In a Consistent System, Can a True Sentence Imply a False One?
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From: MoeBlee on 31 Jul 2010 15:39 To Nam Nguyen: I'd like to ask you a question about your own thoughts on your interaction in this forum. Over and over again, we find you in dramatic disagreement with a number of people who are well informed (some of them professionals and some at a professional level of knowledge even if not actual professionals; by the way, I don't include myself) about logic - sometimes to the point that some of these people just give up trying to reason with you. This is a distinct pattern: one controversy after another after another in which you have your own staunch position on some matter in logic while a certain number of professional logicians and others well informed make post after post with their arguments as to what is incorrect in your position. And this has been going on for years, and not just in a few instances. Now, I'm not arguing that you are incorrect merely on the basis that you disagree with certain people. That's a different matter. What I wonder about though is what thoughts go through your mind when this happens over and over. Do you ever wonder why, over and over and for years, you're so often alone in disagreement with some number of professional logicians? Do you think that it's a fluke in the way things are? Do you think it's just that by some fluke these people happen not to understand these particular matters while you do? Or, Im asking, do you at least think to yourself that it is rather an odd thing that for years and over and over you hold to certain positions while a group of professionals become exasperated trying to convince you that you're burdened with a mistaken notion or even worse that you're quite confused about the subject in general? Again, I'm not arguing that you should change your positions MERELY because some varying number of professional logicians point our their disagreement. Rather, I'm just asking why you think this situation occurs so often. What do you think accounts for it? This post is not just a putdown of you. I'm sincerely interested in what you think about this. MoeBlee
From: MoeBlee on 31 Jul 2010 15:50 On Jul 31, 12:31 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > "Lighten up," said by Aatu Koskensilta, belongs in a class with "Hey, > > I'm feeling you on that, bro," said by Kurt Godel, or something like > > that. > > This is either slander or libel -- I never remember which is which -- > and probably actionable. I expect a prompt and groveling apology. Should > an apology not prove forthcoming, I will be forced to pout in impotent > anger, and possibly stomp my feet. You're such a pushover. Now why don't you go read a logic book and LEARN something about this subject, you veggie eating, mineral water-bathing far-north hemisphere person! Libelously yours, MoeBlee
From: Aatu Koskensilta on 31 Jul 2010 15:51 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > "Waste of time" is rather subjective though and doesn't really > portrait anything much on the surface of it, I'm afraid. Everyone must of course decide for themselves what is or is not a waste of their time. I put it to you the fact that many seemingly intelligent and mathematically competent people appear to have concluded it's a waste of their time to discuss logical matters with you might give you pause, that you might profitably ponder why this should be. > It has been a long past, distant from where I'm standing in this bank of > the time river, it seems. Would it be OK with you if you could refresh my > memory again, either in the forum or my email, what they might be? Certainly. I'll return to this in a later post, in news as my reflections and advice are quite general. In the meanwhile, I suggest you mull over Torkel's wise if somewhat whimsical words: In order to count as a major nuisance, it is of course not enough just to consistently contradict people. You must locate an actual weak spot in their argument, and criticize that weak spot in terms that your opponent will be forced to recognize as justified. Of course this can't always be done. Many posters of a sectarian bent, for example, will not recognize any criticism whatsoever of their argument, or of the argument of their guru, as in any way justified. In such cases you must be content with playing to the gallery. Basically, however, your aim as a major nuisance is to establish intellectual contact with your opponent, opening his eyes to certain facts or difficulties. and something I said to lwalke a while back: [T]here is an important if trivial truth to be appreciated, namely that it is futile to expect people to pay any attention to one's ideas merely as a result of endless bickering about how rotten the accepted ideas are -- rather, what is needed are striking demonstrations of fruitfulness, of conceptual, mathematical, philosophical insights to be gleaned from one's pet ideas. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Sergei Tropanets on 31 Jul 2010 16:05 On Jul 31, 11:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Sergei Tropanets <trop.ser...(a)gmail.com> writes: > > By Godel's second incompleteness theorem we know that PA can't prove > > Con(PA). This implies Con(PA + Con(PA)) and Con(PA + not Con(PA)). > > No it doesn't. It only implies Con(PA + not Con(PA)). You need Sigma-1 > soundness of PA to conclude that PA + Con(PA) is consistent. By Con(T) I understand (and I suppose Charlie also does so) SYNTACTIC consistency and I use the following general result: if it is not the case that T |-- not Q then Con(T+Q). So of course by 2nd incompleteness <<This implies Con(PA + Con(PA)) and Con(PA + not Con(PA)).>> Where I need Sigma-1 soundness of PA there? Is it used in 2nd incompleteness theorem? > > So if we think of PA and its theorems as true statements than we have > > to think so also of PA + not Con(PA). > > We have to do no such thing. PA + not Con(PA) is a consistent theory > that proves arithmetical falsehoods. Sorry, of course I meant "consistent" not "true". (Because "consistent" has been mentioned in the question). > > Then true (and consistent) theory PA + not Con(PA) would prove its own > > inconsistency which may be interpreted as something false. This was > > one of the Godel's key arguments against Hilbert's program: formal > > system may be syntactically consistent but semantically inconsistent! > > No it wasn't. It was. In fact, Godel set forward this argument at his first presentation of 1st incompleteness theorem at philosophy conference. Although he used G="i am unprovable" and PA+not G not the Con(PA) and PA+not Con(PA). (The 2nd theorem was discovered just after that talk by Godel and von Neumann independently). Sergei Tropanets
From: Sergei Tropanets on 31 Jul 2010 16:15
On Jul 31, 11:09 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > So when Andrew Wiles proved Fermat's last theorem he did so by finding a > "syntactical proof through rules of inference"? Are you sure he did prove it? Sergei Tropanets |