
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 nonempty 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 nonempty >>> subset of S" and "x is in a nonempty subset of S"? >> >> Yes there is. >> >> "x is in a nonempty 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 nonempty 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 setstring. >> 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 setstring >> [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 setstring (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 nonempty 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 nonempty 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 Def1 for finite set. > > Iow, would you acknowledge that if a set is finite, we can _encode_ any > of its setmembership 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.

