Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 16 Oct 2006 09:13 David Marcus schrieb: > > > > Consider the binary tree which has (no finite paths but only) infinite > > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > > edges is countable, because we can enumerate them > > > > > > > > 0. > > > > /a \ > > > > 0 1 > > > > /b \c / \ > > > > 0 1 0 1 > > > > ..................... > > > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > > a is inherited by all paths which begin with 0.00, the other half of > > > > edge a is inherited by all paths which begin with 0.01. Continuing in > > > > this manner in infinity, we see that every single infinite path is > > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > > > other path. > > > > > > Are you using "relation" in its mathematical sense? > > > > Of course. But instead of whole elements, I consider fractions. That is > > new but neither undefined nor wrong. > > > > > > Please define your terms "half an edge" and "inherited". > > > > I can't believe that you are unable to understand what "half" or > > "inherited" means. > > I rather believe you don't want to understand it. Therefore an > > explanation will not help much. > > In standard terminology, a "relation between paths and edges" means a > set of ordered pairs where the first element of a pair is a path and the > second is an edge. Is this what you meant? Yes, but this notion is developed to include fractions of edges. > > I am at a loss as to how "half an edge" can be "inherited" by a path. An edge is related to a set of path. If the paths, belonging to this set, split in two different subsets, then the edge related to the complete set is divided and half of that edge is related to each of the two subsets. If it were important, which parts of the edges were related, then we could denote this by "edge a splits into a_1 and a_2". But because it is completely irrelevant which part of an edge is related to which subset, we need not denote the fractions of the edges. > As > far as I know, a path is a sequence of edges. Is this what you mean by > "path"? Yes. A path is a sequence of nodes (bits, 0 or 1). The nodes are connected by edges. > > Is edge a the line connecting 0 in the first row to 0 in the second row? Correct. This edge is related to all paths starting with 0.0. So half of edge a is related to all paths starting with 0.00, and half of it is related to all paths starting with 0.01 > Or, is it the line connecting 0 in the first row to 1 in the second row? > Or, is it something else? I have tried above to make it a bit clearer. Regards, WM
From: mueckenh on 16 Oct 2006 09:23 Virgil schrieb: > > According to the ZFC system: The vase is empty at noon, because all > > natural numbers left it before noon. > > By means of the ZFC system we can formulate sequences and their limits > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > And from this it follows that the vase is not empty at noon. > > By what axiom do you conclude that the limit as t increases towards noon > of any function and the value of that function at noon must be the same? By that or those axiom(s) which lead(s) to the result lim {t-->oo} 1/t = 0. Regards, WM
From: stephen on 16 Oct 2006 09:37 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Dik T. Winter wrote: >> In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> >> Han.deBruijn(a)DTO.TUDelft.NL writes: >> > Virgil schreef: >> ... >> > > I do not object to the constraints of the mathematics of physics when >> > > doing physics, but why should I be so constrained when not doing physics? >> > >> > Because (empirical) physics is an absolute guarantee for consistency? >> >> Can you prove that? > Is it possible to live in a (physical) world that is inconsistent? > Han de Bruijn Perhaps. How could we know? Stephen
From: mueckenh on 16 Oct 2006 09:56 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > > > > > The problem is that you believe in an actually infinite series of > > > > finite numbers. This belief leads to intermingling digit position and > > > > number of digits just according to the needs and, after all, it leads > > > > to such self-contradictive statements as we have seen here with "the > > > > vase". > > > > > > There is nothing self contradictory in "the vase" but there are > > > contradictions between the axiom systems on which "the vase" is based > > > and the assumptions of those who oppose those axiom systems. > > > > > > According to the ZFC system: The vase is empty at noon, because all > > natural numbers left it before noon. > > By means of the ZFC system we can formulate sequences and their limits > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > And from this it follows that the vase is not empty at noon. > > > > You will and must disagree, but the spectators will at least get to > > know this inconsistency. > > ZFC does not talk about "vases". If you say that ZFC is inconsistent, > please give a statement using the language of ZFC. All mathematics is built on ZFC and derived from it. From ZFC we can derive the result of the vase-balls problem too. Hence we can criticize ZFC if its results are contradictory. Mathematics based on ZFC says that the vase at noon is empty or not empty (by the way, what was your result V(12)?). But whatever your result may be: Both statemets are in contradiction with the foundations of ZFC. If the vase is empty then mathematics of limits as derived from ZFC is wrong. If the vase is not empty then the assumption is wrong all natural numbers could be put into a bijection with each other. Regards, WM
From: Randy Poe on 16 Oct 2006 10:03
Tony Orlow wrote: > Randy Poe wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> Randy Poe schrieb: > >> > >>> Han de Bruijn wrote: > >>>>> I merely note that there is no requirement in the problem that > >>>>> the limit be the value at noon. > >>>> The limit at noon - iff it existed - would be the value at noon. > >>> Wrong. That is a flat out incorrect statement showing a > >>> fundamental misunderstanding about what limits mean. > >>> > >>> A CONTINUOUS function at x0 has the property that the > >>> limit of f(x) as x->x0 is f(x0). But not all functions are > >>> continuous. > >> And you are in charge of determining which functions are continuous and > >> which are not? > > > > Where do you get this stuff from? > > > > How do you translate a statement that some functions are not > > continuous into "I am in charge of determining if some functions > > are continuous"? > > >> The function f(t) = 9t is continuous, because the function 1/9t is > >> continuous. > > > > Yes, but that is not the number of balls in the vase. > > After iteration n, there's 9n balls in the vase. To hell with t, anyway. > The Zeno machine is a paradox machine. I'd rather work with t on the real number line, since that makes it easier to talk about continuous functions. There is a function f(t) which describes the number of balls in the vase at time t. On the convention that noon corresponds to t=0, this function increases in steps over t<0, but has a discontinuity at t=0. f(t) = 0 for all t>=0. > > f(t) = sin(t) is continuous also. So what? What does that have > > to do with the vase? > Uh, nothing. In other words, as much as mueck's function. We can name lots of continuous functions. None of them has anything to do with the f(t) defined above, which is not a continuous function. - Randy |