Why can no one in sci.math understand my simple point?
in [Logic]
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From: |-|ercules on 15 Jun 2010 14:55 "Dingo" <dingo(a)gmail.com> wrote ... > 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. What gave it away? Herc
From: |-|ercules on 15 Jun 2010 14:58 "Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote... > |-|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. Yes but HOW does Cantor show that? By producing a NEW SEQUENCE of digits. Like so... 123 456 789 Diag = 159 Anti-diag = 260 Where are you getting a '260' when >the list of computable reals contain every digit of ALL possible infinite >sequences (3) Herc
From: Daryl McCullough on 15 Jun 2010 15:03 WM says... >On 15 Jun., 18:46, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> I'm not sure what you are saying. The fact is, we can prove >> that for every real r_n on the list, d is not equal to r_n. > >Of course. Every real r_n belongs to a finite initial segment of the >list. >That does not yield any result about the whole list On the contrary, the definition of "d is on the list" is that "there exists a natural number n such that r_n = d". We proved "forall n, r_n is not equal to d". So that means "there does not exist a natural number n such that r_n = d", so that means "d is not on the list". We have thus proved something about the whole list. >> That means that d is not on the list. There is no extrapolation >> involved. > >Look here: We can prove for any finite segment >{2, 4, 6, ..., 2n} >of the ordered set of all positive even numbers that its cardinal >number is surpassed by some elements of the set. > >Nevertheless this appears not be a proof that the cardinal number of >the whole set is less than some elements of the set. So there we have an example of an illegitimate extrapolation. If you prove Phi(n) for an arbitrary natural number n, then you are allowed to conclude: forall natural numbers n, Phi(n). So you can conclude: forall natural numbers n > 0, the set of all even numbers less than or equal to 2n has a cardinality less than 2n. That's true. That's a legitimate proof. On the other hand, it is not legitimate to conclude: The set of all even numbers has a cardinality that is less than some even number. That's an unwarranted extrapolation. So there are legitimate proofs, and there are bogus proofs. You have to actually learn logic to be able to tell the difference. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Jun 2010 15:32 WM says... > >On 15 Jun., 18:49, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> I was talking about the list of all *computable* reals. There are >> definable reals that are not computable, and Cantor's proof shows >> how to define one. > >No, it does not show that because it is impossible to define a real by >an infinite definition. It's not an infinite definition. It's a finite definition. >> You can similarly get a list of all definable reals for a specific >> language L. > >Every real that is definable in a specific language is definable in >another specific language. The set does not grow or shrink or change >when the language is changed. It certainly does. Let L be any countable language whose intended model is the naturals. Following Godel, we define a way to code each formula of L as a natural number. Then we introduce a new unary predicate symbol T with the following interpretation: T(x) is true if and only if x is the code of a true sentence of language L. Tarski proved that T(x) is not definable in language L. Using the new predicate T, we can define a real that is provably unequal to any real definable in the original language L. >No, my list contains all words in all languages, even the definitions >of the languages and all the dictionaries. There is nothing else. No language can consistently contains its own truth predicate. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Jun 2010 15:38
WM says... > >On 15 Jun., 18:53, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> For example, we can define a real r as follows: >> >> r = sum from n=0 to infinity of H(n) 2^{-n} >> >> where H(n) = 1 if Turing machine number n halts on input n, >> H(n) = 0 otherwise. >> >> That's definable, but it is not computable. > >Anyhow it is not a definition. It certainly is. It uniquely characterizes a real number, so it's a definition. >It would be more useful to define some number by the legs >of a crowd of unicorns touching the ground at a given time. If you would find that more useful, go ahead. When I talk about mathematical definitions, I'm really talking about definitions that are meaningful to people who understand mathematics. If you prefer to talk about unicorns, then a mathematical definition is probably not appropriate for you. -- Daryl McCullough Ithaca, NY |