The End of an Era - That of The Natural Numbers
in [Math]
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From: Daryl McCullough on 13 Jun 2010 15:09 Nam Nguyen says... > >Daryl McCullough wrote: >> Nam Nguyen says... >>> Daryl McCullough wrote: >>>> No, it doesn't, but I don't actually care what Shoenfield or Tarski >>>> said. What I care about is having a non-stupid definition of "truth >>>> in a model" that applies to models with empty domain. >>> But you shouldn't have worried about that: because that "non-stupid >>> definition" could only render falsehood. >> >> No, a nonstupid definition of "truth in a model" makes >> some statements true, and the rest false. > >The caveat here is when I said "could only render falsehood" I meant >that only in the cases you had refereed as "models with empty domain". Right you have a definition which is nonstupid for models with nonempty domain, but is stupid for models with empty domain. I'd prefer a definition that is nonstupid in all cases. >> I've explained it before. I'm interested in *submodels*. Suppose you >> start with a structure S for a language L and you restrict attention to >> a substructure S' for a sublanguage L' in the following way: >> >> The domain U' of S' consists of all elements of S that satisfy some >> unary predicate D(x). > >We can just top it right here. If the U of S is empty, so is the U' of >S': what is the point of going further? The interesting case is the one in which U is nonempty, but U' may or may not be empty. If U and U' are both nonempty, then we have, for any Phi(x) that does not involve quantifiers: "Ax Phi(x)" is true in the structure with domain U' <-> "Ax D(x) -> Phi(x)" is true in the structure with domain U. There is no reason not to allow this same equivalence to hold in the case in which D(x) is universally false (in which case, U' is empty). There is just no technical justification for your definition. You claim that it's following Tarski and/or Schoenfield. I don't believe you. You misunderstood one or both of them. It's possible that they gave a definition of "truth in a model" that only applies to models with nonempty domain. What I doubt very seriously is that they gave a definition that applies to all models (empty domain or not) but which gives all sentences the value "false" in the case of the empty model. That would be stupid. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 13 Jun 2010 16:52 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > You claim that it's following Tarski and/or Schoenfield. I > don't believe you. You misunderstood one or both of them. Of course he did. I own the Shoenfield book. It is clear that he's confused and thinks that the clause for atomic formulas (either than equations) applies to formulas generally. It's a simple and obvious misunderstanding, but since this is Nam, he will cling to his ignorance as if it were a valuable insight. -- Jesse F. Hughes "Please. I was a philosophy major. Nobody can 'know' anything. And I DO know." -- George Greene embarrasses philosophy majors everywhere.
From: Nam Nguyen on 14 Jun 2010 00:47 Jesse F. Hughes wrote: > stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > >> You claim that it's following Tarski and/or Schoenfield. I >> don't believe you. You misunderstood one or both of them. > > Of course he did. > > I own the Shoenfield book. It is clear that he's confused and thinks > that the clause for atomic formulas (either than equations) applies to > formulas generally. Oh no you're mistaken, Jesse. You're are the one who keeps clinging on that pg. 19 to no avail. That page is _only one part_ of his model definition (even for the case where U is non-empty). The other integral part of Shoenfield's definition of model truth (but he termed it "valid" ) of a _general_ formula is on pg. 18 when he precipitated pg. 19 definition. So no, I was never confused about the clause for atomic formulas on pg. 19: I knew it's an integral part of at least the preceding pg. 18. It's you who got confused what is the precipitating part what is the subordinating part of his presentation between pg 18 - pg 22. > It's a simple and obvious misunderstanding, but > since this is Nam, he will cling to his ignorance as if it were a > valuable insight. Since it's Jesse, it's easy to see how he careless review technical issues in foundations and then clings on attacking his opponent to hide his mistakes.
From: Marshall on 14 Jun 2010 02:11 On Jun 13, 9:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > It's a simple and obvious misunderstanding, but > > since this is Nam, he will cling to his ignorance as if it were a > > valuable insight. > > Since it's Jesse, it's easy to see how he careless review technical > issues in foundations and then clings on attacking his opponent to > hide his mistakes. It's funny how you talk as if you have a sympathetic audience, while the reality is that everyone who has weighed in on the issue thinks you're confused. I have this image of Nam on campus, with a big red rubber nose, orange fright wig, and oversize shoes, standing outside a math building, and solemnly accosting passing students with passages from a coffee-stained, potato chip dusted copy of Schoenfield he found in a dumpster. Marshall
From: Aatu Koskensilta on 14 Jun 2010 08:26
Marshall <marshall.spight(a)gmail.com> writes: > It's funny how you talk as if you have a sympathetic audience, > while the reality is that everyone who has weighed in on the > issue thinks you're confused. I think Nam's dazzled, by very dazzling indeed rays of light. > of Schoenfield he found in a dumpster. Shoenfield, Shoenfield, you brute! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |