Cantor Confusion
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From: mueckenh on 15 Nov 2006 06:46 Franziska Neugebauer schrieb: > set formation by '{' and '}' does effectively not take place in WM's > world. > It does effectively not take place in *the* world, never did so and never will do so. "{" and "}" put some entities together. If you were not suffering from Cantor-hostility, then you could understand his wise words, "daß die Menge P [, die] keinen einzigen Punkt enthält, also streng genommen als solche gar nicht vorhanden ist." And shouldn't we try to reconquer the realm of stringent mathematics? Regards, WM
From: mueckenh on 15 Nov 2006 06:57 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > The suprema are not the same. After havin added one term to each line, > > > > each line has the ordinal number n+1 < omega. And the width of the > > > > matrix has the ordinal number omega. > > > > > > So? Adding one element may or may not change the supremum. > > > > If the supremum is a maximum, it will be changed. If not, it will not > > be changed. > > > > > > The supremum of the lengths of the initial segments of > > > the columns is omega. The supremum of lengths of the lines is > > > omega. The supremums are the same. > > > > > > However, we can make the supremums different by adding elements. > > > > This shows that they were not quite the same before. > > > > > > If we add one element to each of the columns then > > > the supremum of the lengths of the initial segments of the columns > > > changes to omega +1. If we add one elelment to each of the lines > > > the supremum of the lengths of the lines does not change, it > > > remains omega. > > > > Correct. And if we add one element to the diagonal, then its supremum > > changes form omega to omega + 1 > > Correct, but to add one element to the diagonal we > have to add one element to the lines and one > element to the columns. Not necessarily. We add just one element to the diagonal which, according to your assertion is existing ad has ordinal omega. Now it has ordinal omega +1. > If we just add one element > to the lines we no longer have a diagonal. Therefore I recommended to add one element to every line, to every column, and to the diagonal. > . > There is an element of every column with transfinite index. > There is no element of a line with transfinite index. > There is no diagonal. That means, there is no bijection between lines and columns. Therefore, there was no such bijection before. Therefore an infinite set of natural numbers cannot exist without an infinite number. As the infinite number is not a natural number, an infinite set of natural numbers cannot exist at all. Regards, WM
From: mueckenh on 15 Nov 2006 07:05 Lester Zick schrieb: > >> I don't. It's a problematic argument at best. Based once again on a > >> hypothetical finitude of the "physical" universe whatever that means. > > > >Here I cannot understand you. The accessible universe is finite, > > This is not true or at best problematic. It is enforced by the finity of the maximum signal velocity. > >allowing for not more than 10^100 bits (a closer estimation would be > >10^205, but that is irrelevant). Now, to express a number requires at > >least one bit. What more is needed to see? > > The issue seems to be whether reduction to a finite number of bits or > whatever is determinate of the number. Let's assume we have only room > for 2 bits or 4 bits or whatever. Is that determinate of the numbering > capacity of our thoughts and numbers represented in our thoughts? I > don't think so. If there were only 4 bits, then our brain would be included. It would have less capacity than that of an earth worm. Our brain has about 10^11 neurons. This huge number lets us overlook that it is limited. > I see infinitesimal subdivision able to express itself > to any degree necessary for the computation of relationships between > infinitesimal ratios. In other words instead of extending out numbers > infinitely all we're doing is subdividing unity. And this requires no > further finite space than unity regardless of precision or extension. If you get to the subdivision number [pi*10^10^100], you cannot determine his number, neither by any brain nor by all ressources of he universe. Regards, WM
From: mueckenh on 15 Nov 2006 07:12 Virgil schrieb: > >(That's why you think a set is more than the collection of > > its elements.) > > WM may think that, but that is exactly what I do NOT think about a set. > > If a paper sack with one apple in it can exist, why cannot an empty > paper sack exist. Because you do not think that a set is more than its elements, I think. > > The value of an envelope is its contents, and I can, and have, imagined > the set of all finite naturals. How did you imagine [pi*10^10^100] and the many other numbers where the exponent 100 is replaced by 100*n with n e N? I think you have seen only the paper sack and your "NOT" above is false. Regards, WM
From: Franziska Neugebauer on 15 Nov 2006 07:13
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> set formation by '{' and '}' does effectively not take place in WM's >> world. >> > It does effectively not take place in *the* world, never did so and > never will do so. "World" denotes here solely the mathematical universe of discourse. > "{" and "}" put some entities together. Forming new entities which not necessarily inherit properties from their constituents. What's so hard to grasp about that? > If you were not suffering from Cantor-[...] I prefer to discuss your arguments not Cantors. If you have any. So let us refocus our attention on your statements ,----[ <1163507203.468528.168460(a)h48g2000cwc.googlegroups.com> ] | | {1,2,3} is the collection of, and a convenient expression to write | that we are talking about, the numbers 1 ,2, and 3. | `---- ,----[ <1163510216.772155.147810(a)k70g2000cwa.googlegroups.com> ] | > What are we talking about, when we write | > | > { } | > | > ? | | The same as when we write | | | | | | | | `---- F. N. -- xyz |