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

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 Jesse F. Hughes Posts: 9,776 Registered: 12/6/04
Re: Matheology S 224
Posted: Apr 17, 2013 7:56 AM
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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.

> - 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, 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 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).

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 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.

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

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

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