Why can no one in sci.math understand my simple point?
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From: Daryl McCullough on 15 Jun 2010 06:26 |-|ercules says... > >Consider the list of increasing lengths of finite prefixes of pi > >3 >31 >314 >3141 >.... > >Everyone agrees that: >this list contains every digit of pi (1) > >as pi is an infinite digit sequence, this means > >this list contains every digit of an infinite digit sequence (2) > >similarly, as computable digit sequences contain increasing lengths of ALL >possible finite prefixes > >the list of computable reals contain every digit of ALL possible infinite >sequences (3) > >OK does everyone get (1) (2) and (3). If you state it carefully, then yes, everyone gets it: (A) Forall real numbers r, Forall natural numbers n, There exists a computable real r' such that r' agrees with r in the first n decimal places. That fact has nothing to do with Cantor's proof. Cantor's theorem has the consequence: (B) There exists a real number r, Forall computable reals r', there exists a natural number n such that r' and r disagree at the nth decimal place. (B) is relevant to Cantor's theorem, but (A) is not. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Jun 2010 06:39 |-|ercules says... > >"Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote >> "|-|ercules" <radgray123(a)yahoo.com> wrote in message >> news:87om34FahrU1(a)mid.individual.net... >>> "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote >>>> "|-|ercules" <radgray123(a)yahoo.com> wrote in message >>>> news:87ocucFrn3U1(a)mid.individual.net... >>>>> Consider the list of increasing lengths of finite prefixes of pi >>>>> >>>>> 3 >>>>> 31 >>>>> 314 >>>>> 3141 >>>>> .... >>>>> >>>>> Everyone agrees that: >>>>> this list contains every digit of pi (1) >>>>> >>>> >>>> Sloppy terminology, but I agree with what I think you are trying to say. >>>> >>>>> as pi is an infinite digit sequence, this means >>>>> >>>>> this list contains every digit of an infinite digit sequence (2) >>>>> >>>> >>>> Again sloppy, but basically true. >>>> >>>>> similarly, as computable digit sequences contain increasing lengths of >>>>> ALL possible finite prefixes >>>>> >>>> >>>> Not "similarly", but if you are claiming that all Reals which have finite >>>> decimal expansions can be listed, this is correct. >>> >>> You didn't follow the similarity. >>> >>> Given the increasing finite prefixes of pi >>> >>> 3 >>> 31 >>> 314 >>> .. >>> >>> This list contains every digit of the infinite expansion of pi. >>> >> >> But pi doesn't appear on the list. >> >> So? > > >that doesn't matter, because that's a convergent sequence. That's *all* that matters, for Cantor's theorem. The claim is that for every list of reals, there is another real that does not appear on the list. -- Daryl McCullough Ithaca, NY
From: Dingo on 15 Jun 2010 07:16 On Tue, 15 Jun 2010 18:29:43 +1000, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >But pi doesn't appear on the list. > >So? This subject is irrelevant to aus.tv.
From: WM on 15 Jun 2010 07:28 On 15 Jun., 08:15, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> > No. You cannot form a list of all computable Reals. If you could do this, > then you could use a diagonal argument to construct a computable Real not in > the list. Twice no. First, a number cannot be defined by an infinite sequence of digits, because of practical reasons. (To define means to let somebody know what is meant.) Second the list of all real definitions cannot have a diagonal because at least two lines have only one symbol. Here is a list that contains not only every computable real number but also every possible definition of every item that can be defined. 0 1 00 01 10 11 000 .... Regards, WM
From: WM on 15 Jun 2010 07:35
On 15 Jun., 06:13, "|-|ercules" <radgray...(a)yahoo.com> wrote: > Consider the list of increasing lengths of finite prefixes of pi > > 3 > 31 > 314 > 3141 > .... > > Everyone agrees that: > this list contains every digit of pi (1) There is no "every digit of pri". If it were, this must be proved by showing the last digit. > > as pi is an infinite digit sequence, this means > > this list contains every digit of an infinite digit sequence (2) If you write every digit in the same line, no set theorist will disagree. If you write the next digit always in the next line, they will diagree. You can see that they have restricted capabilities of thinking. > > similarly, as computable digit sequences contain increasing lengths of ALL possible finite prefixes > > the list of computable reals contain every digit of ALL possible infinite sequences (3) > > OK does everyone get (1) (2) and (3). > > There's no need for bullying (George), He has reasons enough. He does not even know that countable sets, according to Fraenkel, Cantor and others, can be constructed. And he his incompetent to know what it means "to construct" without receiving a definition that says exactly as much as the word "to construct". Regards, WM |