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

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 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: Matheology S 224
Posted: Apr 17, 2013 12:38 AM

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?

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

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