "if there are infinite sets, there must be infinite > natural numbers (since nat. numbers are sets)" becomes "if there are > German mortals, there must be German Greeks (since Greeks are > mortals)". > Aristotle must be rolling in his grave. About the sillyness a man could shows like you do? Don't forget Einstein: "The universe and the stupidity of men are infinite. About the universe I'm not really shure." > > -William Hughes > > [1] Deutchland, Deutchland, ueber alles [2] William Hughes before the fuehrer Regards AS |