'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 27 Mar 2010 14:14 Alan Smaill says... >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). Is that because they feel that there is something different about the Godel sentence than other sentences of arithmetic of similar complexity, or because they would equally well refrain from saying that other arithmetic statements are true? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 27 Mar 2010 14:15 Nam Nguyen says... > >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. Well, it seemed perfectly relevant (and precise) to me. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 27 Mar 2010 14:26 Daryl McCullough wrote: > Nam Nguyen says... >> 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. > > Well, it seemed perfectly relevant (and precise) to me. Well, you know that's not a basis to further discussion about foundation of FOL reasoning. In any rate, do you think if there's any _valid_ context that "G(PA) can be arithmetically false" is true? (Hope that you'd agree this is a Yes/No question.)
From: Alan Smaill on 27 Mar 2010 14:52 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > Alan Smaill says... > >>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). > > Is that because they feel that there is something different about > the Godel sentence than other sentences of arithmetic of similar > complexity, or because they would equally well refrain from saying > that other arithmetic statements are true? I don't think it's exactly either of these -- they have a general tendency towards formalisation as first and foremost about proving theorems, so that truth only matters insofar as it's a problem if the axioms are inconsistent (which even then can be expressed in terms of provability). So Goedel shows a limitation on our ability to provide complete consistent axiomatisations -- no need to mention truth, & better to avoid foundational squabbles. > -- > Daryl McCullough > Ithaca, NY > -- Alan Smaill email: A.Smaill at ed.ac.uk School of Informatics tel: 44-131-650-2710 University of Edinburgh
From: Nam Nguyen on 27 Mar 2010 14:55
Nam Nguyen wrote: > Daryl McCullough wrote: >> Nam Nguyen says... >>> 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. >> >> Well, it seemed perfectly relevant (and precise) to me. > > Well, you know that's not a basis to further discussion about foundation > of FOL reasoning. In any rate, do you think if there's any _valid_ context > that "G(PA) can be arithmetically false" is true? (Hope that you'd agree > this is a Yes/No question.) There's reason why the word "cranks" has a different meaning than "standard mathematicians, logicians" and I believe the difference is genuine. It's just that the later somehow believe that they're aways invincible in their methods of reasoning and they'd would slam the door shut on a slightest hint their methods could be wrong. They know the 1-many problem and yet somehow they could convince themselves they'd fully understand the infinite complexity of the natural numbers. Aren't there any conservative, objective, and rational mathematicians/logicians left in these forums to further discussions about the current state of FOL reasoning? |