
Re: Matheology S 224
Posted:
Apr 16, 2013 11:31 PM


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.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

