
Re: Matheology S 224
Posted:
Apr 17, 2013 7:56 AM


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.
>  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, 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.
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).
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.
> 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.
>> >> 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.
I suspect that things will take a turn for the worse by the time we get there.
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.
 Jesse F. Hughes "To [mathematicians] amateur mathematicians are worse than scum, and scarier than nuclear bombs."  James S. Harris on mathematicians' phobias

