Cantor Confusion
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From: Virgil on 15 Nov 2006 16:08 In article <1163590497.338210.196620(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No all the natural cardinal numbers have already been used up for > > sets of the form {1,2,3,...,n}. > > Correct. > > > There are no natural numbers > > left for the set {1,2,3,...}. > > Then, what does this set consist of, if all natural numbers have been > used up by finite sets? Its members are naturals, but there is no "last" natural left to be its name. > > > So it is not a question of maybe > > not knowing what the natural cardinal number is. We know that > > there is no natural cardinal number. > > Why then do you assert that this set exists? ZF asserts it. > > > Let N be the set of natural numbers. Then induction is valid > > for all elements of the set N. "the set N" is not an element of N. > > Induction > > is not valid for "the set N". > > I do not want to prove anything for this chimera. All I say it that > induction is valid for all natural numbers. Call this a collection or a > sequence if the notation "set" leads you to astray. The "all natural > numbers" (whatever you may call it) has not an actually infinite > number of elements. It has /more/ than any finite number of them. If WM does not wish to allow this to be called infinite, let us merely call it not-finite. > > Sets of the form {1,2,3,...,n} are not elements of N either. > > However, we can associate > > each set with its cardinal, a unique element of N, and then use > > induction > > on the elements of N to show something about the sets of this form. > > However, > > the set {1,2,3,...} does not have a cardinal which is an element > > of N. So we cannot use induction to tell us anything about this set. > > You say: However we cannot use induction to tell us anything about this > set so we cannot use induction to tell us anything about this set. Let > the set be what it may. Talk about all the natural numbers. In ZF or NBG, the set of all "natural numbers" exists. What is the complete axiom system WM proposes in which it does not?
From: Virgil on 15 Nov 2006 16:12 In article <1163591197.967744.192690(a)h48g2000cwc.googlegroups.com>, 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. "{" and "}" put some entities together. If you were > not suffering from Cantor-hostility, then you could understand his wise > words, "da? die Menge P [, die] keinen einzigen Punkt enth?lt, also > streng genommen als solche gar nicht vorhanden ist." And shouldn't we > try to reconquer the realm of stringent mathematics? > > Regards, WM WM cannot have it both ways. Either he must accept everything Cantor says if he is to use Cantor as authority, or nothing Cantor says, unless otherwise supported, if he does not use CAntor as authority.
From: Virgil on 15 Nov 2006 16:16 In article <1163591830.120817.169230(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But it is no longer a diagonal, so is irrelevant. > > > 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. The problem is where is such "addition" to take place? If one inserts the new bits in finite places, no problem, but if one wants to put them at the "ends" of endless sequences, one no longer has lists.
From: Virgil on 15 Nov 2006 16:22 In article <1163592771.959998.60900(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. I think, because the axioms tell me so, that the identity of a set is determined by what objects are, or are not, members of it. If (for all x in y x is also in z) and (for all x in z x is also in y) then y = z. This works quite as properly for empty sets as for non-empty ones. > > > > > 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? As one of many others.
From: Virgil on 15 Nov 2006 16:27
In article <1163593387.013209.260550(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > We discuss just this triangle as our marix. > > > > Prima facie "Adding x to each column" ("at the end") is not "possible", > > since the columns have no end (no last element). > > If 1,2,3,... has the ordinal number omega and if it is possible to > construct 2,3,4,...,1 and if it is meaningful to denote the ordinal > number of 2,3,4,...,1 by omega + 1, then your argument fails. But just > these antecedents are assumed. Wrong. If you have an endless (omega type) list, and you append something to it , you no longer have an endless list. So appending something to the end of a omega type list does not give an omega type list any more. |