The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 29 May 2010 19:40 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Alan Smaill wrote: >>>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>>> >>>>>> Alan Smaill wrote: >>>>>>> Fine, take U = {0,1,2}, and take everything else as above. >>>>>> Well, so far you've only spelled out U (and in effect <'A',U>). >>>>>> You've not spelled out the mapping (ordered pair) <'blue',p_blue> >>>>>> where p_blue is an actual _set_. Iow, if R is p_blue, can you >>>>>> spell out the predicate-set R? >>>>> The set is the extension of the relation R: >>>>> >>>>> { x in U | R(x) } = {1} >>>> Now then, let's extend the language L(t4) to L(T4b) so it >>>> has another 1-ary symbol 'non-blue', and extend T4 into >>>> T4b so it has another axiom: non-blue(c1) <-> ~blue(c1). >>>> >>>> Can we keep the model M4 for T4b? If not what can we keep, >>>> and what should we add? >>> You need to say what the meaning of the new predicate is. >> What would you think "non-blue" usually mean? > > It's your predicate, I wouldn't introduce it myself. > If you want to express the property of "not being blue" when > you already have a predicate blue, then why not use the > negation of FOL? What happens if tomorrow I or somebody else would like to change mind and would the predicate "non-blue" instead of "blue"? Would FOL _forbid_ that? > >>> This can be done by extending the old structure, eg by takinh >>> extension of not-blue as { x in U | not R(x) } = {0,2}, >>> which then provides a different structure which is a model for T4b. >> Right. But to be precise M4b now would have <'non-blue',C({1})> >> where C({1}) is the complemantary set of {1} (in U), which is >> {0,2}. The point is, Alan, the 1-ary predicate {0,2} is a common >> part of _both_ M4 and M4b, serving the same purpose: to interpret >> ~blue(c1) - or any equivalent formula - as true. Iow, difference >> on this between M4 and M4b is just the _name_ of first component >> of <x,{0,2}>: which name x should have? You could choose a "foreign" >> name by extending the language, or a "domestic" using FOL symbol >> redefinition, or in my case a variance of redefinition: an "intrinsic" >> name such as "the-complementary-set-in-U-of-the-predicate-symbolized- >> by-'blue'", which in this case I use an short alias '~blue'. >> >> And everything would still conform with the definition of structure >> (model). > > I have no idea where this is taking us, but, yes, you can extend > the initial structure in this way to get a model in the extended > syntax with your extra axiom. The point is if you agree that whatever Shoenfield meant to say in defining truth can be captured in the terminologies of basic unformalized set, then we can have better understanding of the key concepts using these _technical terminologies_. Textbooks of course do use natural language hence you can't expect them being without some degree of "glossing over", unintended and intended omission, persnal choice of harmless wordings, etc... If you use strictly set termilogies here then you'd see I'm not being astray from the definition of structure mention in his book. But you got to use the terminologies for the most part of the conversation, at least the key parts. > > Why on earth would Shoenfield say: > > "In particular, for each constant e of L, e_a is an individual of > *A*." > > if he meant that individuals could denote the empty set??? Are you saying it's forbidden in FOL that U = {} ? Did you, btw, read pg. 10 about his "0-tuple", "0-ary function"? > > That's not *his* usage at all. Well, "formulas ... be the expressions which assert some fact" is one of his usages (pg. 4). So (x=x) /\ (~x=x) isn't a formula because such expression would assert no fact, right? The point is don't read his usage only: that got to be balanced with the _technical essence_ of his usages! > And it leads to contradictions with > basic principles, like Tarski's recursive definition > of truth, If you do have reasons but don't present that with technical terminologies (such as set) then I don't know what you mean here. and important claims, such as the completeness theorem. Well, Completeness is a downstream subject and we've been debating something upstream. Let's have a closure on whether or not x=x is true in all contexts of FOL first, before moving on to others, I'd think.
From: Marshall on 29 May 2010 20:39 On May 29, 2:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On May 29, 1:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> On May 29, 10:58 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> Marshall wrote: > >>>>> On May 29, 10:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> Note my "the only way" in the question. If FOL, by the technicalities > >>>>>> vested in all of its layers, doesn't insist that that's the only way, > >>>>>> then technically other ways are equally possible > >>>>> Yes, but: > >>>>>> and Marshall's counter > >>>>>> stipulation that x=x is true in all contexts of FOL is incorrect in one > >>>>>> of those possible ways > >>>>> doesn't follow. FOL puts *some* restrictions on what the > >>>>> mapping can be. One such restriction is that x=x must > >>>>> be true in all contexts. > >>>> And I've refuted this in the same post, via the truth of some > >>>> meta statements A and B. > >>> No you didn't. > >> And your _technical reasons_ for saying that is ...? > > > My technical reason for saying that you didn't refute > > that x=x is true in all FOL contexts is that you didn't. > > If that's what you call "technical reason" then let me borrow > what AK said before and say it _to you_: > > > thank you for the helpful reminder that it's useless to attempt to > > discuss logic with some people. It's certainly true that it's useless for you to attempt to ply your nonsense onto any minimally competent individual. > > Have you figured out yet that you are wrong about > > vacuous truth yet? You dropped the ball on that > > subthread. > > Don't bet on it. Just reopen subthread and I'm sure you'd be shown > again there's nothing there that I was wrong that's worth mentioning. Wrong. You get the concept of vacuous truth wrong. I gave plain-as-day references. Of course, you being a talentless bufoon and all, it's hard for you to understand even the most basic and obvious bits. > I'm a bit tired of your "technical reasons", > to be frank. Being constantly shown to be wrong by everyone all the time must indeed be tiring, I'd imagine. Marshall
From: Marshall on 29 May 2010 20:42 On May 29, 2:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > >>>>> On May 29, 2:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> But when U = {}, there's no flexibility at all > >>>>> So your claim is that > >>>>> There does not exist an x such that blue(x) > >>>>> must be false? You can refer to as many mappings > >>>>> and definitions of "truth" as you want. At the end > >>>>> of the day if all formula are false in a model with > >>>>> empty universe, then > >>>>> There does not exist an x such that blue(x) > >>>>> must be considered false. > >>>> It must have been the case you either didn't read or wasn't > >>>> paying attention or wasn't able to understand what I said > >>>> about the truth preemptive characteristics of the meta statement > >>>> B in the post. > >>>> It doesn't matter what I "want" here: that's Tarski's definition > >>> Ok, rephrase. > >>> At the end of the day your claim is that, > >>> using Tarski's defintion of truth, > >>> all formula are false in a model with empty universe > >>> Then > >>> There does not exist an x such that blue(x) > >>> must be considered false. > >> Are you saying that > > >> "There does not exist an x such that blue(x)" > > >> is a FOL formula of L(T4)? > > > Yes. Are you claiming it is not. > > Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a > just FOL and therefore is false in the structure in which U = {}, > due to B is false, correct? It's false by your personal "every formula is false" stricture. However the actual fact of the matter is that it is true. And it's true according to FOL. This is a simple, technical demonstration of the unsoundness of your approach. Marshall
From: spudnik on 29 May 2010 21:06 textbooks are often *generically* bad glosses on the discoveries in the original monographs, or simply pedantic workbooks. the realempty set, to me, is those who attempt proofs, without any grounding in elementary geometrical & numbertheory proofs -- see wlym.com. and, recall, it was Liebniz who gave the generic format of "iff," which is necessity & sufficiency, used meaningfully in various ways in natural language. the New Math following upon General Bourbaki was a silly thing, since you *need* natural language (and diagrams etc.) to make ready analogies & metaphors for your work. such that, the glaring example of Borubakism was perhaps Russell's illinguistic "paradoxes" -- whence "silly" deploys from over-reliance on Aristotle's syllogisms! --Stop BP's capNtrade rip-off; call Waxman & tell him, we need a small *tax* on carbon emmissions, instead! http://wlym.com > Textbooks of course do use natural > language hence you can't expect them being without some degree of > "glossing over", unintended and intended omission, persnal choice > of harmless wordings, etc...
From: William Hughes on 29 May 2010 22:30
On May 29, 6:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > >>>>> On May 29, 2:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> But when U = {}, there's no flexibility at all > >>>>> So your claim is that > >>>>> There does not exist an x such that blue(x) > >>>>> must be false? You can refer to as many mappings > >>>>> and definitions of "truth" as you want. At the end > >>>>> of the day if all formula are false in a model with > >>>>> empty universe, then > >>>>> There does not exist an x such that blue(x) > >>>>> must be considered false. > >>>> It must have been the case you either didn't read or wasn't > >>>> paying attention or wasn't able to understand what I said > >>>> about the truth preemptive characteristics of the meta statement > >>>> B in the post. > >>>> It doesn't matter what I "want" here: that's Tarski's definition > >>> Ok, rephrase. > >>> At the end of the day your claim is that, > >>> using Tarski's defintion of truth, > >>> all formula are false in a model with empty universe > >>> Then > >>> There does not exist an x such that blue(x) > >>> must be considered false. > >> Are you saying that > > >> "There does not exist an x such that blue(x)" > > >> is a FOL formula of L(T4)? > > > Yes. Are you claiming it is not. > > Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a > just FOL and therefore is false in the structure in which U = {}, > due to B is false, correct? Nope. If there is no x then There is no x such that x is blue is true. - William Hughes |