In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 30 Jul 2010 22:37 Marshall wrote: > On Jul 30, 6:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Jul 30, 6:31 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> MoeBlee wrote: >>>>> Oh, lordy lordy, I SWORE to you...please, God in heaven, make me >>>>> strong to resist temptation... >>>>> Your mortal servant, >>>> It's kind of hard for us, mortal beings, to know exactly how God would >>>> help one but I'd venture to guess that, in the context of making mathematical >>>> arguments, HE'd advise something like: >>>> Be honest, straight forward, to the points, logical, conforming to the 4 >>>> Principles (Consistency, Compatibility, Symmetry, and Humility). But most >>>> important of all, be kind in wordings and not attacking thy opponents just >>>> because thou are about to loose thy arguments. >>>> Hope that would help you somehow. >>> Since you pretty much don't do any of those things, >> As I advised MoeBlee about "straight forward, to the points", >> what "those things" did you _actually know_ that I don't know? > > I didn't say anything about you not knowing anything. I don't think you understand my simple question. Let me repeat: >> As I advised MoeBlee about "straight forward, to the points", >> what "those things" did you _actually know_ that I don't know? Did I use the word "anything" in the question? (I only asked you to be specific and spell out which "those things", right?) > > >>> your advice comes across as insincere. >> You haven't given credible reasons why! > > Yes I have. You just can't read. Hint: look earlier > in the sentence. Since you are the one who couldn't see that my question doesn't have the word "anything", it'd seem you're the one who couldn't read! > > I wasn't surprised. In fact, I thought you were acting > completely in character: insincere, arrogant, clueless, > smug, etc. etc. What happens to the common sense that I mentioned above: >>>> Be honest, straight forward, to the points, logical, ? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Sergei Tropanets on 30 Jul 2010 22:52 Charlie-Boo wrote: > In a consistent system, can a true sentence imply a false one? I have no precise answer to this question, but the following example may be of some interest in that respect (as far as Charlie believes in consistency of Peano Arithmetic). 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)). But PA + not Con(PA) |-- not Con(PA) and, since |-- not Con(PA) => not Con(PA + not Con(PA)) (exercise) we have PA + not Con(PA) |-- not Con(PA + not Con(PA)). So if we think of PA and its theorems as true statements than we have to think so also of PA + not Con(PA). 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! So just syntactical consistency is not enough for foundation of mathematics! I've known this corollary of 1st incompleteness theorem from Joel David Hamkins. Sergei Tropanets
From: Nam Nguyen on 30 Jul 2010 23:42 Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: >>> Nam Nguyen says... >>> >>>> If you clarified so. Now then, as I asked before, if e.g. I tell you I have >>>> a T that has a disprovable formula in it, would you be able to tell if that >>>> T is consistent, or not? >>> No, because *every* theory (inconsistent or not) has disprovable formulas >>> in it. >> Right. So, would you see why it's odd, not making a lot of sense, >> not to restrict the definition only to the case of consistent theories? > > No, I don't see that at all. OK. I hear you. In a sense definitions, especially in the sense of naming, are very subjective and I could only do as best as I could to convince one. I did provide some reasons why the restriction be warranted or even necessary. Would you care to comment on those reasons? > I can see why you would not want to > spend a lot of time on inconsistent theories---they aren't very > interesting, after all, and they're not good for anything much. This is the fundamental characteristics of the miscommunication here. For years somehow people got the wrong notion that I fall in love, so to speak, with inconsistent theories because I'd believe somehow these theories are good! Again they're wrong: that's simply not the case. At the heart of the matter is I'd like to point out to them that they (notably AK, CM, MoeBlee, etc...) would use the word "proof" of a 1st order formula in 2 different senses: there's the syntactical sense via the rules of inference for sure but they have been talking about proofs through the intuitive knowledge of the natural numbers. What I've tried to say to them is that, among other consequences of having another definition of proof other than the one using the rules, there's an incompatibility (as mentioned in the suggested Principle of Compatibility) between the 2 and this incompatibility would cast down or even render our reasoning incorrect, logically speaking. By going back and forth about what's possible to prove in a syntactically inconsistent theory, and what's impossible to prove in a syntactically consistent one, I'd like to contrast the strength and weakness of both kinds of proof-definition. The hope is that they'd understand more about the Principles Compatibility and Symmetry, for the better of reasoning. So for the n-th time, anyone including I would know what an inconsistent theory is and _NO ONE_ would care to use it, as a FOL system. But as far as matters of foundation are concerned, they have ignored the meaning of inconsistency too long to the point they couldn't recognize what the foundational issues are, in the edifice of FOL reasoning. > But it doesn't make any sense to me to make definitions so that > they don't apply to inconsistent theories. What is the point of that? What does it mean when we say we "prove" a theorem? Of course that's an easy question: to prove a theorem is to find a syntactical proof through rules of inference. Now then, what does it mean to not-be-able-even-in-principle-to-prove a formula doesn't have a proof? If used as is, that's too long a verb, right? But if you use "unprove" that's not quite correct because to "un-prove" could mean to simply show that the proof is incorrect, but doesn't necessarily mean the formula doesn't have a proof. So, among that very long verb, the verb "un-prove", and the verb "dis-prove", what would you pick as the opposite of the verb to "prove"? If you could see this difficulty of choice here, you'd understand why I said "disprove" should be used only in the context of a consistent theory. Because in the case of an inconsistent theory by definition all formulas have proofs already, and it's extremely not making sense to give them another name that has the word "proof" and yet the name doesn't indicate a proof! After all, to "disprove" and to "un-prove" are opposite to the verb to "prove" semantically. And any of these 2 shouldn't be mixed up with to "prove", if we can help it. And we can! -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Aatu Koskensilta on 31 Jul 2010 03:28 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > For example, if I tell you of a theory T and say there's a > "disprovable" formula in T, would you know if T is consistent, or not? No. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 03:33
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > If it stays "just as it was" why do _you_ need to rename/re-characterize > that to something else that would characterize a consistent theory? What on Earth are you on about? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |