halting problem proof, via diagonalization?
in [Theory]
|
Prev: Integer factorization reduction to SAT
Next: Solutions manual to Microeconomic Theory Solution Manual - Mas-Colell
From: cplxphil on 8 Aug 2008 17:29 I have no comment on the argument itself; I'm sure it's incorrect, I just haven't read it in detail. I just wanted to say that while I haven't been on this forum very long, this is by far the most vicious discussion of theoretical computer science I've ever seen! I'm glad no one reacts that strongly to the incorrect arguments I've posted.... -Phil
From: julio on 8 Aug 2008 18:09 On 8 Aug, 19:10, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 8, 10:23 am, ju...(a)diegidio.name wrote: > > > > To make a long story short: we are not interested in the "limit entry" > > > (whatever that may be), but in the fact that the diagonal differs from > > > each and any entry in the list. > > > Interested or not, you cannot just dismiss it. The "limit" entry makes > > just as much sense as the above (or the below) "[for] every entry in > > the list". Is the list "infinite" or is it not? I am saying, yours is > > not a valid objection. > > Whether the list is infinite or NOT, the anti-diagonal is not on the > list. You keep saying so as if it were the starting point, while that's the result to get. The first problem is indeed in your general way of argumentation. > Otherwise, you are free to mention EXACTLY what premises in the > proof or what rule of logic used you disagree with. The diagonal argument is broken in its essence, and this comes before your axioms, another problem of yours. But I am not trying to disprove *you*. Here I have provided the simpliest of the arguments based on the naturals and induction, to build the very same list and the very same diagonal as Cantor's, with quite opposite results: and they are very natural results. You have not been able to provide a meaningful objection that is one. Somewhere else, though, you have finally admitted that there exists such a thing as constructive theories. So all the fuzz for nothing? > Then we could at > least say, "Yes, given that you don't accept those premises and rules > of logic, you don't have to accept the theorem that is proved from > them. Now, what premises and logic DO you accept for the purpose of > deriving mathematical conclusions?" See, that would be productive. But > we can't get there if all you do is use UNDEFINED hocus pocus about > "limit induction", and amphiboly in your symbolisms, along with > recriminations about what dishonest idiots mathematicians are. Let's be serious. I have given the simpliest of the arguments. Can you spot any significant flaw? If not, then I guess it's really the time to revisit Cantor's... and then I can see what is the fuzz for. How do you dismiss constructive theories? I'd be very interested in learning what eventually there is to lose. -LV > MoeBlee
From: MoeBlee on 8 Aug 2008 18:41 On Aug 8, 3:09 pm, ju...(a)diegidio.name wrote: > On 8 Aug, 19:10, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Aug 8, 10:23 am, ju...(a)diegidio.name wrote: > > > > > To make a long story short: we are not interested in the "limit entry" > > > > (whatever that may be), but in the fact that the diagonal differs from > > > > each and any entry in the list. > > > > Interested or not, you cannot just dismiss it. The "limit" entry makes > > > just as much sense as the above (or the below) "[for] every entry in > > > the list". Is the list "infinite" or is it not? I am saying, yours is > > > not a valid objection. > > > Whether the list is infinite or NOT, the anti-diagonal is not on the > > list. > > You keep saying so as if it were the starting point, while that's the > result to get. The first problem is indeed in your general way of > argumentation. We state exactly the premises and rules of inference. I've been that saying over and over, because YOU are not responsive to the poing (instead you just complain about the repetition of my answer while you don't ADDRESS my answer). One more time: Theorems of Z set theory are from an explicit set of premises (axioms) and an explicit logic (the inference rules). Yes, if you fault the axioms and/or inference rules, then you don't have to accept the theorems. But that is a separate question from what is or is not a theorem from whatever given set of axioms and inference rules. PLEASE LISTEN THIS TIME: You and I have NO DISAGREEMENT that if you fault the axioms and rules, then you don't have to accept the conclusions. BUT, then an INTELLEGENT discussion starts with you saying just which axioms and rules you don't accept, and then perhaps to tell us what axioms and rules that you instead propose to derive theorems of mathematics. > > Otherwise, you are free to mention EXACTLY what premises in the > > proof or what rule of logic used you disagree with. > > The diagonal argument is broken in its essence, and this comes before > your axioms, another problem of yours. You see. You didn't answer the question. You won't say what axioms or rules you disagree with. Okay, fine, so we just leave it at that. Since the axioms and rules entail a conclusion you disagree with, there is at least one, though unspecified axiom or rule, that you don't accept. > But I am not trying to disprove > *you*. Here I have provided the simpliest of the arguments based on > the naturals and induction, to build the very same list and the very > same diagonal as Cantor's, with quite opposite results: No, you have not. Please, we keep telling you, but you keep evading, that your "argument" doesn't work because it uses an UNDEFINED notion of a 'limit'. We DEFINE the use of limits EXACTLY. But your just SAYING "in the limit case" is not a mathematical argument. It is using mathematical sounding terminology, but without defintion of the sense of a limit that would apply. > and they are > very natural results. You have not been able to provide a meaningful > objection that is one. Please, it is INSULTING that you say that when people have taken the time to SPECIFICALLY articulate the objections to your "argument". > Somewhere else, though, you have finally > admitted that there exists such a thing as constructive theories. So > all the fuzz for nothing? Because, as far as I know, such constructive mathematics does not work by just going around using mathematical sounding terminology such as vacuously spouting about some UNDEFINED "limit case". > > Then we could at > > least say, "Yes, given that you don't accept those premises and rules > > of logic, you don't have to accept the theorem that is proved from > > them. Now, what premises and logic DO you accept for the purpose of > > deriving mathematical conclusions?" See, that would be productive. But > > we can't get there if all you do is use UNDEFINED hocus pocus about > > "limit induction", and amphiboly in your symbolisms, along with > > recriminations about what dishonest idiots mathematicians are. > > Let's be serious. I have given the simpliest of the arguments. Can you > spot any significant flaw? So you will never actually READ the posts given to you, I guess. We've said over and over and over that the flaw is using an UNDEFINED notion of 'limit'. > If not, then I guess it's really the time > to revisit Cantor's... and then I can see what is the fuzz for. > > How do you dismiss constructive theories? I DON'T dismiss well presented constructive theories. And I very much DO appreciate the importance of constructivism in mathematics. What I do dismiss is an argument that just uses mathematical sounding terminology without properly defining it. > I'd be very interested in > learning what eventually there is to lose. Constructivism has drawbacks and advantages, while classical mathematics also has its drawbacks and advantages. That is a rich area for discussion, but it requires at least two things you are not willing to bring: (1) An understanding of the basics of the subject, and (2) a willingness to rise from mindless recriminations about mathematicians and instead pay attention at least long enough so that you understand what various mathematicians - constructivst and non- constructivist - are actually saying. MoeBlee
From: MoeBlee on 8 Aug 2008 18:49 On Aug 8, 3:41 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > constructive theories? P.S. It is decidedy NOT constructivist to assert the existence of a "limit" without CONSTRUCTING it (and just saying "the limit case" is decidedly not a construction), let alone, not even defining what possible sense you might mean by a "limit" in such a context. MoeBlee
From: julio on 8 Aug 2008 18:50
On 8 Aug, 23:41, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 8, 3:09 pm, ju...(a)diegidio.name wrote: > > > > > > > On 8 Aug, 19:10, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Aug 8, 10:23 am, ju...(a)diegidio.name wrote: > > > > > > To make a long story short: we are not interested in the "limit entry" > > > > > (whatever that may be), but in the fact that the diagonal differs from > > > > > each and any entry in the list. > > > > > Interested or not, you cannot just dismiss it. The "limit" entry makes > > > > just as much sense as the above (or the below) "[for] every entry in > > > > the list". Is the list "infinite" or is it not? I am saying, yours is > > > > not a valid objection. > > > > Whether the list is infinite or NOT, the anti-diagonal is not on the > > > list. > > > You keep saying so as if it were the starting point, while that's the > > result to get. The first problem is indeed in your general way of > > argumentation. > > We state exactly the premises and rules of inference. I've been that > saying over and over, because YOU are not responsive to the poing > (instead you just complain about the repetition of my answer while you > don't ADDRESS my answer). > > One more time: > > Theorems of Z set theory are from an explicit set of premises (axioms) > and an explicit logic (the inference rules). Yes, if you fault the > axioms and/or inference rules, then you don't have to accept the > theorems. But that is a separate question from what is or is not a > theorem from whatever given set of axioms and inference rules. > > PLEASE LISTEN THIS TIME: > > You and I have NO DISAGREEMENT that if you fault the axioms and rules, > then you don't have to accept the conclusions. BUT, then an > INTELLEGENT discussion starts with you saying just which axioms and > rules you don't accept, and then perhaps to tell us what axioms and > rules that you instead propose to derive theorems of mathematics. > > > > Otherwise, you are free to mention EXACTLY what premises in the > > > proof or what rule of logic used you disagree with. > > > The diagonal argument is broken in its essence, and this comes before > > your axioms, another problem of yours. > > You see. You didn't answer the question. You won't say what axioms or > rules you disagree with. Okay, fine, so we just leave it at that. > Since the axioms and rules entail a conclusion you disagree with, > there is at least one, though unspecified axiom or rule, that you > don't accept. > > > But I am not trying to disprove > > *you*. Here I have provided the simpliest of the arguments based on > > the naturals and induction, to build the very same list and the very > > same diagonal as Cantor's, with quite opposite results: > > No, you have not. Please, we keep telling you, but you keep evading, > that your "argument" doesn't work because it uses an UNDEFINED notion > of a 'limit'. We DEFINE the use of limits EXACTLY. But your just > SAYING "in the limit case" is not a mathematical argument. It is using > mathematical sounding terminology, but without defintion of the sense > of a limit that would apply. > > > and they are > > very natural results. You have not been able to provide a meaningful > > objection that is one. > > Please, it is INSULTING that you say that when people have taken the > time to SPECIFICALLY articulate the objections to your "argument". > > > Somewhere else, though, you have finally > > admitted that there exists such a thing as constructive theories. So > > all the fuzz for nothing? > > Because, as far as I know, such constructive mathematics does not work > by just going around using mathematical sounding terminology such as > vacuously spouting about some UNDEFINED "limit case". > > > > Then we could at > > > least say, "Yes, given that you don't accept those premises and rules > > > of logic, you don't have to accept the theorem that is proved from > > > them. Now, what premises and logic DO you accept for the purpose of > > > deriving mathematical conclusions?" See, that would be productive. But > > > we can't get there if all you do is use UNDEFINED hocus pocus about > > > "limit induction", and amphiboly in your symbolisms, along with > > > recriminations about what dishonest idiots mathematicians are. > > > Let's be serious. I have given the simpliest of the arguments. Can you > > spot any significant flaw? > > So you will never actually READ the posts given to you, I guess. We've > said over and over and over that the flaw is using an UNDEFINED notion > of 'limit'. > > > If not, then I guess it's really the time > > to revisit Cantor's... and then I can see what is the fuzz for. > > > How do you dismiss constructive theories? > > I DON'T dismiss well presented constructive theories. And I very much > DO appreciate the importance of constructivism in mathematics. What I > do dismiss is an argument that just uses mathematical sounding > terminology without properly defining it. > > > I'd be very interested in > > learning what eventually there is to lose. > > Constructivism has drawbacks and advantages, while classical > mathematics also has its drawbacks and advantages. That is a rich area > for discussion, but it requires at least two things you are not > willing to bring: (1) An understanding of the basics of the subject, > and (2) a willingness to rise from mindless recriminations about > mathematicians and instead pay attention at least long enough so that > you understand what various mathematicians - constructivst and non- > constructivist - are actually saying. So you cannot add anything significant, you just reiterate your insults to my intelligence. Never mind, it was anyway very "instructive". Now, from Cantor to Goedel... See you soon. -LV > MoeBlee |