'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 27 Mar 2010 13:25 Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: > >>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>> of the language of PA. >> So you've agreed "G(PA) can be arithmetically false"? > > It is false in nonstandard models of PA. Why don't we make it more precise. When we say F, a formula written in the language of arithmetic, is true or false _by default_ we mean it's being arithmetically true or false: i.e. true or false in the natural numbers. So we're *not* talking about F is being true/false in any general kind of models here. The point is Alan said he wouldn't know what an absolute truth of G(PA) would mean and I've implicitly defined it for him, and here is the explicit version: There's _no other_ context in which the meta statement "G(PA) is arithmetically true in the natural number" would be false. The question I was hoping you'd answer one way or another is whether or not there's a context in which the meta statement: "G(PA) is arithmetically true in the natural number" would be _false_ ? If your answer is "yes", then the [arithmetically-in-the-natural- number] truth of G(PA) is a relative notion. Otherwise it's an absolute notion. Which answer would you have? And perhaps why?
From: Daryl McCullough on 27 Mar 2010 13:48 Nam Nguyen says... > >Daryl McCullough wrote: >> Nam Nguyen says... >>> Daryl McCullough wrote: >> >>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>> of the language of PA. >>> So you've agreed "G(PA) can be arithmetically false"? >> >> It is false in nonstandard models of PA. > >Why don't we make it more precise. What I said was already perfectly precise. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 27 Mar 2010 14:03 Jim Burns wrote: > Nam Nguyen wrote: >> Jim Burns wrote: >>> Nam Nguyen wrote: >>>> Marshall wrote: >>>>> On Mar 26, 8:57 pm, Nam Nguyen >>>>> <namducngu...(a)shaw.ca> wrote: >>> >>>>>> Or you're just full of babbling words with >>>>>> no technical substance, as usual? [It seems >>>>>> like a habit of yours that when you couldn't >>>>>> technically counter your opponent's argument >>>>>> then you just call him a mad dog!] >>>>> >>>>> I have never called you a mad dog that I can >>>>> recall. My term for you is "talentless bufoon." >>>>> So much more apropos. >>>> >>>> "A mad dog" is just an idiom expression. There's >>>> a saying like "To kill a dog just call it a mad >>>> dog". So your "talentless bufoon", here, is the >>>> same as "mad dog". >>>> >>>> But you missed my point: in argument here, Marshall >>>> has been "full of babbling words with no technical >>>> substance", like an *intellectual clown*. >>> >>> So, are you calling Marshall a mad dog here? >>> Apparently, you only object to that /sometimes/. >> >> There are real mad dogs and there are real non-mad dogs. >> There difference is in the real symptoms they really do >> or do not exhibit. Naturally. > > Please correct me if I misinterpret what you are saying: > > Marshall called you a talentless buffoon, and that > was wrong, because you are not a talentless buffoon. I don't know know exactly what talent or talentless be, it's subjective definition. But keeping attributing people with subjective names in a middle of technical argument while not talking about the technical matters at hand _is insincere and is wrong_. And that's my opinion and that's how I'd would react. Of course you have the right to have a different opinion. > You called Marshall an intellectual clown, and that > was okay, because he is an intellectual clown. > > I'm a little disappointed, because I thought your > argument went a little deeper, that it was an > objection to shouting down unpopular views by > burying them under a pile of nasty accusations. Your reaction seems to be quick here. The _only_ sentence that he posted about conversations I had (not with him) but with another poster about foundational issues was: >> That Moe has failed to do so in a way that you can >> understand is a failing, but it's not Moe's failing. Since that's the lone sentence in that post (meaning he didn't even have any thing to back up what he posted), it's a pile of accusation (my alleged "stupidity") he made. would that be a "deeper" argument you're looking for? For me, I do have reason to call him an unintellectual clown. And Iirc, this isn't the first time he has exhibited such character. > That would have made you a hypocrite, of course, > for trying to do the very same thing to Marshall. As I've justed mentioned, I do have reasons and I've presented the reason (and I could present more reasons if you would like). Your alluding me as a hypocrite is *not* warranted! If you want to defend him you should look at his conversations with me in this thread (if not in other threads as well) more closely. > > So all this is just a difference of opinion as to > whether you are a talentless buffoon, on the one > hand, and Marshall is an intellectual clown, > on the other? Then I guess it doesn't matter. > > Jim Burns
From: Alan Smaill on 27 Mar 2010 14:10 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Daryl McCullough wrote: >> Nam Nguyen says... >>> Daryl McCullough wrote: >> >>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>> of the language of PA. >>> So you've agreed "G(PA) can be arithmetically false"? >> >> It is false in nonstandard models of PA. > > Why don't we make it more precise. When we say F, a formula written > in the language of arithmetic, is true or false _by default_ we > mean it's being arithmetically true or false: i.e. true or false > in the natural numbers. So we're *not* talking about F is being > true/false in any general kind of models here. You haven't (and can't) give me an effective way to use this definition to decide truth or falsity in the natural numbers. > The point is Alan said he wouldn't know what an absolute truth > of G(PA) would mean > and I've implicitly defined it for him, and > here is the explicit version: > > There's _no other_ context in which the meta statement > "G(PA) is arithmetically true in the natural number" would > be false. I'm not obliged to accept any particular definition of yours. FWIW you can look at Bourbaki's account of Goedel's incompleteness theorem to note that they studiously avoid saying that the goedel sentence is true (arithaally, absolutely, or in any other way). > The question I was hoping you'd answer one way or another is > whether or not there's a context in which the meta statement: > > "G(PA) is arithmetically true in the natural number" > > would be _false_ ? > > If your answer is "yes", then the [arithmetically-in-the-natural- > number] truth of G(PA) is a relative notion. Otherwise it's an > absolute notion. Why should I believe the only answers are no or yes? Your realist assumptions are showing. > Which answer would you have? And perhaps why? you need a bit more Zen. -- Alan Smaill
From: Nam Nguyen on 27 Mar 2010 14:11
Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Daryl McCullough wrote: >>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>>> of the language of PA. >>>> So you've agreed "G(PA) can be arithmetically false"? >>> It is false in nonstandard models of PA. >> Why don't we make it more precise. > > What I said was already perfectly precise. If you ask me whether or not Pythagoras is provable in some T and I answer you "2+2=4" is true, then what I answer might be precise in certain context but is completely _irrelevant_ in the underlying discussion. Whatever you thought you said "precise" is _not relevant_ in the conversation I had with Alan about the absolute [arithmetical] truth of G(PA). |