Cantor Confusion
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From: Andy Smith on 12 Jan 2007 05:10 OK. Thanks. Ultimately all this is trying to square my (undoubtedly primitive) understanding of "infinity" with what you lot have locked down tight. I am still brooding about it .... > > >Like pointing to e.g. > > some definition of a number on the real line, and > saying > > "there it is". Can you give any concrete example of > > an actually realised set of transfinite numbers? > (bearing > > in mind that I am an earthy physicist, I want some > > explicit points that I can see on a calculator, > making up > > the transfinite set). > > Can you give any concrete example of an actually > realised number 3? I > don't mean an example of three objects, but an > instance of the actual > number 3. > To try to be a bit more explicit, I can see a diagram on a page illustrating the set {1/2,1/4,1/8, ...} and mentally visualise the interval [0,1] as having all those, in a bag as it were. So no problem visualising N. (I can't visualise N as an infinite unbounded sequence {0,1,2,..} but that is psychology, not maths.) As I understand it, ordinals are constructed abstractly as a set of successive elements - then one can say, oh, and there is the set of ALL of them, then another with that and a successor, and so on. Cardinality relates to demonstrating a correspondence (or not) between elements of infinite sets. But how would one visualise the abstract concept of a transfinite number - in a comparable way to {1/2,1/4,1/8, ...}? --- Andy
From: Dave Seaman on 12 Jan 2007 15:38 On Fri, 12 Jan 2007 15:10:38 EST, Andy Smith wrote: > OK. Thanks. Ultimately all this is trying to square > my (undoubtedly primitive) understanding of "infinity" > with what you lot have locked down tight. > I am still brooding about it .... >> >> >Like pointing to e.g. >> > some definition of a number on the real line, and >> saying >> > "there it is". Can you give any concrete example of >> > an actually realised set of transfinite numbers? >> (bearing >> > in mind that I am an earthy physicist, I want some >> > explicit points that I can see on a calculator, >> making up >> > the transfinite set). >> >> Can you give any concrete example of an actually >> realised number 3? I >> don't mean an example of three objects, but an >> instance of the actual >> number 3. >> > To try to be a bit more explicit, I can see a diagram on > a page illustrating the set {1/2,1/4,1/8, ...} and > mentally visualise the interval [0,1] as having all > those, in a bag as it were. So no problem visualising N. > (I can't visualise N as an infinite unbounded sequence > {0,1,2,..} but that is psychology, not maths.) > As I understand it, ordinals are constructed abstractly as a set of successive elements - then one can say, oh, and there is the set of ALL of them, then another with that and a successor, and so on. Cardinality relates to demonstrating a > correspondence (or not) between elements of infinite sets. The first ordinal is the empty set, denoted by 0. If x is any ordinal, then the successor of x is the set x U {x}. So the first few ordinals are: 0 = { } 1 = { 0 } 2 = { 0, 1 } 3 = { 0, 1, 2 } and so on. > But how would one visualise the abstract concept of a > transfinite number - in a comparable way to {1/2,1/4,1/8, ...}? If you can visualize {1/2, 1/4, 1/8, ... }, then why can't you visualize { 0, 1, 2, ... }? We have an axiom (the axiom of infinity) that guarantees such a set exists. Specifically, the axiom says there is a set that contains 0 and is closed under the successor operation. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: cbrown on 12 Jan 2007 15:53 Andy Smith wrote: > OK. Thanks. Ultimately all this is trying to square > my (undoubtedly primitive) understanding of "infinity" > with what you lot have locked down tight. > > I am still brooding about it .... > > > > >Like pointing to e.g. > > > some definition of a number on the real line, and > > saying > > > "there it is". Can you give any concrete example of > > > an actually realised set of transfinite numbers? > > (bearing > > > in mind that I am an earthy physicist, I want some > > > explicit points that I can see on a calculator, > > making up > > > the transfinite set). > > > > Can you give any concrete example of an actually > > realised number 3? I > > don't mean an example of three objects, but an > > instance of the actual > > number 3. > > > > To try to be a bit more explicit, I can see a diagram on > a page illustrating the set {1/2,1/4,1/8, ...} and > mentally visualise the interval [0,1] as having all > those, in a bag as it were. So no problem visualising N. > (I can't visualise N as an infinite unbounded sequence > {0,1,2,..} but that is psychology, not maths.) > > As I understand it, ordinals are constructed abstractly as > a set of successive elements - then one can say, oh, and there is > the set of ALL of them, then another with that and a successor, > and so on. Cardinality relates to demonstrating a > correspondence (or not) between elements of infinite sets. > > But how would one visualise the abstract concept of a > transfinite number - in a comparable way to {1/2,1/4,1/8, ...}? > Since you're having trouble "visualizing" N, this is a bit of a tall order; but... Consider the set of all ordered pairs of natural numbers, (m, n). Now define the ordering: (m, n) <= (x,y) if n < y OR n=y and m<=x. Then you should be able to see that: This is a total order: (m,n) < (x,y) or (m,n) = (x,y) or (m,n) > (x,y) There is a smallest element: (0,0) <= (m,n) for all m,n The ordering is a well-order: If S is any set of such elements {(x,y)}, there exists (m,n) in S such that for all (x,y) in S, (m,n) <= (x,y). It contains a subset whose ordering is the same as the ordering of the naturals: (m, 0) <= (n,0) iff m <= n There is no largest element /in/ that /subset/: (m, 0) < (m+1,0) for all m There is an element which is "larger than any natural": (m,0) <= (0,1) for all m In some sense, (0,1) is "like" the ordinal w (roughly - for visualization purposes only!) Think of w+1 as (1,1), w+2 as (2,1), w + w = 2*w as (0,2), etc. HTH! Cheers - Chas
From: David Marcus on 12 Jan 2007 16:12 cbrown(a)cbrownsystems.com wrote: > Consider the set of all ordered pairs of natural numbers, (m, n). > > Now define the ordering: > > (m, n) <= (x,y) if > n < y > OR > n=y and m<=x. > > Then you should be able to see that: > > This is a total order: > (m,n) < (x,y) or (m,n) = (x,y) or (m,n) > (x,y) > > There is a smallest element: > (0,0) <= (m,n) for all m,n > > The ordering is a well-order: > If S is any set of such elements {(x,y)}, there exists (m,n) in S such > that for all (x,y) in S, (m,n) <= (x,y). > > It contains a subset whose ordering is the same as the ordering of the > naturals: > (m, 0) <= (n,0) iff m <= n > > There is no largest element /in/ that /subset/: > (m, 0) < (m+1,0) for all m > > There is an element which is "larger than any natural": > (m,0) <= (0,1) for all m > > In some sense, (0,1) is "like" the ordinal w (roughly - for > visualization purposes only!) Think of w+1 as (1,1), w+2 as (2,1), w + > w = 2*w as (0,2), etc. That's a good example. It wouldn't be hard to make your "like" precise, i.e., as the two being order isomorphic. -- David Marcus
From: David Marcus on 12 Jan 2007 16:18
Dave Seaman wrote: > On Fri, 12 Jan 2007 11:48:38 EST, Andy Smith wrote: > > > that you can have an ordered > > infinite sequence which has a finite start and end point. > > For infinite, read "countably infinite". > > Of course. The rationals in [0,1], for example. Normally, a "sequence" doesn't have a last element. So, the rationals are an example of a countably infinite set (not sequence) that has a finite start and end point (where "start" and "end" mean smallest and largest). Andy seems to be confusing sequence with countably infinite. Not sure why he is doing this or how to make him stop doing it. -- David Marcus |