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Re: Matheology S 224
Posted:
Apr 16, 2013 2:40 AM
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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 Def-1, Def-2, 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 Def-1 and Def-2.
>>>>>>
>>>>>> I'm serious in saying that it's crucial to my thesis about
>>>>>> cGC. If such a simple definition of set-membership truth-relativity
>>>>>> 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 Def-1 and Def-2
>>>>> 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 non-technical 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:
>>>> |
>>>> | Def-1 - 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.
>>>> |
>>>> | Def-2 - 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 Def-1 and Def-2 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 proof-steps_ 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 Def-1 says in the finite case is that given a set S, if an
>>> element x is proven, verified to be in a non-empty _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 meta-statement fact/truth we have
> to _establish_.
>
> For a finite set, such a meta-statement fact can be established by
> way of meta level encoding.
>
> For example, if I define the singleton set S of 2-tuple 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).
>
--
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There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
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