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

