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Topic: Matheology § 224
Replies: 84   Last Post: Apr 20, 2013 4:43 PM

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 Jesse F. Hughes Posts: 9,776 Registered: 12/6/04
Re: Matheology S 224
Posted: Apr 17, 2013 7:58 AM

Nam Nguyen <namducnguyen@shaw.ca> writes:

> On 16/04/2013 9:31 PM, Nam Nguyen wrote:
>> On 16/04/2013 6:08 AM, Jesse F. Hughes wrote:
>>> Nam Nguyen <namducnguyen@shaw.ca> writes:
>>>

>>>>
>>>> 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_ .

>>>
>>> Fine.
>>>
>>> So, let M be a language structure for ZF, and let x and S be elements
>>> of |M|. Then,
>>>
>>> if x is proven/verified to be in a non-empty subset of S, then
>>> x in S is an /absolute truth/.
>>>
>>> So, some questions:
>>>
>>> Is there any difference between "x is proven to be in a non-empty
>>> subset of S" and "x is in a non-empty subset of S"?

>>
>> Yes there is.
>>
>> "x is in a non-empty subset of S" could be _expressed_ as a FOL language
>> expression: x e S' /\ Ay[ y e S' -> y e S].
>>
>> On the other hand, in "x is proven to be in a non-empty subset of S",
>> the _meta phrase_ "is proven" can not be expressed by a FOL language:
>> "is proven" pertains to a meta truth, which in turns can't be equated
>> to a language expression: truth and semantics aren't the same.
>>
>> Let me put it in a more precise way. In meta level, _if a set is finite_
>> _then it we can encode the set_ , say as a finite set-string.
>> For instance, the below string:
>>
>> (1) { [], [[]] }
>>
>> would _encode a finite set_ of 2 elements. We then would have the
>> following meta definitions:
>>
>> - To create, construct a finite set is to to write down a set-string
>> [as we've done in (1)] that would encode (represent) the set.
>>
>> - To verify a element x to be in a set S is to verify that a portion
>> of the set-string (representing S) would represent x being a member
>> of the set S.
>>
>> For example in (1), if we let x be [], S be { [], [[]] }, then we can
>> verify that x is in the set S.

>>>
>>> Why the non-empty subset of S stuff?
>>>
>>> x in S <-> x in {x} & {x} c S,

>>
>> First, I'd put it this way:
>>
>> true(x e S) <=> true({x} c S)
>>
>> Secondly, the "subset of S stuff" is the key phrase in meta level
>> we'd use to define infinite sets, _incomplete_ sets later.
>>

>>>
>>> so *every* element of S is in some non-empty subset of S.

>>
>> Right. It looks "funny" to phrase it that way, but it's instrumental
>> to define _incomplete_ sets, which we'd need shortly, after we agree
>> on Def-1 for finite set.

>
> Iow, would you acknowledge that if a set is finite, we can _encode_ any
> of its set-membership truth or falsehood?

No idea what your question means.

--
Jesse F. Hughes
"If anything is true in general about Usenet, it's that people can go
on and on about just about anything." -- James Harris speaks the
truth.

Date Subject Author
4/12/13 Alan Smaill
4/12/13 namducnguyen
4/12/13 Frederick Williams
4/12/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Peter Percival
4/14/13 fom
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/16/13 namducnguyen
4/16/13 namducnguyen
4/16/13 Jesse F. Hughes
4/16/13 namducnguyen
4/16/13 fom
4/17/13 namducnguyen
4/17/13 fom
4/17/13 namducnguyen
4/17/13 Jesse F. Hughes
4/17/13 Jesse F. Hughes
4/17/13 namducnguyen
4/20/13 namducnguyen
4/17/13 Frederick Williams
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 fom
4/18/13 namducnguyen
4/18/13 Frederick Williams
4/18/13 namducnguyen
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Peter Percival
4/15/13 Peter Percival
4/14/13 namducnguyen
4/14/13 namducnguyen
4/13/13 Frederick Williams
4/13/13 Peter Percival
4/13/13 Peter Percival
4/13/13 namducnguyen
4/15/13 Peter Percival
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Frederick Williams
4/14/13 Frederick Williams
4/14/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 namducnguyen