Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 11 Nov 2006 13:50 stephen(a)nomail.com wrote: > I think a lot of this "opposition" would go away if the word > "transfinite" instead of "infinite" had been used to describe > a set that can be put into a one-to-one correspondence with > a proper subset of itself. The word "infinite" sends people > down strange philosophical paths, as does the word "infinity" > despite the fact that it is not really even used in set theory. > Noone would argue about "transfinity". You could be right. Although, it seems unfair of the cranks to dictate what words mathematicians can appropriate. It is hard to make up good names. We have enough names like "second category" as it is. -- David Marcus
From: David Marcus on 11 Nov 2006 13:53 William Hughes wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Therefore omega cannot appear in the sequence. > > > > So we have the irrevocable dilemma: > > 1) omega is the number of countable many numbers like 1,2,3,... or > > 7,8,9,... and as such a single term in the sequence following the > > counted terms like 1,2,3,...,omega or 7,8,9,...,omega > > 2) omega + 1 = 2,3,4,...,1 > > If we allow representations like the above, then omega+1 > does not have a unique representaion > > > and therefore omega cannot not be a term in > > the sequence > > It can however be a term in another sequence representing > omega+1. There is no single sequence which represents > omega+1. I.e., there are lots of ordered sets that have the same order type. -- David Marcus
From: Lester Zick on 11 Nov 2006 14:33 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. Let me see if I can simplify how the issue is or ought to be argued. You posit certain properties of an infinite however you define it. So the question then becomes whether your claim is or can be true. Now one way to show it's actually true would be to produce some entity with the properties you posit of an infinite. Otherwise you'd have to find some other way to get at the truth of what you claim unless you just intend to claim it's true because you or others say so. Now as I understand WM's argument he suggests you can never actually produce any physical infinite because the physical universe is finite. However he then apparently concludes from this that there can be no infinites at all because there can be no physical infinites if the universe is finite. Now personally I find most of the arguments disingenuous on both sides.And I see no special merit to your definition for the properties of infinites you recommend to the exclusion of others. But they are specific properties you can't demonstrate through exemplification so if you wish to show that the characteristics you assign to infinites can be true you have to approach the proof some other way than empirical exemplification. ~v~~
From: Lester Zick on 11 Nov 2006 14:41 On Sat, 11 Nov 2006 17:42:12 +0000 (UTC), stephen(a)nomail.com wrote: >Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: >> In article <4rjeevFrhvopU2(a)mid.individual.net> Bob Kolker writes: >> > Dik T. Winter wrote: >> > > What surprises me is that opposition against it in this way comes from >> > > especially the German posters in this group. They all want to stress >> > > the difference between actual infinity and potential infinity, that >> > > are indeed not mathematical terms, and most other posted do not know >> > > the difference. Is it still the old religious issue? I think so, >> > > based on the postings of at least one of the German posters. >> > >> > If Cantor's last name had been Shickelgrueber or von Richthoffen, none >> > of this would be written. > >> Perhaps. Though I do not think so. But what I am wondering about is the >> preoccupation with "actual infinity" and "potential infinity" with some >> of the posters. I think you are objecting to my "German posters". >> I can shrink this to "German physicists". In this newsgroup there are >> actually three posters that refer to these terms. You may try to find out >> who those three are. > >I think a lot of this "opposition" would go away if the word >"transfinite" instead of "infinite" had been used to describe >a set that can be put into a one-to-one correspondence with >a proper subset of itself. The word "infinite" sends people >down strange philosophical paths, as does the word "infinity" >despite the fact that it is not really even used in set theory. >Noone would argue about "transfinity". Oh I dunno, Stephen. I certainly might. A rose by any other name. Of course you might duck the issues here with such a strategem of verbal regression. It seems every time modern mathematikers get into trouble conceptually they just adopt a different name and pretend the problem has gone away. On another thread I call it The Transfinite Zen Abacus. The only reason mathematikers don't define infinity is that they can't and prefer to sublimate the problem by referring to infinites instead. ~v~~
From: Lester Zick on 11 Nov 2006 14:48
On Fri, 10 Nov 2006 17:48:43 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> It's amazing that 21st century mathematikers cannot analyze the truth >> of contentions without playing cards:the race card, the religion card, >> the homeland card, etc. My last name is Zick. I'm American. Period. >> When it comes to mathematics I'm universalist. > >A pity you don't know any mathematics, though. I don't know any mathematics you know. And vice versa. I try to stick to the truth and let math fall where it may. Others prefer to assume the truth of what they can't demonstrate. It's exactly how they got into playing cards instead of mathematics in the first place. A pity. >> Patriotism and religion >> among others are the last resort of scoundrels. Perhaps I should add set theory and modern math to the list. > In mathematics these >> kinds of issues are historical anachronisms unless you can't establish >> truth in universal terms which it would seem mathematikers can't. ~v~~ |