'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 10 Mar 2010 01:37 MoeBlee wrote: > (1) Anyone who has studied mathematical logic already recognizes that > a sentence that is neither logically true nor logically false has > models in which it is true and models in which it is false. Did you yourself quantifying-ly inspect uncountably many models of, say, PA to know that? > (2) As far > as calling one model 'the standard model', if you object to lack of > neutrality, then we could just as easily refer to it as 'the blandard > model' or whatever. I suppose so. > But what would be ridiculous would be to require > every mathematician to be as interested in EVERY possible model as > much as he or she is interested in certain particular models. You've missed the key point and that's why you don't seem to understand the objection to lack of neutrality: each model has its own "peculiarity", hence _each equally deserves_ the name "standard". We're *not* talking requiring anybody anything! At least the 4 principles don't require so. > People > focus on certain mathematical objects, questions, etc. for a variety > of reasons. It is not, and should not be required that mathematicians > promise to do what is not even humanly, not even finitely, possible to > do, such as study EVERY SINGLE model with as much interest and > attention as every other single model. But the concept of the naturals number is "not even humanly, not even finitely, possible", don't you think? Why do people then force themselves to study it? > (3) Such matters as this kind > of neutrality are not even themselves formal mathematical questions > but rather heuristic and ... ethical questions. Iirc, I introduced the word "symmetry", not "neutrality". So I'm not sure why the word "ethical" has to be here. > > In any given context, all two mathematicians have to do is agree as to > which particular terminology, axioms, and rules are in play in that > context. If some other mathematician wishes to call some other model > the 'standard model' then all he has to do is make it clear that the > terminology 'the standard model' now refers to such and such. So then you'd agree the following meta theorem: For any formal system T capable to carry basic notions of arithmetic there's a formula that's false but not provable in T. is true? > Of > course, this may make the discourse awkward (at least at first), but > in principle there is nothing stopping it. Agree!
From: Nam Nguyen on 10 Mar 2010 02:40 Nam Nguyen wrote: > MoeBlee wrote: > >> (3) Such matters as this kind >> of neutrality are not even themselves formal mathematical questions >> but rather heuristic and ... ethical questions. > > Iirc, I introduced the word "symmetry", not "neutrality". So I'm > not sure why the word "ethical" has to be here. My mistake: I did use the word "_neutral_" in a post. But the sense I used the word had nothing to do with "ethical", so I don't know the relevance of "ethical" here.
From: Nam Nguyen on 10 Mar 2010 02:45 Nam Nguyen wrote: > Transfer Principle wrote: > >> The point I'm trying >> to make is that if ZFC proves a formula F, there's no reason why >> there can't be another _dual_ theory, one which is just as powerful >> and elegant as ZFC, which proves ~F. > > I wouldn't call that "another _dual_ theory": just "another theory". > Also, "elegant" is a subjective notion, not a mathematical (FOL) notion. > > Principle 3. actually is just a generalized reflection of a known > fact in FOL: if a formal system is consistent, then there exists a > different consistent formal system, all of which has nothing to > do with "elegance". or with "powerful".
From: Daryl McCullough on 10 Mar 2010 06:58 Nam Nguyen says... > >MoeBlee wrote: > >> (1) Anyone who has studied mathematical logic already recognizes that >> a sentence that is neither logically true nor logically false has >> models in which it is true and models in which it is false. > >Did you yourself quantifying-ly inspect uncountably many models >of, say, PA to know that? The beauty of mathematical proof is that you can be certain of the truth of a universal statement without checking every instance. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Mar 2010 07:01
Nam Nguyen says... >MoeBlee wrote: >> What I mean is: What mathematical theorems can you prove? Come on, if >> I don't have to prove any mathematical theorems, then I wouldn't need >> to propose any principles AT ALL! > >You still could prove a lot of theorems under an edifice that complies >to these principles. In fact if you take the current FOL and strip off >anything that are (or _depends on_) the naturals or models from the >role of _inference_ then you'd obtain an edifice that would conform >to these 4 principles (at least from a first glance at the matter). Give an example of a nontrivial theorem in such a system. I don't think anyone would be interested in it, not even you. -- Daryl McCullough Ithaca, NY |