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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 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 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.

"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.

- 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.

For example in (1), if we let x be [], S be { [], [[]] }, then we can
verify that x is in the set S.
>
> 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)

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.

--
----------------------------------------------------
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