
Re: Matheology S 224
Posted:
Apr 16, 2013 2:40 AM


On 15/04/2013 11:12 PM, Nam Nguyen wrote: > On 14/04/2013 9:29 PM, Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen@shaw.ca> writes: >> >>> On 14/04/2013 7:40 PM, Jesse F. Hughes wrote: >>>> Nam Nguyen <namducnguyen@shaw.ca> writes: >>>> >>>>> On 14/04/2013 4:58 PM, Nam Nguyen wrote: >>>>>> On 14/04/2013 4:28 PM, Nam Nguyen wrote: >>>>>>> On 14/04/2013 3:41 PM, fom wrote: >>>>>>>> On 4/14/2013 3:40 PM, Nam Nguyen wrote: >>>>>>>>> On 14/04/2013 9:19 AM, Nam Nguyen wrote: >>>>>>>>>> On 14/04/2013 12:44 AM, Nam Nguyen wrote: >>>>>>>>>>> >>>>>>>>>>> Can you or they give me a straightforward statement of >>>>>>>>>>> understanding >>>>>>>>>>> or not understanding of Def1, Def2, F, F' I've requested? >>>>> >>>>>> Also, if you'd like to help the debate about my cGC thesis, >>>>>> why don't you offer a closure on my Def1 and Def2. >>>>>> >>>>>> I'm serious in saying that it's crucial to my thesis about >>>>>> cGC. If such a simple definition of setmembership truthrelativity >>>>>> is technically wrong, inconsistent, or what have you, of course my >>>>>> entire thesis would falter to pieces. And you will never hear me >>>>>> attempt >>>>>> on the relativity of cGC truth anymore. >>>>>> >>>>>> But I do need a closure on these 2 definitions. >>>>> >>>>> Naturally _everyone_ who could constructively contribute to the >>>>> closure >>>>> would be welcomed. And if I miss anyone in the below list I'd like >>>>> to apologize in advance. >>>>> >>>>> In particular, with some reasons no so important of my own, I'd >>>>> like appreciate in advance if Chris Menzel, Herman Rubin, Franz >>>>> Fritsche, Aatu Koskensilta, George Greene, Dave Seaman, Rupert, >>>>> Jim Burns, could offer some analysis and closure on my Def1 and Def2 >>>>> as presented in: >>>>> >>>>> http://groups.google.com/group/sci.math/msg/e6f47fad548fbb97?hl=en >>>> >>>> You sure seem eager for some comments. I know I'm not on the list, >>>> but I'll bite. >>> >>> First is my nontechnical caveat about the list of names. It obviously >>> can only be a finite list so any in which way I would have listed, >>> an apparent "offending" would occur however unintentionally. And I >>> did apologize in advance for that. To prove the point, after I sent >>> out the list I realized I forgot to mention Mike Oliver. The reason >>> for the finite list, as I've said is of _my own reason but isn't an_ >>> _important one_ . >>> >>> Based on some dialogs in the past, I believe correctly or incorrectly >>> those I mentioned _might_ be aware of some of the technical motivation >>> behind my presenting for years. That's all: just the motivation; I >>> certainly did _not_ chose the list based on who I'd think be "on my >>> side", so to speak. >>> >>> In any rate, I did invite "_everyone_ who could constructively >>> contribute ...". >>> >>>> >>>> , >>>>  Given a set S: >>>>  >>>>  Def1  If an individual (element) x is defined to be in S in a >>>> finite >>>>  manner or inductively, then x being in S is defined an >>>> absolute >>>>  truth. >>>>  >>>>  Def2  If an individual (element) x isn't defined to be in S in a >>>>  finite manner or inductively, then then x being in S, or >>>> not, >>>>  is defined as a relative truth, or falsehood, respectively >>>>  >>>>  Would you et al. understand Def1 and Def2 definitions now? >>>> ` >>>> >>>> I don't understand the definitions at all, because I don't know what >>>> it means that "x is defined to be in S in a finite manner or >>>> inductively." >>> >>> Let's do the finite case first, and I will address the "inductively" >>> case after. >>> >>> The finite case is the quite similar to the definition of FOL >>> syntactical theorems, where a proof of a formula is a _finite_ >>> _sequence of proofsteps_ conforming to a certain patterns (of >>> application of rules of inference). We have no choice but take >>> for granted what certain priori, "finite", "sequence", "steps", >>> etc... would mean. >>> >>> What Def1 says in the finite case is that given a set S, if an >>> element x is proven, verified to be in a nonempty _finite subset_ >>> of S then x being a member of S is defined to be an absolute truth. >>> Naturally here we also take for granted what it'd mean by "finite >>> subset", an element being in a set or not, to know or to verify an >>> element to be or not to be in a finite set, etc... >> >> It seems to me that we are mixing syntactic and semantic notions here. > > It only appears so; and there's no mixing or mixed up. >> >> Do you mean: let t and s be any terms of the language of ZF (or >> whatever) and suppose that >> >> ZF  (E x)(x c s & x != {} & x is finite & t in x) >> >> then "t in s" is an absolute truth. > > As the presenter, and for clarity purposes, I'm going to reiterate what > I emphasized with Rupert in a related thread: my presentation _is about_ > _language structures_ and _not about formal systems_ . > > Hence, any counter argument using formal system, formal system > provability, such as "" or the like, will _not be within_ the > parameters, the assumptions, the _context of my presentation_ > and I'd have an option not to address it. > > If I use part of the language L(ZF), I'll only use it as a _shorthand_ > _notation_ for _what I present in meta level_ . >> >>>> In fact, I understand almost none of that phrase. I don't know what >>>> it means for x to be "defined to be in S", much less so defined "in a >>>> finite manner or inductively". >>> >>> I'm not sure I understand your objection here: isn't defining a set S >>> is defining certain elements x's _to be in S_ ? >> >> That's certainly not how I would put it. Is there any difference >> between the following two statements? >> >> x is defined to be in S. >> >> x is in S. > > There's a difference in meta level. > > "x is in S" is translatable to a FOL language formula: x e S, > hence it's just a FOL language expression. In brief it _is_ a formula > stated in an informal way: "x is in S" <> x e S > > "x is defined to be in S" is a _meta assertion_ _corresponding to a_ > _fact_ . Iow, in stating "x is defined to be in S" we have to: > >  define exactly what the element x be. >  define that the set {x} be indeed a subset of S. > > In brief, "x is in S" is a formula which could be true or false, > but "x is defined to be in S" is a metastatement fact/truth we have > to _establish_. > > For a finite set, such a metastatement fact can be established by > way of meta level encoding. > > For example, if I define the singleton set S of 2tuple as: > > S = { ([],[[]]) } > > then _in meta level_ I've used the meta level symbols '{', '}', > '[', ']', '=', 'S' to encode the singleton set S where there's > _the fact or truth_ that x is in the set S, should we let the > meta symbol x to stand for, say, [].
Sorry for a typo, I meant: "let the meta symbol x to stand for, say, ([],[[]])".
> > On the other hand, x e S is a FOL which could be true or false, > _depending on the context_ . In the context of S as defined above, > if the FOL symbol x stands for ([[[]]],[[[]]]) then obviously x e S > would be false. > > In summary, I hope you would understand that it's critical that > we would: > >  Leave out any mentioning of FOL formal system and provability > thereof. > >  While we _borrow_ expressibility of FOL language expressions > as much as possible (such as the use of the epsilon symbol 'e'), > quite a few underlying meta structure assertions, statements, we're > using are just that: _meta statements_ . I use these FOL language > symbols mainly _for shorthand notation at meta level_ . > > If you could translate a meta statement into a FOL expression, that > might be a bonus for my presentation, should I _choose_ to use the > translation. But should I not choose so, _it would still be a meta_ > _statement_ . > >> >>>> So, there you have it  a response >>> >>> Sure. Likewise I think I've explained your concern, in the finite case. >>> >>> Are you with me so far? >> >> Not particularly, but we'll see how it goes. > > For the sake of time, so as not to go back and forth from post to post > unnecessarily, I'd wait until we have a clear understanding here, > before going further. (Hope that you'd understand). >
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

