
Re: Matheology S 224
Posted:
Apr 16, 2013 1:12 AM


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, [].
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 

