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From: David C. Ullrich on 14 Apr 2005 07:45 On 13 Apr 2005 08:53:43 -0700, worldsofsolution(a)yahoo.com wrote: >There is something I've been wondering about cantor's proof: the >decimal number generated to prove the contradiction, was taken to be a >real number. There is a tacit assumption that all decimal numbers >represent a real number. Does that not require a proof? Yes. It's not hard to prove - exactly _how_ you prove it depends on exactly how you defined "real number". ************************ David C. Ullrich
From: Matt Gutting on 14 Apr 2005 09:52 Eckard Blumschein wrote: > > On 4/13/2005 3:33 PM, Matt Gutting wrote: > > > >>Okay, but I was intending to be able to refer both to finite and to infinite >>sets. The statements I made, although they didn't specifically refer to finite >>or to infinite sets, are applicable to both. > > > On costs of violation to basic principles. > > > >>>>results as a quantitative comparison does for those sets to which "quantitative >>>>comparison" applies. Having found such a comparison, >>> >>> >>>I wonder if you really found it. Concerning infinite sets I am only >>>aware of the possiblity do decide whether or not there is a bijection >>>being synonymous to countable. >> >>Very specifically, here is my method of comparison: Given two (for the moment, >>let's say non-empty) sets A and B, if one can find an injection, but not a >>surjection, from A to B, say that A "comes before" B or that A is "smaller >>than" B. Similarly, if one can find an injection, but not a surjection, from B >>to A, say that A "comes after" B or that A is "bigger than" B. Finally, if >>one can find both an injection and a surjection (that is, a bijection) between >>A and B, say that A "occupies the same position as" B or that A is "as big as" >>B. >> >>(I put "smaller than","bigger than", and "as big as" in quotes in the paragraph >>above to make it clear that these concepts are not to be literally interpreted >>in the usual sense. This is also why I provided equivalent phrasings not as >>laden with specifically mathematical meaning.) >> >>Since injections and surjections can be found between infinite sets as well as >>between finite ones, > > > Are you sure about that? > Positive. For the truth of that statement *as I phrased it*, it is sufficient to show the existence of at least one bijection (injection + surjection) between infinite sets. Here is one f:N -> N, where f(x) = x. > > >>To extend some of Will Twentyman's work here, the existence of a bijection >>between sets demonstrates that the two sets occupy the same position in an >>order of cardinalities. The lack of a bijection indicates that the two sets >>are in different positions in this order. One can find two infinite sets which >>occupy different positions in this order. > > > I will relpy tomorrow. > > > > >>>Just an example. Children at school must not be taught Cantor's nonsense >>>infinite whole numbers. There are no infinite numbers. >> >>No one is saying that there are numbers which are infinite. > > > Except for Cantor. > He is saying that there are cardinalities which describe infinite sets, not that there are infinite numbers. > >>There are >>cardinalities that don't describe finite sets. And there are different >>cardinalities which describe different infinite sets. > > > Originally Cantor replaced oo by omege. Later on he veiled the traces of > his thinking. > > Eckard >
From: Willy Butz on 14 Apr 2005 10:00 Matt Gutting wrote: > Eckard Blumschein wrote: > >> >> On 4/13/2005 3:33 PM, Matt Gutting wrote: >>> No one is saying that there are numbers which are infinite. >> >> Except for Cantor. > > He is saying that there are cardinalities which describe infinite sets, not > that there are infinite numbers. In his early papers Cantor actually refered to cardinalities as infinite numbers. That term was later changed by Cantor himself to cardinalities. Nevertheless that is one of the major reasons why Eckard is incapable to understand that cardinalities are not numbers. Best wishes, Willy
From: Eckard Blumschein on 14 Apr 2005 11:53 On 4/14/2005 11:51 AM, Barb Knox wrote: > In article <425D55DA.80001(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > [snip] > >>The set of real numbers between 0 and 1 is finite. > > Surely you jest. Stop mocking. As I explained elsewhere, I just wrote the wrong word. Simultaneous elucidation of mathematics to mathematician by an old non-mathematician is demanding.
From: Eckard Blumschein on 14 Apr 2005 12:36
On 4/14/2005 1:45 PM, David C. Ullrich wrote: > On 13 Apr 2005 08:53:43 -0700, worldsofsolution(a)yahoo.com wrote: > >>There is something I've been wondering about cantor's proof: the >>decimal number generated to prove the contradiction, was taken to be a >>real number. There is a tacit assumption that all decimal numbers >>represent a real number. Does that not require a proof? > > Yes. > > It's not hard to prove - exactly _how_ you prove it depends > on exactly how you defined "real number". > Because the discussion mostly deals with Cantor, I would like to suggest preferring his definition. |