Why can no one in sci.math understand my simple point?
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From: Peter Webb on 19 Jun 2010 03:11 "Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1omjv.jrj.tim(a)soprano.little-possums.net... > On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >> If you give me a purported list of all computable Reals I can use a >> diagonal argument to form a computable Real not on the list. > > OK, here you go: define f:N->R by the method of my previous post. > That is, it is the real computed by the n'th Turing machine within the > set of Turing machines that compute a real, ordered by Godel numbering > of their specifications. The function f defines a list of real > numbers. > > Your task is to find a real not on that list, and prove that it is a > computable real. > > You haven't specified the list. I have to guess at some of the entries in the list, as I don't actually know and cannot determine which TMs halt. Cantor's proof demands that you provide the list in a form whereby the nth digit of the nth entry can be determined. My proof has exactly the same requirement. Cantor's proof doesn't work unless the list is explicitly provided, and nor does mine. Same same. > - Tim
From: Peter Webb on 19 Jun 2010 03:17 "Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1oka3.jrj.tim(a)soprano.little-possums.net... > On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >> Which of these two statements do you agree with: >> >> 1. You cannot form a list of computable Reals. >> >> 2. The computable Reals are countable. > > Answering the same questions for the fourth time now, 1 is false, 2 is > true. > OK, how about you give me a list of computable Reals - in exactly the same format as Cantor requires, ie each item in the list can be evaluated to any desired degree of accuracy - and I will produce a Real not on the list. > >> You have failed to explain why Cantor's diagonal proof "proves" the >> Reals are uncountable, but the same proof applied to computable >> Reals does *not* prove the Computable Reals are uncountable. > > I have, multiple times now. The antidiagonal of a list of computable > reals is not necessarily a computable real. yes it is. There is a very simple, explicit construction that will calculate the anti-diagonal number to any required degree of accuracy. Take the list, determine the nth digit of the nth item, blah blah. Remember, just as in Cantor's proof, you have to provide the list first. If you don't believe me, give me a list of all computable Reals in the same format as Cantor uses in his proof, and I will produce a computable number not on the list. > Your insistence that it > must be is almost certainly because you don't know what a computable > real is. (There are other possibilities, even less flattering to your > math skills) > > In particular, the antidiagonal of a list of all computable reals is > never a computable real. > I am still waiting to see your list of all computable Reals. In the same format as Cantor used in his proof; after all, I want the two proofs to be identical other than one considers all Reals and the other only computable Reals. Go ahead. > - Tim
From: Sylvia Else on 19 Jun 2010 03:55 On 19/06/2010 4:14 PM, |-|ercules wrote: > "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >> On 19/06/2010 1:40 PM, |-|ercules wrote: >>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >>>> On 19/06/2010 12:45 PM, |-|ercules wrote: >>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote >>>>>> On 19/06/2010 6:50 AM, WM wrote: >>>>>>> On 18 Jun., 09:37, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>>>>> On 18/06/2010 5:31 PM, |-|ercules wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>>>>>>> On 18/06/2010 4:52 PM, |-|ercules wrote: >>>>>>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>>>>>>>>> On 18/06/2010 3:03 PM, |-|ercules wrote: >>>>>>>>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote >>>>>>>>>>>>>> On 18/06/2010 10:40 AM, Transfer Principle wrote: >>>>>>>>>>>>>>> On Jun 17, 6:56 am, Sylvia Else<syl...(a)not.here.invalid> >>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>> On 15/06/2010 2:13 PM, |-|ercules wrote: >>>>>>>>>>>>>>>>> the list of computable reals contain every digit of ALL >>>>>>>>>>>>>>>>> possible >>>>>>>>>>>>>>>>> infinite sequences (3) >>>>>>>>>>>>>>>> Obviously not - the diagonal argument shows that it >>>>>>>>>>>>>>>> doesn't. >>>>>>>> >>>>>>>>>>>>>>> But Herc doesn't accept the diagonal argument. Just because >>>>>>>>>>>>>>> Else accepts the diagonal argument, it doesn't mean that >>>>>>>>>>>>>>> Herc is required to accept it. >>>>>>>> >>>>>>>>>>>>>>> Sure, Cantor's Theorem is a theorem of ZFC. But Herc said >>>>>>>>>>>>>>> nothing about working in ZFC. To Herc, ZFC is a "religion" >>>>>>>>>>>>>>> in which he doesn't believe. >>>>>>>> >>>>>>>>>>>>>> Well, if he's not working in ZFC, then he cannot make >>>>>>>>>>>>>> statements >>>>>>>>>>>>>> about >>>>>>>>>>>>>> ZFC, and he should state the axioms of his system. >>>>>>>> >>>>>>>>>>>>> Can you prove from axioms that is what I should do? >>>>>>>> >>>>>>>>>>>>> If you want to lodge a complaint with The Eiffel Tower that >>>>>>>>>>>>> the >>>>>>>>>>>>> lift is >>>>>>>>>>>>> broken >>>>>>>>>>>>> do you build your own skyscraper next to the Eiffel Tower to >>>>>>>>>>>>> demonstrate >>>>>>>>>>>>> that fact? >>>>>>>> >>>>>>>>>>>> That's hardly a valid analogy. >>>>>>>> >>>>>>>>>>>> If you're attempting to show that ZFC is inconsistent, then say >>>>>>>>>>>> that >>>>>>>>>>>> you are working within ZFC. >>>>>>>> >>>>>>>>>>>> If you're not working withint ZFC, then you're attempting to >>>>>>>>>>>> show that >>>>>>>>>>>> some other set of axioms is inconsistent, which they may be, >>>>>>>>>>>> but >>>>>>>>>>>> the >>>>>>>>>>>> result is uninteresting, and says nothing about ZFC. >>>>>>>> >>>>>>>>>>>> Sylvia. >>>>>>>> >>>>>>>>>>> That would be like finding a fault with the plans of The Leaning >>>>>>>>>>> Tower >>>>>>>>>>> Of Piza. >>>>>>>> >>>>>>>>>>> I might look at ZFC at some point, but while you're presenting >>>>>>>>>>> Cantor's >>>>>>>>>>> proof >>>>>>>>>>> in elementary logic I'll attack that logic. >>>>>>>> >>>>>>>>>>> Instead of 'constructing' a particular anti-diagonal, your proof >>>>>>>>>>> should >>>>>>>>>>> work equally >>>>>>>>>>> well by giving the *form* of the anti-diagonal. >>>>>>>> >>>>>>>>>>> This is what a general diagonal argument looks like. >>>>>>>> >>>>>>>>>>> For any list of reals L. >>>>>>>> >>>>>>>>>>> CONSTRUCT a real such that >>>>>>>>>>> An AD(n) =/= L(n,n) >>>>>>>> >>>>>>>>>>> Now to demonstrate this real is not on L, it is obvious that >>>>>>>>>>> An AD(n) =/= L(n,n) >>>>>>>> >>>>>>>>>>> Therefore >>>>>>>>>>> [ An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) ] proves >>>>>>>>>>> superinfinity! >>>>>>>> >>>>>>>>>>> And THAT is Cantor's proof! >>>>>>>> >>>>>>>>>>> Want to see his other proof? That no box contains the box >>>>>>>>>>> numbers >>>>>>>>>>> (of >>>>>>>>>>> boxes) that >>>>>>>>>>> don't contain their own box number? >>>>>>>>>>> That ALSO proves superinfinity! >>>>>>>> >>>>>>>>>>> Great holy grail of mathematics you have there. >>>>>>>> >>>>>>>>>>> Herc >>>>>>>> >>>>>>>>>> What are you trying to prove? >>>>>>>> >>>>>>>>> There is only one type of infinity. >>>>>>>> >>>>>>>> Infinity is a mathematical construct. Before you can even being to >>>>>>>> discuss it, you have to have a set of axioms. >>>>>>> >>>>>>> What was the set that Cantor used? >>>>>>> Nevertheless he "proved". >>>>>> >>>>>> He certainly was using some. For example, the diagonal argument falls >>>>>> apart if the axioms don't permit the construction of a number by >>>>>> choosing digits different from those on the diagonal. >>>>>> >>>>>> It isn't even clear whether Herc is tying to invalidate Cantor's >>>>>> proof >>>>>> by finding a mistake in it, or to prove the inverse, which wouldn't >>>>>> invalidate Cantor's proof, but would only show that the axioms on >>>>>> which it is based are inconsistent. >>>>>> >>>>>> Herc cannot avoid the need to specify the set of axioms. >>>>>> >>>>>> Sylvia. >>>>> >>>>> How would one dispute axiomatic deductions if that were the case? >>>> >>>> What do you mean by "axiomatic deduction"? >>>>> >>>>> Are you saying all mathematical facts are either axioms or the >>>>> result of >>>>> (X & X->Y) -> Y >>>>> ? >>>> >>>> Mathematics consists of axioms and statements (theorems) that can be >>>> proved from those axioms. The axioms cannot themselves be proved, nor >>>> disproved, though they may be shown to be inconsistent with one >>>> another. >>>> >>>> Sometimes the axioms seem so self-evidently true that people aren't >>>> even aware that they're there. But they are, if you look. >>>> >>>> Sylvia. >>> >>> blah blah blah... >>> >>> you skipped my question, but don't bother I wasn't arguing anything >>> just seeing if you knew what you were talking about. >> >> Your question was typically vague. They you dived into some notation >> which might be construed to mean "if X and X implies Y, then Y", which >> is itself unproveable, and needs to be introduced as an axiom. >> >> None of which eliminates your need to specify the axioms in which >> you're making claims about Cantor's proof. >> >> Sylvia. > > HAHAHAHA > > You never studied theorem provers. You're like Wally Anglesea, one of > the thickest morons I've ever come across, but he has in innate ability > to regurgitate > the arguments of other skeptics, in the right places, imitating > intelligence. > > Herc It's pattern. When someone gets too close to pinning you down, you abandon any kind of argument and shift to abuse instead. I take it, then, that you're not willing to specify the axioms you're working with, because you know very well that doing so will make your claims disprovable, rather than merely undecidable, which they are in the absence of axioms. Sylvia.
From: Sylvia Else on 19 Jun 2010 04:06 On 19/06/2010 4:11 PM, |-|ercules wrote: > To support your argument you should at least show that you've formed a > new sequence of digits. I'll explain it simply then. The first digit of the created number differs from the first digit of the first number in the list. The second digit differs from the second digit of the second number in the list. In general, digit n differs from digit n of the nth number in the list. So for all n, the created number differs from number n. Therefore the created number is not in the list - it is a new sequence of digits. > > If you actually read my derivation of herc_cant_3 instead of blindly > dismissing it, > you'll see it holds, just like all digits of PI appear in order below > this line, if interpreted > correctly. > > Herc > > ___________________ > > 3 > 31 > 314 > 3141 > ... > > herc-cant-3 is not a derivation. It's a wild leap of faith. Nothing is proved therein. Sylvia.
From: |-|ercules on 19 Jun 2010 05:07
"Sylvia Else" <sylvia(a)not.here.invalid> wrote ... > On 19/06/2010 4:11 PM, |-|ercules wrote: > >> To support your argument you should at least show that you've formed a >> new sequence of digits. > > I'll explain it simply then. The first digit of the created number > differs from the first digit of the first number in the list. The second > digit differs from the second digit of the second number in the list. > > In general, digit n differs from digit n of the nth number in the list. > > So for all n, the created number differs from number n. Therefore the > created number is not in the list - it is a new sequence of digits. No I've told you all 20 times that does not create any new sequence at all. All you've done is CONSTRUCT a digit sequence like so An AD(n) =/= L(n,n) And then you say, it's different to each number like so PROOF An AD(n) =/= L(n,n) But you have not demonstrated a NEW SEQUENCE OF DIGITS. All you've done is this [ An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) ] -> Superinfinity Your actual 'proof' is a specific example of the above 'proof'! [ An AD(n) = (L(n,n) + 1) mod 9 -> An AD(n) =/= L(n,n) ] -> Superinfinity Do you agree with the above version of Cantor's proof? > >> >> If you actually read my derivation of herc_cant_3 instead of blindly >> dismissing it, >> you'll see it holds, just like all digits of PI appear in order below >> this line, if interpreted >> correctly. >> >> Herc >> >> ___________________ >> >> 3 >> 31 >> 314 >> 3141 >> ... >> >> > > herc-cant-3 is not a derivation. It's a wild leap of faith. Nothing is > proved therein. > > Sylvia. Then which step do you disagree with? defn(herc_cant_3) The list of computable reals contains every digit (in order) of all possible infinite sequences. Derivation Given the increasing finite prefixes of pi 3 31 314 ... This list contains every digit (in order) of the infinite expansion of pi. Given the increasing finite prefixes of e 2 27 271 ... This list contains every digit (in order) of the infinite expansion of e. Given the increasing finite prefixes of ALL infinite expansions, that list contains every digit (in order) of every infinite expansion. So herc_cant_3 is true. The list of computable reals contains every digit (in order) of all possible infinite sequences. Herc -- "There are more things in Cantor's paradise, Horatio, than are dreamt of by your computers." ~ Barb Knox |