'When Are Relations Neither True Nor False?
in [Math]
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From: Transfer Principle on 30 Mar 2010 16:29 On Mar 27, 7:02 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Any rate, enough talk. Do you have even a single absolute truth you > > could show me so that I'd realize I've been wrong all along? Let's > > begin with the natural numbers: which formula in the language of > > arithmetic could _you_ demonstrate as absolutely true? > There is a fairly straightforward construction that can yield both > boolean logic and continuous higher forms, and even a lower form that > I will call universal. > Constrain the real numbers to those values whose magnitude is unity. > We see two options > +1, -1 . > Using polysign numbers extend this system to P3. Ah yes, the polysign numbers. I still remember Golden's constuction of these sets. > One might initially consider there to be a three verticed logic here, > but in the general form we see that the unity values now form a > continuous circle. There was a discussion of alternate-valued logic back when tommy1729 proposed using three-valued logic (tommy1729 being, of course, one of Golden's biggest supporters). But we found out that usually, standard theorists object to these alternate forms of logic. For one thing, standard theorists obviously accept two-valued Boolean logic (FOL), and they appear to be open to continuum-valued logic (also called "fuzzy" logic). But they tend to object to kappa-valued logic, where kappa is a cardinal that is strictly between two and the cardinality of the continuum. > In two dimensions we see that the same procedure yields a continuum of > values, though there are arguably those three unique positions > -1, +1, *1 . But notice that Golden does acknowledge a continuum of values. So perhaps this could be a form of fuzzy logic that the standard theorists might accept as well. There's a huge difference between fuzzy logic and Golden's though. For fuzzy logic usually considers the values to lie in the interval [0,1], with 0 being false and 1 being true. Golden's fuzzy logic is decidedly _not_ described by the interval [0,1] at all. Golden regularly points out that multiplication on the set {-1,+1,*1} in P3 is isomorphic to addition in the group Z/3Z. In general, Golden wishes to construct an n-valued logic by considering a subset of Pn that's isomorphic to addition in the group Z/nZ, which is also isomorphic to the multiplicative set of nth roots of unity in C. (As was discussed in many previous Golden threads, _addition_ in Pn is not isomorphic to _addition_ in C, but this current subthread only deals with multiplication, not addition.) Thus, our continuum-valued logic can be described by the entire unit circle in C, not the unit interval [0,1]. Now we ask ourselves, is such a logic even possible. Back in the tommy1729 three-valued logic threads, the standard theorists often pointed out that what they needed to see were the laws of inference for any proposed logic. Without laws of inference, one can't really call it a logic at all. I wouldn't mind taking at the laws of inference for fuzzy logic and modifying them so that they work for Golden's logic. But of course, I don't own a textbook on fuzzy logic or its laws of inference, nor do I plan on owning such a book anytime soon. > By leaving the Euclidean and working the sphere these forms exist > naturally. Hmmm, non-Euclidean geometry and the sphere. This reminds me of AP's work as well. I wonder whether the AP-adics might work better if we used Golden's logic instead of FOL. (And before Jesse Hughes or anyone else protests, I'm fully aware that the link between Golden's post here and AP's work is even flimsier than that between Newberry and Clarke. I'm the one who's trying to unify the so-called "cranks" as best as I can in order to find a theory that will satisfy at least two of them, which saves me work from having to find a different theory for each and every "crank.")
From: Jesse F. Hughes on 30 Mar 2010 18:02 Transfer Principle <lwalke3(a)lausd.net> writes: > There was a discussion of alternate-valued logic back when tommy1729 > proposed using three-valued logic (tommy1729 being, of course, one > of Golden's biggest supporters). But we found out that usually, > standard theorists object to these alternate forms of logic. No, you found no such thing. Multi-valued logic is a perfectly respectable field. (It does not follow that Tommy's blatherings are perfectly respectable. I don't read Timothy's posts, so I don't offer an opinion on them. They might be blatherings, too, for all I know.) Of course, most mathematicians would bristle at the suggestion that they ought to do their work in three-valued logic. But that's not the same as objecting to three-valued logic per se. -- Jesse F. Hughes "Radicals are interesting because they were considered 'radical' by the people who played with them who wrote a lot of math work that modern mathematics depends on." --Another JSH history lesson
From: MoeBlee on 30 Mar 2010 18:11 On Mar 30, 2:42 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Multiplication of ordinals is defined using > transfinite recursion -- which requires the Replacement Schema! See your next paragraph: > At this point, one might point out that we only need to define the > operations on the _finite_ ordinals, and so we ought to use finite, > rather than _trans_finite, recursion. But finite recursion -- though > it > apparently avoids Replacement Schema -- requires the Axiom of > _Infinity_ in its proof. (1) I must have overlooked that you're not allowing the axiom of infinity. (2) Of course, per ordinary set theory, without the axiom of infinity, it is not possible to have a function whose domain is the set of ordered pairs of finite ordinals. But still, if I am not mistaken, even without the axiom of infinity, we can define a 2-place operation symbol "+" that yields the ordinary arithmetic sum of two finite ordinals (and yields "junk" 0 otherwise). This is done by using such techniques as the ancestral, as I recall. It's mentioned in various texts in set theory. > "1 is the multiplicative identity" and all that. But MoeBlee and > Ullrich > were explicitly working in Z(-Regularity), I mentioned Ullrich in connection with ".999..=1"; I didn't say anything about Ullrich in connection with multiplicative identity. Also, in the proof of ".999...=1", I don't recall whether I mentioned the explicit set theoretic axioms, but, in any case, yes, no axioms not in Z-regularity were used. > I'm not sure whether MoeBlee and Ullrich wish to avoid Replacement > in the same way that a finitist/"crank" wishes to avoid Infinity. No, I don't think replacement is to be avoided in the manner that some people think infinity ought to be avoided. > It > may > be that MoeBlee and Ullrich are open to using Replacement when it > is absolutely inevitable, but as long as they can avoid the schema -- > and one can construct R without it -- they'll do so. I have no idea what Ullrich thinks about this, but the above is okay for me. Sure, if I can prove something without replacement then I prefer to do it that way. But it's not much of a philosophical issue for me (I have no philosophical quibbles with replacement); rather it's more an admiration of the relative economy and power of Z- regularity. MoeBlee
From: MoeBlee on 30 Mar 2010 18:17 On Mar 30, 2:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 26, 10:04 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Mar 26, 12:42 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > > as powerful > > This cries for a definition or explanation of what is meant by 'as > > powerful'. In mathematical logic we have various notions such as > > 'intepretability' and 'conservative extension'. But some of your > > comments seem not to use 'as powerful' in such technical senses > > (otherwise some of your questions in this regard would be non- > > starters). > > Marshall Spight was the first to use the phrase in this thread. I only > use the phrase in reponse to Spight's post. > > Here is a quote from the post from earlier in this thread, back on the > 2nd of March, at 4:11PM Greenwich time: > > "One thing that the cranks and crankophiles never understand > is that the systems they come up with add a lot of complexity > while actually removing functionality or utility. To do so > merely to avoid some counterintuitive but harmless > property (such as vacuous truth, in Newberry's case) is > a huge waste of time. > Less powerful; more work to use: that's a crank theory for you." > > So Spight criticizes so-called "crank" theories as less powerful than > some other theory -- presumably the standard theory (ZFC). My > goal, therefore, is to find a theory that's _as_ powerful as ZFC, so > that Spight would have less reason to criticize it. Whether you're adopting his terminology or not, for such remarks as the previous one to have much meaning, you'd need to say what YOU mean (or what you think Spight might mean, or SOMETHING) by 'as powerful' . For if you mean it in it's most natural sense in mathematical logic (i.e., the sense of 'as strong'), then your project is a non-starter, since any theory at least as strong as ZFC is not one that would be accepted by someone who eschews ZFC (let alone the axiom of infinity itself), since such a theory INCLUDES all the theorems of ZFC. MoeBlee
From: Nam Nguyen on 30 Mar 2010 23:52
Tim Golden BandTech.com wrote: > On Mar 27, 11:25 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Tim Golden BandTech.com wrote: >>> On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Nam Nguyen wrote: >>>>> Alan Smaill wrote: >>>>>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>>>>>> Seriously, if you could demonstrate a truly absolute abstract truth >>>>>>>> in mathematical reasoning, I'd leave the forum never coming back. >>>>>>> If you can't (general "you") then I'm sorry: my duty to the Zen council, >>>>>>> so to speak, is to see to it that "absolute" truths such as G(PA) is a >>>>>>> thing of the past, if not of oblivion. >>>>>> one day you will realise that your duty to the Zen council >>>>>> is to overcome your feeling of duty to what is purely subjective ... >>>>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which >>>>> you'd need to overcome - someday. Each of us (including Godel) coming to >>>>> mathematics and reasoning has our own subjective "baggage". >>>>> Is it FOL, or FOL=, that you've alluded to? For example. >>>> Note how much this physical reality has influenced and shaped our >>>> mathematics and mathematical reasonings. Euclidean postulates had their >>>> root in our once perception of space. From P(a) we infer Ex[P(x)] >>>> wouldn't be an inference if the our physical reality didn't support >>>> such at least in some way. And uncertainty in physics is a form >>>> relativity. >>>> The point is relativity runs deep in reality and you're not fighting >>>> with a lone person: you're fighting against your own limitation! >>>> Any rate, enough talk. Do you have even a single absolute truth you >>>> could show me so that I'd realize I've been wrong all along? Let's >>>> begin with the natural numbers: which formula in the language of >>>> arithmetic could _you_ demonstrate as absolutely true? >>> There is a fairly straightforward construction that can yield both >>> boolean logic and continuous higher forms, and even a lower form that >>> I will call universal. >>> Constrain the real numbers to those values whose magnitude is unity. >>> We see two options >>> +1, -1 . >> It's relative as to how many real numbers one could "constrain". So >> "constraint" is a relative notion, not an absolute one. >> >> In any rate, in all the below (including the URL) I still couldn't >> see an absolute truth. Could you state such truth here? > > By accepting the generalization of sign the existence of dimension > follows directly. > That is the most absolute truth that I've come up with. So what would happen if one doesn't accept the "the generalization of sign"? Would we get a relative truth, or an absolute falsehood? |