The End of an Era - That of The Natural Numbers
in [Math]
|
From: William Hughes on 8 May 2010 21:19 On May 8, 2:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 7, 1:56 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 7, 1:13 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>> <snip> > >>>> There's a meta theorem stating that if GC is true then it'd be > >>>> undecidable in PA (or any system T "as strong as arithmetic"). > >>>> Would you still make the same "guess" A, in light of this meta > >>>> theorem? > >>> Since this meta theorem is obviously idiotic > >>> I would ignore it completely. > >> Are you saying that GC is decidable in, say, PA then? > > > No, it is not known if GC is decidable or not. > > I am saying that it is possible for GC > > to be true and decidable. > > <footnote> > As well as true but not decidable. So GC's being true would > correspond to 2 mutually exclusive conditions of the very same > underlying theory T. Nope. We now that exactly one of four possiblities must be true 1. GC is true and decidable 2. GC is true and not decidable 3. GC is false and decidable 4. GC is false and not decidable. Based on present knowledge we can rule out 4, but not 1,2, or 3. It is possible that 1 is true and 2 false, or that 2 is true and 1 false. It is not possible that 1 and 2 are both true. <snip> > At this stage, I still haven't had a complete articulation of the title > of the thread and my observation about (1), but I think I have enough > "concrete intuitions" to believe it's a matter of time only, not of "if" > that the articulation would be done. Nope. Your First Observation is wrong, Throw as many "concrete intuitions" and/or as much time as you want at it. It will remain wrong. - William Hughes
From: Nam Nguyen on 9 May 2010 18:28 Nam Nguyen wrote: > > So I'm not going to rush it (to reduce the chance of be incorrect), > and would go slow step, by step. > > **** > > The first step then would be to demonstrate the end of the monopoly of > Godel-Incompleteness that's based on the knowledge of the natural > numbers. In this step, we'll demonstrate that if we define a formula > being true in a purely intuitive manner, as we do in "being arithmetically > true" viz-a-viz the naturals, then we could have formal systems in which > a certain formula would be _true but not provable_ and these systems > aren't in the class of "as strong as arithmetic. We'll give 2 examples > of such systems: Tni0, and Tni1 whose languages are sub-languages of L(PA); > ("ni" as in "new [kind of] incompleteness"). > > Definitions of formal systems and (language) models: > > Tni0 <-> {Ex[~(0=x)] \/ Ax[0=x]} > Tni1 <-> {Ax[~(Sx=0)] \/ Ex[Sx=0]} > > [Note: in the definitions below, "f" would be as in "finite" and > "i" as in "infinite".] > > M0f <-> {<'=',{<e0,e0>}>} > M0i <-> {<'=',{<e0,e0>,...}>} > > M1f <-> {<'=',{<e0,e0>,<e1,e1>}>, > <'S',{<e0,e0>,<e1,e1>}} and with some correction: > M1i <-> {<'=',{<e0,e0>,<e1,e1>,...}>, > <'S',{<e0,e1>,...}>} > > where ",..." means there is _at least_ zero more elements in the set. > > A few notions about these 2 formal systems: > > (a) They're of the form {A \/ ~A), which is tautological and which, > as the lone-axiom systems would reveal NO information (knowledge) > about the individual component formulas (A, ~A). > > (b) Semantically (and naturally) A and ~A would reflect mutually exclusive > concepts. In case of Tni0, these 2 concepts are "singularity" and its > negation, and in the of Tni1 they are "finite cardinality" and its > negation. [It should be noted that to the extend that a wff F has any > semantic, the formula (F /\ ~F) has a semantics as well.] > > (c) We can not know - it's impossible to know - the decidability of A or > ~A. > > (d) If we _intuit_ M0i or M1i as the "standard" model of the perspective > formal system above (Tni0, or Tni1 respectively), then either the > corresponding A and ~A would be come a formula of the status "true but > not provable". Hence both of these 2 systems are _incomplete_, though > _not a la Godel_. > > In summary, Tni0 and Tni1 together with their corresponding purported "the > standard" models (M0i, M1i), are the template, the driving force, of my > thread title and my observation about (1). I think in the case of the > arithmetic formal systems, such as PA, the mutually exclusive concepts > mentioned in (b) are those of Induction (as exemplified by addition) and > of non-Induction (as exemplified by the Fundamental Theorem of Arithmetic > which stipulates basically all numbers a generated by multiplicative-primes > [and 0 and 1 of course]). This post is the next step (step 2) which is still an introductory step but in which we'll explain what (1) and its negation would semantically convey, as 2 mutually exclusive concepts alluded to in (b). A Canadian Lotto 6/49 "number" is a sequence of 6 unique numbers, each is in between 1 and 49. For example, the 6/49 winning "number" for May 8th, 2010 is: 04 10 21 39 45 48 Suppose we now have a similar lottery but in which a wining ticket would consist of an infinite sequence of unique non-zero natural numbers (note: that doesn't mean _necessarily_ all non-zero numbers). Suppose further that it's officiated the weekly winning ticket be the one that the 2 official sources A and B have agreed, but that while A would need to generate an "induction" sequence (e.g. "2*n"), B would only need to stipulate the winning sequence in a "non-induction" manner such as "There exists a countably infinite sequence of unique non-zero naturals", so that if the officials from source B don't agree with A then the winning number has to be re-drawn in the sense A has to come up with another "induction" sequence until B would agree to it! In details, an example of A's drawn (generated) winning ticket would be (*) 10 12 100 112 1000 1012 10000 10012 ... while the way B's "drawn" winning number could be described as: (**) x1 x2 x3 x4 .... where x1, x2, x4, of ... are of _unknown_ values though constrained by some criteria such as being unique and being non-zero. Setting aside the ridiculousnesses of this lottery winning-ticket decision-making definition, suppose now we make a meta-level _prediction_ statement: (***) <-> ((*) is the winning ticket) xor ((**) is the winning ticket) From what we know about this kind of lottery, _it's impossible to know_ the truth or falsehood of (***). The long and short of this analogy-example is that pGC would play the role of the "induction" formula while cGC would be the "non-induction" counter part, and our (1) formula would play the role of (***). Semantically, (1) and (***) would mean "[certain counterpart] knowledge of induction is different from that of non-induction", the truth value of which would be impossible to know. Again, this step is an introductory step, introducing the semantics of (1) and the motivation why (1) is of the form it is. In the next step (step 3) we'll review the what would be available for us in meta level, to make certain inferences and conclusions about (1).
From: Aatu Koskensilta on 11 May 2010 09:42 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > So everything boils down to define term such as "intuition" in a > technical debate? Great! Great! But as you have told us, anything and everything except knowledge of specific formal derivations is "intuitive". There's not much to discuss or debate on such arbitrary (and bizarre) posits. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 11 May 2010 22:40 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> So everything boils down to define term such as "intuition" in a >> technical debate? Great! > > Great! But as you have told us, anything and everything except knowledge > of specific formal derivations is "intuitive". So, what would you call reasoning by rules of inference? Counter intuitive reasoning? > There's not much to > discuss or debate on such arbitrary (and bizarre) posits. Not much? How about the end of GIT's monopoly of the perceived incompleteness? Can you for instance discuss the similarity between the perceived standard model M1i of the system Tni1 and the perceived natural number as the standard model of arithmetic systems such as PA?
From: Alan Smaill on 12 May 2010 05:57
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> So everything boils down to define term such as "intuition" in a >>> technical debate? Great! >> >> Great! But as you have told us, anything and everything except knowledge >> of specific formal derivations is "intuitive". > > So, what would you call reasoning by rules of inference? Counter intuitive > reasoning? Just as worthwhile as the rules of inference and axioms in question are -- not to be trusted on the sole grounds that there is a formal system; and so open to dispute. Just like reasoning about natural numbers. Why should I think that every statement of the form "A or not A" is true? Do you think that all such statements are true? You said that LEM "appears to be true" IIRC. -- Alan Smaill |