The End of an Era - That of The Natural Numbers
in [Math]
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From: MoeBlee on 7 May 2010 14:13 On May 7, 10:18 am, Transfer Principle <lwal...(a)lausd.net> wrote: > This marks the second time that I've seen a poster who criticized a > standard theory (last time it was ZFC, this time it's PA) and then was > asked to define "is." And then the person asking for the definition of > "is" actually defends Clinton, a national laughingstock during that > particular case. It appears you are entirely confused as to the course of this particular digression in the convesation. MoeBlee
From: MoeBlee on 7 May 2010 14:51 On May 7, 1:10 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On May 6, 5:15 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > is a very good example for using a demand for definition > > to block debate. > > No, it doesn't. It wasn't a DEBATE. It wasnt' anything LIKE a debate > even. Clinton was in no POSITION even to block any debate that was not > even occurring. Rather, Clinton was under examination in an > ADVERSARIAL context, in which context he has the right to draw > whatever fine distinctions (and this distinction was not just > gratuiously fine) he feels are needed to answer the question > accurately and even in as best light to himself as he can. When he > made the remark about 'is' he was drawing attention to an important > point regarding the particular question. P.S. He didn't even demand or even SUGGEST a need for a definition. Rather, he pointed out that there were at least to senses to the word, thus he made a point that he was answering the original question was with respect to the sense of the word he had in mind. All he really said is that according to one definition (one sense) of the word 'is' we get one thing but with another definition (sense) we get something else. He didn't demand that the prosecutor provide a definition before the questioning could continue or even demand or suggest that anyone at all provide a definition. MoeBlee
From: Nam Nguyen on 8 May 2010 13:52 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. Interesting isn't it? But I suppose if the Moon were made of cheese then anything would go, and similarly if we claimed to know the naturals precisely, while we actually don't, then we'd arrive such correspondence. Suppose I define being true as being provable then it's possible for a formula to be true and decidable. It all depends on each intuition of what being true means, doesn't it? But let's defer this to later part of the thread. </footnote> > [I think you may have your meta theorems mixed up. > There is a meta theorem that says that if GC is > undecidable it must be true, i.e. it is not possible > for GC to be false and undecidable] Yes. Sorry for the mix-up: I thought I had a reference to it but I actually don't. I do have intuitions to defend the title of the thread, to back up my observation about (1), or my belief if GC is true it's impossible to know, given our intuitive knowledge of the natural numbers. What I'm conceding though is I haven't been able to transform my intuitions into more technically articulated arguments, yet: there are more subtleties surrounding the knowledge about the naturals than my intuitions quickly scanned through, to allow me to make certain conclusions in meta level. 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>}} M1i <-> {<'=',{<e0,e0>,<e1,e1>}>, <'S',{<e0,e0>,<e1,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]). 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.
From: Nam Nguyen on 8 May 2010 18:12 Nam Nguyen 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. Interesting isn't it? But I suppose if the > Moon were made of cheese then anything would go, and similarly > if we claimed to know the naturals precisely, while we actually > don't, then we'd arrive such correspondence. > > Suppose I define being true as being provable then it's possible > for a formula to be true and decidable. It all depends on each > intuition of what being true means, doesn't it? But let's defer > this to later part of the thread. > </footnote> > >> [I think you may have your meta theorems mixed up. >> There is a meta theorem that says that if GC is >> undecidable it must be true, i.e. it is not possible >> for GC to be false and undecidable] > > Yes. Sorry for the mix-up: I thought I had a reference to it > but I actually don't. I do have intuitions to defend the title > of the thread, to back up my observation about (1), or my belief > if GC is true it's impossible to know, given our intuitive knowledge > of the natural numbers. What I'm conceding though is I haven't been able > to transform my intuitions into more technically articulated arguments, > yet: there are more subtleties surrounding the knowledge about the > naturals than my intuitions quickly scanned through, to allow me > to make certain conclusions in meta level. > > 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>}} > > M1i <-> {<'=',{<e0,e0>,<e1,e1>}>, > <'S',{<e0,e0>,<e1,e1>,...}} Correction M1i should have been: > > M1i <-> {<'=',{<e0,e0>,<e1,e1>,...}>, > <'S',{<e0,e0>,<e1,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]). > > 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. >
From: Nam Nguyen on 8 May 2010 18:29
Nam Nguyen wrote: > Nam Nguyen wrote: >> >> M1i <-> {<'=',{<e0,e0>,<e1,e1>}>, >> <'S',{<e0,e0>,<e1,e1>,...}} > > Correction M1i should have been: > > > > > M1i <-> {<'=',{<e0,e0>,<e1,e1>,...}>, > > <'S',{<e0,e0>,<e1,e1>,...}>} My apology for making the same typo twice. M1i should really be: M1i <-> {<'=',{<e0,e0>,<e1,e1>,...}>, <'S',{<e0,e1>,...}} |