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Topic: Matheology § 224
Replies: 84   Last Post: Apr 20, 2013 4:43 PM

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 namducnguyen Posts: 2,777 Registered: 12/13/04
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 non-empty 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 non-empty
>>> subset of S" and "x is in a non-empty subset of S"?

>>
>> Yes there is.

>
> I'll see what I can see here.
>

>> "x is in a non-empty 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 non-empty 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 set-string.
>> 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 set-string
>> [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 eliminated-able 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 set-string (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 Def-1 _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 well-formed strings involving "{","}" and
> ",",[1] and we can also interpret them as elements of |M|. Let's call
> the set of all such well-formed 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 1-1 mapping relevant to
elements of the set that (1) encodes, but the key concept you seem
to have missed here is the well-formedness 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 Def-1 _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 non-empty 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 non-empty 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 Def-1 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 set-string:

(1') { {, [[]] }

the second occurrence of '{' encodes/represents an individual, and
has nothing to do with set, the way the 1st occurrence would.

> Iow, would you acknowledge that if a set is finite, we can _encode_
> any of its set-membership 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
----------------------------------------------------

Date Subject Author
4/12/13 Alan Smaill
4/12/13 namducnguyen
4/12/13 Frederick Williams
4/12/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Peter Percival
4/14/13 fom
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/16/13 namducnguyen
4/16/13 namducnguyen
4/16/13 Jesse F. Hughes
4/16/13 namducnguyen
4/16/13 fom
4/17/13 namducnguyen
4/17/13 fom
4/17/13 namducnguyen
4/17/13 Jesse F. Hughes
4/17/13 Jesse F. Hughes
4/17/13 namducnguyen
4/20/13 namducnguyen
4/17/13 Frederick Williams
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 fom
4/18/13 namducnguyen
4/18/13 Frederick Williams
4/18/13 namducnguyen
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Peter Percival
4/15/13 Peter Percival
4/14/13 namducnguyen
4/14/13 namducnguyen
4/13/13 Frederick Williams
4/13/13 Peter Percival
4/13/13 Peter Percival
4/13/13 namducnguyen
4/15/13 Peter Percival
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Frederick Williams
4/14/13 Frederick Williams
4/14/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 namducnguyen