
Re: Cantor's first proof
Posted:
Nov 8, 2012 1:33 PM


On 08/11/2012 2:12 PM, Zuhair wrote: > On Nov 8, 8:15 pm, gus gassmann <g...@nospam.com> wrote: >> On 08/11/2012 12:00 PM, Zuhair wrote: >> >> >> >> >> >> >> >> >> >>> On Nov 8, 2:10 pm, "LudovicoVan" <ju...@diegidio.name> wrote: >>>> "Zuhair" <zaljo...@gmail.com> wrote in message >> >>>> news:dac67778bb434846996b0cd984b03922@s12g2000vbw.googlegroups.com... >> >>>>> From what I knew after surfing about this proof is that Cantor >>>>> presented it to Dedekind first containing infinite number of gaps, but >>>>> what was published was the customary one. >> >>>>> A nice proof that I saw online is this one: >> >>>>> Quote: >> >>>>> Cantor's 1874 proof. To show that the real numbers are uncountable, we >>>>> must show that given any countable sequence of distinct real numbers, >>>>> there exists another real number not in the sequence. >> >>>>> Like the diagonalization argument, we will do so by providing >>>>> an explicit algorithm which produces such a number; unlike the >>>>> diagonalization argument, we will employ not decimal expansions but >>>>> order properties of the real numbers. >> >>>>> Let (a_n) be a countable sequence of distinct real numbers. Suppose >>>>> that there are two distinct terms a_j and a_k such that no term a_l >>>>> lies strictly between a_j and a_k in other words, suppose that (a_n) >>>>> does not possess the Intermediate Value Property. Let L be any real >>>>> number strictly between a_j and a_k, for example (a_j + a_k) /2 . Then >>>>> L is not in the sequence (a_n). >> >>>>> Now suppose that (a_n) does have the Intermediate Value Property. >> >>>> Which is equivalent to assuming that (a_n) is an enumeration of (all) the >>>> real numbers, is it not? >> >>> I already explained it is not, but because this is an important point, >>> I'll expand on showing that it is not! >> >>> Take any countable enumeration of all rationals, it does have the >>> Intermediate Value Property, yet it is not an enumeration of (all) >>> real numbers. >> >> It seems you should define what you mean by "Intermediate Value >> Property". In analysis one talks about the intermediate value theorem >> for continuous functions, but this is not what you mean here. > > Here it is about enumeration of reals, so such an enumeration would be > said to possess an Intermediate Value Property iff for every two > distinct reals it enumerates there is a real that is strictly between > them that it enumerates. Actually it is obvious from the quote.
It clearly was not obvious to LV.
Cheers

