THE CANTOR ARGUMENT SO FAR
in [Logic]
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From: Sylvia Else on 20 Jun 2010 21:22 On 21/06/2010 11:17 AM, |-|ercules wrote: > "Sylvia Else" <sylvia(a)not.here.invalid> wrote... >> On 21/06/2010 2:40 AM, |-|ercules wrote: >>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote >>>> On 20/06/2010 7:42 PM, |-|ercules wrote: >>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >>>>>> On 20/06/2010 5:03 PM, |-|ercules wrote: >>>>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >>>>>>>> On 20/06/2010 12:42 PM, |-|ercules wrote: >>>>>>>>> "George Greene" <greeneg(a)email.unc.edu> wrote >>>>>>>>>>> Is it a *NEW DIGIT SEQUENCE* or not? >>>>>>>>>> >>>>>>>>>> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was >>>>>>>>>> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! >>>>>>>>> >>>>>>>>> >>>>>>>>> But you keep saying the anti-diagonal is NEW and ignoring me when >>>>>>>>> I say it's not a new digit sequence. >>>>>>>>> >>>>>>>>> Then you repeat Cantor's proof again and again that it's NEW. >>>>>>>>> >>>>>>>>> You use terms NEW and NOT ON THE LIST, but evade me when I >>>>>>>>> challenge >>>>>>>>> whether it contains any new digit sequence. >>>>>>>> >>>>>>>> And you just ignore the point that it must be new because it >>>>>>>> demonstrably isn't in the list. >>>>>>>> >>>>>>>> Sylvia. >>>>>>> >>>>>>> All you demonstrated was >>>>>> >>>>>> In what sense could a number be said to be in a list if it doesn't >>>>>> match any element of the list? >>>>> >>>>> >>>>> What sense is a number that's only definition is to not be on a list? >>>>> >>>>> That may seem less concrete than your question, but it's worth >>>>> thinking >>>>> what "anti-diagonals" entail. >>>>> >>>>> You're not just constructing 0.444454445544444445444.. a 4 for >>>>> every non >>>>> 4 digit and a 5 for a 4. >>>>> >>>>> You're constructing ALL 9 OTHER DIGITS to the diagonal digits. >>>> >>>> Why? We only need one number that's not in the list. Doing what you >>>> suggest just creates many more numbers that are not in the list. >>>> >>>>> >>>>> And it's not just the diagonal, it's the diagonal of ALL >>>>> PERMUTATIONS OF >>>>> THE LIST. >>>> >>>>> >>>>> So, the first digit of the list can be... well anything, so the >>>>> antidiagonal starts with anything.. >>>>> then the second digit of the second real can be anything, so the >>>>> antidiagonals next digit is anything.. >>>> >>>> The problem with that is before you can permute a list, you have to >>>> prove that a list exists. If it doesn't exist, you can't permuted it. >>>> Cantor assumes that a list exists, and then proceeds to show that that >>>> leads to a contradiction. Which is fair enough. But you can't just >>>> assume the list exists, and then use arguments about permuting the >>>> list to prove that it exists, or to negate the proof that it doesn't >>>> exist. That's circular. >>>> >>>> <snipped rest based on false premise> >>>> >>>> Sylvia. >>> >>> So you can form an anti-diagonal on a hypothetical list like the >>> computable reals >>> but you can't reorder that list? >>> >>> If you take the 2nd digit of the 2nd real, then the 1st digit of the 1st >>> real, then >>> the remaining nth digit of the nth reals, the remainder of the usual >>> diagonal, that won't work? >>> >>> 0. _ x _ _ _ _ _ >>> 0. x _ _ _ _ _ _ >>> 0. _ _ x _ _ _ _ >>> 0. _ _ _ x _ _ _ >>> 0. _ _ _ _ x _ _ >>> 0. _ _ _ _ _ x _ >>> 0. _ _ _ _ _ _ x >>> >>> Does not an anti-diagonal make! >>> >>> Herc >> >> It would also be a number that is not in the list. What difference >> would that make? >> >> As usual, it's unclear where you think you're going with this. >> >> OK.... >> >> You can't permute a list of reals, because there is no such list. >> >> If could construct a list of all computable reals, then you could >> permute the list, generating as many anti-diagonals as you like, and >> construct thereby numbers that are not in the list of computable >> reals. All you would be doing is demonstrating that that there are >> reals that are not computable. >> >> Similarly, you can permute a list of computable reals so that any >> desired number, computable or otherwise, lies on the diagonal, but the >> existence of the number on the diagonal doesn't mean that it's >> computable. >> >> Or you can construct a list, finite or otherwise, of a subset of the >> reals, and then use anti-diagonalisation to contruct humbers that are >> not in the list. You could then add them to the list, and repeat. You >> end up with an arbitrary countable subset of the reals, which is >> totally uninteresting. >> >> Sylvia. >> > > > Actually if a real list contains 0.111... it's difficult to get a > diagonal = 0.222... Did I say a subset of computable reals? No, I didn't. Did I use the word "subset" in the following paragraph? Yes, I did. So am I drawing a distinction? Yes, most likely. > If you COULD set the diagonal to ANYTHING then it would prove my point. Why would it? A number is not computable merely because it appears on a diagonal. > > Anyway you were going on about a false premise, now it's not interesting, > so it doesn't matter what the claim is you refute it illogically. > > At any rate, your interest is gone so we are finished. Actually, what appears to happen at this point is that you start a new thread on exactly the same topic. Sylvia.
From: |-|ercules on 20 Jun 2010 21:37 "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... > On 21/06/2010 11:17 AM, |-|ercules wrote: >> "Sylvia Else" <sylvia(a)not.here.invalid> wrote... >>> On 21/06/2010 2:40 AM, |-|ercules wrote: >>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote >>>>> On 20/06/2010 7:42 PM, |-|ercules wrote: >>>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >>>>>>> On 20/06/2010 5:03 PM, |-|ercules wrote: >>>>>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >>>>>>>>> On 20/06/2010 12:42 PM, |-|ercules wrote: >>>>>>>>>> "George Greene" <greeneg(a)email.unc.edu> wrote >>>>>>>>>>>> Is it a *NEW DIGIT SEQUENCE* or not? >>>>>>>>>>> >>>>>>>>>>> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was >>>>>>>>>>> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> But you keep saying the anti-diagonal is NEW and ignoring me when >>>>>>>>>> I say it's not a new digit sequence. >>>>>>>>>> >>>>>>>>>> Then you repeat Cantor's proof again and again that it's NEW. >>>>>>>>>> >>>>>>>>>> You use terms NEW and NOT ON THE LIST, but evade me when I >>>>>>>>>> challenge >>>>>>>>>> whether it contains any new digit sequence. >>>>>>>>> >>>>>>>>> And you just ignore the point that it must be new because it >>>>>>>>> demonstrably isn't in the list. >>>>>>>>> >>>>>>>>> Sylvia. >>>>>>>> >>>>>>>> All you demonstrated was >>>>>>> >>>>>>> In what sense could a number be said to be in a list if it doesn't >>>>>>> match any element of the list? >>>>>> >>>>>> >>>>>> What sense is a number that's only definition is to not be on a list? >>>>>> >>>>>> That may seem less concrete than your question, but it's worth >>>>>> thinking >>>>>> what "anti-diagonals" entail. >>>>>> >>>>>> You're not just constructing 0.444454445544444445444.. a 4 for >>>>>> every non >>>>>> 4 digit and a 5 for a 4. >>>>>> >>>>>> You're constructing ALL 9 OTHER DIGITS to the diagonal digits. >>>>> >>>>> Why? We only need one number that's not in the list. Doing what you >>>>> suggest just creates many more numbers that are not in the list. >>>>> >>>>>> >>>>>> And it's not just the diagonal, it's the diagonal of ALL >>>>>> PERMUTATIONS OF >>>>>> THE LIST. >>>>> >>>>>> >>>>>> So, the first digit of the list can be... well anything, so the >>>>>> antidiagonal starts with anything.. >>>>>> then the second digit of the second real can be anything, so the >>>>>> antidiagonals next digit is anything.. >>>>> >>>>> The problem with that is before you can permute a list, you have to >>>>> prove that a list exists. If it doesn't exist, you can't permuted it. >>>>> Cantor assumes that a list exists, and then proceeds to show that that >>>>> leads to a contradiction. Which is fair enough. But you can't just >>>>> assume the list exists, and then use arguments about permuting the >>>>> list to prove that it exists, or to negate the proof that it doesn't >>>>> exist. That's circular. >>>>> >>>>> <snipped rest based on false premise> >>>>> >>>>> Sylvia. >>>> >>>> So you can form an anti-diagonal on a hypothetical list like the >>>> computable reals >>>> but you can't reorder that list? >>>> >>>> If you take the 2nd digit of the 2nd real, then the 1st digit of the 1st >>>> real, then >>>> the remaining nth digit of the nth reals, the remainder of the usual >>>> diagonal, that won't work? >>>> >>>> 0. _ x _ _ _ _ _ >>>> 0. x _ _ _ _ _ _ >>>> 0. _ _ x _ _ _ _ >>>> 0. _ _ _ x _ _ _ >>>> 0. _ _ _ _ x _ _ >>>> 0. _ _ _ _ _ x _ >>>> 0. _ _ _ _ _ _ x >>>> >>>> Does not an anti-diagonal make! >>>> >>>> Herc >>> >>> It would also be a number that is not in the list. What difference >>> would that make? >>> >>> As usual, it's unclear where you think you're going with this. >>> >>> OK.... >>> >>> You can't permute a list of reals, because there is no such list. >>> >>> If could construct a list of all computable reals, then you could >>> permute the list, generating as many anti-diagonals as you like, and >>> construct thereby numbers that are not in the list of computable >>> reals. All you would be doing is demonstrating that that there are >>> reals that are not computable. >>> >>> Similarly, you can permute a list of computable reals so that any >>> desired number, computable or otherwise, lies on the diagonal, but the >>> existence of the number on the diagonal doesn't mean that it's >>> computable. >>> >>> Or you can construct a list, finite or otherwise, of a subset of the >>> reals, and then use anti-diagonalisation to contruct humbers that are >>> not in the list. You could then add them to the list, and repeat. You >>> end up with an arbitrary countable subset of the reals, which is >>> totally uninteresting. >>> >>> Sylvia. >>> >> >> >> Actually if a real list contains 0.111... it's difficult to get a >> diagonal = 0.222... > > Did I say a subset of computable reals? No, I didn't. Did I use the word > "subset" in the following paragraph? Yes, I did. So am I drawing a > distinction? Yes, most likely. > >> If you COULD set the diagonal to ANYTHING then it would prove my point. > > Why would it? A number is not computable merely because it appears on a > diagonal. > >> >> Anyway you were going on about a false premise, now it's not interesting, >> so it doesn't matter what the claim is you refute it illogically. >> >> At any rate, your interest is gone so we are finished. > > Actually, what appears to happen at this point is that you start a new > thread on exactly the same topic. > > Sylvia. What? You said you could make ANY diagonal with selective permutation. >> Similarly, you can permute a list of computable reals so that any >> desired number, computable or otherwise, lies on the diagonal, I corrected this, and you backpeddle and make further ad homs. I compared you to Wally which is "abuse" but you ridicule my posts every time. Having discussion with you is fruitless, you duck and weave and make general criticisms dismissing points you can't address, change you mind and don't correlate anything with your previous posts. You just forget what the discussion is about and shoot from the hip on the current paragraph. Herc
From: Mike Terry on 20 Jun 2010 21:48 "Sylvia Else" <sylvia(a)not.here.invalid> wrote in message news:887pi5F9g6U1(a)mid.individual.net... > On 21/06/2010 2:40 AM, |-|ercules wrote: > > "Sylvia Else" <sylvia(a)not.here.invalid> wrote > >> On 20/06/2010 7:42 PM, |-|ercules wrote: > >>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... > >>>> On 20/06/2010 5:03 PM, |-|ercules wrote: > >>>>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... > >>>>>> On 20/06/2010 12:42 PM, |-|ercules wrote: > >>>>>>> "George Greene" <greeneg(a)email.unc.edu> wrote > >>>>>>>>> Is it a *NEW DIGIT SEQUENCE* or not? > >>>>>>>> > >>>>>>>> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was > >>>>>>>> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! > >>>>>>> > >>>>>>> > >>>>>>> But you keep saying the anti-diagonal is NEW and ignoring me when > >>>>>>> I say it's not a new digit sequence. > >>>>>>> > >>>>>>> Then you repeat Cantor's proof again and again that it's NEW. > >>>>>>> > >>>>>>> You use terms NEW and NOT ON THE LIST, but evade me when I challenge > >>>>>>> whether it contains any new digit sequence. > >>>>>> > >>>>>> And you just ignore the point that it must be new because it > >>>>>> demonstrably isn't in the list. > >>>>>> > >>>>>> Sylvia. > >>>>> > >>>>> All you demonstrated was > >>>> > >>>> In what sense could a number be said to be in a list if it doesn't > >>>> match any element of the list? > >>> > >>> > >>> What sense is a number that's only definition is to not be on a list? > >>> > >>> That may seem less concrete than your question, but it's worth thinking > >>> what "anti-diagonals" entail. > >>> > >>> You're not just constructing 0.444454445544444445444.. a 4 for every non > >>> 4 digit and a 5 for a 4. > >>> > >>> You're constructing ALL 9 OTHER DIGITS to the diagonal digits. > >> > >> Why? We only need one number that's not in the list. Doing what you > >> suggest just creates many more numbers that are not in the list. > >> > >>> > >>> And it's not just the diagonal, it's the diagonal of ALL PERMUTATIONS OF > >>> THE LIST. > >> > >>> > >>> So, the first digit of the list can be... well anything, so the > >>> antidiagonal starts with anything.. > >>> then the second digit of the second real can be anything, so the > >>> antidiagonals next digit is anything.. > >> > >> The problem with that is before you can permute a list, you have to > >> prove that a list exists. If it doesn't exist, you can't permuted it. > >> Cantor assumes that a list exists, and then proceeds to show that that > >> leads to a contradiction. Which is fair enough. But you can't just > >> assume the list exists, and then use arguments about permuting the > >> list to prove that it exists, or to negate the proof that it doesn't > >> exist. That's circular. > >> > >> <snipped rest based on false premise> > >> > >> Sylvia. > > > > So you can form an anti-diagonal on a hypothetical list like the > > computable reals > > but you can't reorder that list? > > > > If you take the 2nd digit of the 2nd real, then the 1st digit of the 1st > > real, then > > the remaining nth digit of the nth reals, the remainder of the usual > > diagonal, that won't work? > > > > 0. _ x _ _ _ _ _ > > 0. x _ _ _ _ _ _ > > 0. _ _ x _ _ _ _ > > 0. _ _ _ x _ _ _ > > 0. _ _ _ _ x _ _ > > 0. _ _ _ _ _ x _ > > 0. _ _ _ _ _ _ x > > > > Does not an anti-diagonal make! > > > > Herc > > It would also be a number that is not in the list. What difference would > that make? > > As usual, it's unclear where you think you're going with this. > > OK.... > > You can't permute a list of reals, because there is no such list. > > If could construct a list of all computable reals, then you could > permute the list, generating as many anti-diagonals as you like, and > construct thereby numbers that are not in the list of computable reals. > All you would be doing is demonstrating that that there are reals that > are not computable. > > Similarly, you can permute a list of computable reals so that any > desired number, computable or otherwise, lies on the diagonal, but the > existence of the number on the diagonal doesn't mean that it's computable. This isn't right. If you could do that, you could arrange for the antidiagonal of the permuted list to be in the list, which would be a contradiction. (So you can't do it!) Of course by permuting the list before taking the antidiagonal you can create a large supply of different numbers which are ALL not in the original list, but there ARE constraints on the numbers you can produce this way. As you say, none of this matters for Cantors proof, as it only needs ONE such number - producing 100 missing numbers could be called "misplaced mathematical enthusiasm". > > Or you can construct a list, finite or otherwise, of a subset of the > reals, and then use anti-diagonalisation to contruct humbers that are > not in the list. You could then add them to the list, and repeat. You > end up with an arbitrary countable subset of the reals, which is totally > uninteresting. > > Sylvia. >
From: |-|ercules on 20 Jun 2010 22:03 "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote >> Similarly, you can permute a list of computable reals so that any >> desired number, computable or otherwise, lies on the diagonal, but the >> existence of the number on the diagonal doesn't mean that it's computable. > > This isn't right. If you could do that, you could arrange for the > antidiagonal of the permuted list to be in the list, which would be a > contradiction. (So you can't do it!) I pointed this out twice, apparently we misunderstood the woman. Herc
From: Sylvia Else on 20 Jun 2010 22:29
On 21/06/2010 11:48 AM, Mike Terry wrote: > "Sylvia Else"<sylvia(a)not.here.invalid> wrote in message > news:887pi5F9g6U1(a)mid.individual.net... >> On 21/06/2010 2:40 AM, |-|ercules wrote: >>> "Sylvia Else"<sylvia(a)not.here.invalid> wrote >>>> On 20/06/2010 7:42 PM, |-|ercules wrote: >>>>> "Sylvia Else"<sylvia(a)not.here.invalid> wrote ... >>>>>> On 20/06/2010 5:03 PM, |-|ercules wrote: >>>>>>> "Sylvia Else"<sylvia(a)not.here.invalid> wrote ... >>>>>>>> On 20/06/2010 12:42 PM, |-|ercules wrote: >>>>>>>>> "George Greene"<greeneg(a)email.unc.edu> wrote >>>>>>>>>>> Is it a *NEW DIGIT SEQUENCE* or not? >>>>>>>>>> >>>>>>>>>> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was >>>>>>>>>> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! >>>>>>>>> >>>>>>>>> >>>>>>>>> But you keep saying the anti-diagonal is NEW and ignoring me when >>>>>>>>> I say it's not a new digit sequence. >>>>>>>>> >>>>>>>>> Then you repeat Cantor's proof again and again that it's NEW. >>>>>>>>> >>>>>>>>> You use terms NEW and NOT ON THE LIST, but evade me when I > challenge >>>>>>>>> whether it contains any new digit sequence. >>>>>>>> >>>>>>>> And you just ignore the point that it must be new because it >>>>>>>> demonstrably isn't in the list. >>>>>>>> >>>>>>>> Sylvia. >>>>>>> >>>>>>> All you demonstrated was >>>>>> >>>>>> In what sense could a number be said to be in a list if it doesn't >>>>>> match any element of the list? >>>>> >>>>> >>>>> What sense is a number that's only definition is to not be on a list? >>>>> >>>>> That may seem less concrete than your question, but it's worth > thinking >>>>> what "anti-diagonals" entail. >>>>> >>>>> You're not just constructing 0.444454445544444445444.. a 4 for every > non >>>>> 4 digit and a 5 for a 4. >>>>> >>>>> You're constructing ALL 9 OTHER DIGITS to the diagonal digits. >>>> >>>> Why? We only need one number that's not in the list. Doing what you >>>> suggest just creates many more numbers that are not in the list. >>>> >>>>> >>>>> And it's not just the diagonal, it's the diagonal of ALL PERMUTATIONS > OF >>>>> THE LIST. >>>> >>>>> >>>>> So, the first digit of the list can be... well anything, so the >>>>> antidiagonal starts with anything.. >>>>> then the second digit of the second real can be anything, so the >>>>> antidiagonals next digit is anything.. >>>> >>>> The problem with that is before you can permute a list, you have to >>>> prove that a list exists. If it doesn't exist, you can't permuted it. >>>> Cantor assumes that a list exists, and then proceeds to show that that >>>> leads to a contradiction. Which is fair enough. But you can't just >>>> assume the list exists, and then use arguments about permuting the >>>> list to prove that it exists, or to negate the proof that it doesn't >>>> exist. That's circular. >>>> >>>> <snipped rest based on false premise> >>>> >>>> Sylvia. >>> >>> So you can form an anti-diagonal on a hypothetical list like the >>> computable reals >>> but you can't reorder that list? >>> >>> If you take the 2nd digit of the 2nd real, then the 1st digit of the 1st >>> real, then >>> the remaining nth digit of the nth reals, the remainder of the usual >>> diagonal, that won't work? >>> >>> 0. _ x _ _ _ _ _ >>> 0. x _ _ _ _ _ _ >>> 0. _ _ x _ _ _ _ >>> 0. _ _ _ x _ _ _ >>> 0. _ _ _ _ x _ _ >>> 0. _ _ _ _ _ x _ >>> 0. _ _ _ _ _ _ x >>> >>> Does not an anti-diagonal make! >>> >>> Herc >> >> It would also be a number that is not in the list. What difference would >> that make? >> >> As usual, it's unclear where you think you're going with this. >> >> OK.... >> >> You can't permute a list of reals, because there is no such list. >> >> If could construct a list of all computable reals, then you could >> permute the list, generating as many anti-diagonals as you like, and >> construct thereby numbers that are not in the list of computable reals. >> All you would be doing is demonstrating that that there are reals that >> are not computable. >> >> Similarly, you can permute a list of computable reals so that any >> desired number, computable or otherwise, lies on the diagonal, but the >> existence of the number on the diagonal doesn't mean that it's computable. > > This isn't right. If you could do that, you could arrange for the > antidiagonal of the permuted list to be in the list, which would be a > contradiction. (So you can't do it!) I don't see that. If a number isn't in a list, then it's not in any permutation of the list either. > > Of course by permuting the list before taking the antidiagonal you can > create a large supply of different numbers which are ALL not in the original > list, but there ARE constraints on the numbers you can produce this way. Well, it's true that I rather assumed, without proof, that any number can be created in the diagonal. But I don't think anything relevant turns on that. > > As you say, none of this matters for Cantors proof, as it only needs ONE > such number - producing 100 missing numbers could be called "misplaced > mathematical enthusiasm". > Herc certainly seems to have that :) Sylvia. |