Cantor Confusion
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From: mueckenh on 14 Nov 2006 15:30 Dik T. Winter schrieb: > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > > So lets count the set of all natural numbers {1,2,3,...} > > There are no natural number left. So we stop > > using natural numbers and use ordinals > > (and to nobody's surprise a few things change). > > This is wrong. There is no ordinal needed to count the elements of the > set of all natural numbers. You can count until you weigh an ounce (;-)) > but you will never finish. In this way you can also count the reals without ever finishing. Therefore this cannot be the meaning of Cantor's "countable". > Neither the elements you wish to count will > be exhausted nor the numbers with which you count. I think this is > potential infinity. Correct. > On the other hand, when you ask "how many" elements > there are in N, you need an infinity (and this is, I think, actual > infinity). Correct. > But all this hinges quite closely on the semantics of the > word "count". If seen as a process, you do not need an infinity; when > seen as the result of a process, you do need an infinity. and, above all, you do not need different infinities. This had been known already before Cantor. Regards, WM
From: mueckenh on 14 Nov 2006 15:36 William Hughes schrieb: > > The cardinal number aleph_0 is the same as the cardinal number omega. > > Already Cantor in his later years use omega as a cardinal. > > Names do not matter. All what matters is whether you consider the cardinal omega or the ordinal omega. > Let kumquat be the first infinite > ordinal. kumqual + 1 is not equal to kumqual. Please remember that when talking about columns and lines of the matrix each one extended by one element. > > > This cardinal (actually ordinal) number is not an element of the set. > > > > May be so with some sets. A set of natural numbers includes its > > cardinal number. > > > No. (not even if we change this to initial segment of natural numbers > which is what you mean). An initial segment of natural numbers that > has a maximum contains its cardinal. There is one initial segment > of natural numbers, {1,2,3,...} that does not have a maximum. > {1,2,3,...} does not include its cardinal number. There are two possibilities: Either the set has no cardinal number. Or, it has a natural cardinal number, but we cannot know it. > > > > > As long as the "..." denote nothing but natural numbers, your statement > > is obviously false, as induction proves. > > No. Induction can only prove things about initial segements > of the natural numbers that have a maximum. Who said so? > Induction cannot prove > anything about an initial segment of natural numbers that > does not have a maximum. Induction is valid for all natural numbers. That the set does not have a maximum can be proved by induction. As long as it contains ony natural numbers, the laws of natural numbers remain valid. Regards, WM
From: William Hughes on 14 Nov 2006 16:10 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > The cardinal number aleph_0 is the same as the cardinal number omega. > > > Already Cantor in his later years use omega as a cardinal. > > > > Names do not matter. > > All what matters is whether you consider the cardinal omega or the > ordinal omega. > > > Let kumquat be the first infinite > > ordinal. kumqual + 1 is not equal to kumqual. > > Please remember that when talking about columns and lines of the matrix > each one extended by one element. > > > > > This cardinal (actually ordinal) number is not an element of the set. > > > > > > May be so with some sets. A set of natural numbers includes its > > > cardinal number. > > > > > > No. (not even if we change this to initial segment of natural numbers > > which is what you mean). An initial segment of natural numbers that > > has a maximum contains its cardinal. There is one initial segment > > of natural numbers, {1,2,3,...} that does not have a maximum. > > {1,2,3,...} does not include its cardinal number. > > There are two possibilities: > Either the set has no cardinal number. > Or, it has a natural cardinal number, but we cannot know it. No all the natural cardinal numbers have already been used up for sets of the form {1,2,3,...,n}. There are no natural numbers left for the set {1,2,3,...}. 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. > > > > > > > > As long as the "..." denote nothing but natural numbers, your statement > > > is obviously false, as induction proves. > > > > No. Induction can only prove things about initial segements > > of the natural numbers that have a maximum. > > Who said so? > > > Induction cannot prove > > anything about an initial segment of natural numbers that > > does not have a maximum. > > Induction is valid for all natural numbers. 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". 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. - William Hughes - William Hughes
From: Franziska Neugebauer on 14 Nov 2006 16:43 William Hughes wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> William Hughes schrieb: [...] >> > Induction cannot prove >> > anything about an initial segment of natural numbers that >> > does not have a maximum. >> >> Induction is valid for all natural numbers. > > 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". > > 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. As WM has pointed out in ,----[ <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. | `---- and in ,----[ <1163510216.772155.147810(a)k70g2000cwa.googlegroups.com> ] | > What are we talking about, when we write | > | > { } | > | > ? | | The same as when we write | | | | | | | | `---- set formation by '{' and '}' does effectively not take place in WM's world. F. N. -- xyz
From: Lester Zick on 14 Nov 2006 17:08
On 13 Nov 2006 14:32:01 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 12 Nov 2006 16:28:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> On 10 Nov 2006 15:18:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >> >> >Lester Zick wrote: >> >> >> Your way or the highway huh, Moe(x). >> >> >> >If he has an argument that he thinks can be put in set theory, then I'm >> >> >interested in his argument; If he doesn't think his argument can be put >> >> >in set theory, then I'm not interested. He can post about his argument >> >> >all he wants, but I'm not obligated to study his argument. >> >> >> >> No one suggests you are. The problem I see is that one might cast an >> >> argument in such terms as are acceptable to you and still not satisfy >> >> exactly the same criteria on the part of others. I mean unless you are >> >> the generally acknowledged expert in the field. Otherwise it would >> >> look to me like you're just trying to take control of the discussion >> >> in terms you find acceptable whether or not others do. >> > >> >Nothing of the kind. HE suggested to ME that I can see if his argument >> >can be put into set theory. If I am take MY time and effort to do that, >> >then I have every prerogative to set my own terms for doing it. >> >> Oookay. Then the question still occurs whether your comprehension of >> set theory is sufficient unto the task. If not the issue is moot. > >It's very possible that someone else would succeed where I would fail. >But when I am invited to see whether I can put an argument into set >theory, then I can only bring the knowledge that I do have. If I fail >to put the argument into set theory, then I would not claim that as a >demonstration that the argument cannot be put into set theory, but if I >do succeed in putting the argument into set theory, then the argument >can be put into set theory, so a confirmation may come from me that the >argument can be put into set theory, even though I would not claim that >a failure by me would be refutation of a claim that the argument can be >put into set theory. Yeah. Well put. I don't know as any argument from set theory would be definitive since WM's arguments seem to run counter to the tenets of set theory. But since I misjudged the context of the discussion any comments from me would be irrelevant. ~v~~ |