
Re: Matheology S 224
Posted:
Apr 17, 2013 11:53 PM


On 17/04/2013 5:56 AM, Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen@shaw.ca> writes: > >> 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. > > I'll see what I can see here. > >> "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. > > Surely, this is not so. > > Here's a construction of an obviously finite set: > > Let X be a set consisting of a single element, namely the > 10^10^10^10th digit of pi. > > Arguably, I've "constructed" a finite set via that definition. In > fact, I can specify that it is one of ten possible singletons, {0}, > {1}, ..., {9}. I cannot tell you which one (due to my own, personal > ignorance), but it's one of those.
_Your singleton set is simply encoded as_ :
{ d }
where d is simply an alias, a _definable symbol_ , standing for, and is eliminatedable by, the 10^10^10^10th digit of pi. So, we've encoded a singleton set.
Why did you say "Surely, this is not so"?
> >>  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. > > Puzzling! I thought we were dealing with semantics here,
I'm not puzzling: I already gave the caveat that "truth and semantics aren't the same".
> but now you > want to take two objects (i.e., sets) in a language structure, convert > them to strings and establish a certain relation holds between the > strings in order to say that it is an "absolute truth" that one set is > an element of the other.
Let me give some caveats here:
 First, we're still talking about Def1 _general_ finite set and the concept of structure is _not_ required here; (1) has nothing with structure.
 secondly, (1) isn't only about set objects: it's also about individual objects, such as the _strings_ [], [[]]. (Note an individual might or might _not_ be a set). > > So, we have a complicated picture. On the one hand, we have the > language of ZF, and terms of the language are interpreted as elements > of M, the carrier of the language structure, i.e., > >  : Term(ZF) > M
> > On the other hand, we have wellformed strings involving "{","}" and > ",",[1] and we can also interpret them as elements of M. Let's call > the set of all such wellformed strings "SN" for "Set Notation". So, > we can map > > SN > M. > > The definition of that map isn't really so obvious to me, but I reckon > it can be specified. Note, however, that the image of the map > SN > M will be (a subset of?) the finite or countable "sets" of > M. This map is *not* onto (neither, of course, is the map  > mentioned above).
Yes, in the string (1) you could discern an 11 mapping relevant to elements of the set that (1) encodes, but the key concept you seem to have missed here is the wellformedness of (1) as a structure theoretical _formula_ written in SN. > > And you're claiming that, if x and S are elements of M, then "x in S" > is an /absolute truth/ iff there are strings s and t in SN such that s > is "an element of" t (a relation between s and t that can be spelled > out, but I won't do so presently), and > > s > x > t > S.
See. Your "x and S are elements of M" is of some degree of confusion. We're still at Def1 _general_ finite set and we're only talking about x and S in relation to (1): M should not be mentioned at all here.
>> For example in (1), if we let x be [], S be { [], [[]] }, then we can >> verify that x is in the set S. > > Of course, x is *not* literally [], nor S literally { [], [[]] }, but > rather x is the object in the language structure corresponding to the > string [], and similarly for S.
Like defined symbols in a formal language, x and S are aliases [like 0, prime(x)] and could/should be literally replaced, in this case, by [] and { [], [[]] }, respectively. They're used for _shorthand notation_ .
> >>> >>> 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) > > I don't know why you'd put it that way, but never mind. > >> 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. > > Sounds to me like you're anticipating doing something remarkably > fishy. Language structures for ZF don't have "incomplete" sets in any > sense I know.
No. It's just your confused analysis that is fishy. (A) we're not talking about "Language structures for ZF" in (1)m and (B) "incomplete set" is a topic we have yet to define!
So far, we're only talking about encoding finite sets using (certain) SN [set notation].
If so far you don't understand what I've said then basically you wouldn't understand that the notation {} would stand for, encode the concept of the empty set.
_It is that simple_ !
> I suspect that things will take a turn for the worse by the time we > get there.
I suspect your confusion would be cleared up not long from now.
> Footnotes: > [1] I won't use "[", since it has no different meaning than "{" as > near as I can figger, and serves only to confuse me.
See. You're confused there: '[' encodes/represents an individual and has nothing to do with set. In fact, in the following setstring:
(1') { {, [[]] }
the second occurrence of '{' encodes/represents an individual, and has nothing to do with set, the way the 1st occurrence would.
I asked you previously:
> Iow, would you acknowledge that if a set is finite, we can _encode_ > any of its setmembership truth or falsehood?
Let me ask you a simpler question:
Would you now acknowledge that if a set is finite, we can _encode_ the set?
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

