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Topic: Q on incompleteness proof
Replies: 7   Last Post: Jun 30, 2013 6:48 AM

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Posts: 2,777
Registered: 12/13/04
Re: Q on incompleteness proof
Posted: Jun 29, 2013 2:46 PM
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On 22/06/2013 10:49 AM, Nam Nguyen wrote:
> On 22/06/2013 10:39 AM, Rupert wrote:
>> On Saturday, June 22, 2013 6:27:28 PM UTC+2, Nam Nguyen wrote:
>>> On 22/06/2013 10:00 AM, Rupert wrote:

>>>> On Saturday, June 22, 2013 5:41:53 PM UTC+2, Nam Nguyen wrote:
>>>>> You don't understand that the string "[]" is an abstract object,
>>>>> in a mathematical logic discussion?


>>>> I see. So it's an abstract object. But not a set?
>>> Right. (It doesn't have to be, in any rate).

>> Is there some alphabet that the symbols "[" and "]" come from?

> Who knows. All I know is they're abstract objects and that's all it'd
> matter here.

>>> Again, can we get back to the main topic of defining structure
>>> verification?

>> Okay, so what's the definition?

> It would be in conjunction with the rules I mentioned earlier.
> But I'll give the definition not long from now. [I have to be
> away from my keyboards].

I will summarize in the steps below and you (general "you") could
ask for clarification or what have you in any steps.

-------------> Step 1 - Defining the individuals.

Each individual would be an _abstract_ "string", finite or infinite.

"Abstract" because if a string is infinite, it does _NOT_ have to be
_just countably infinite_ . The 2 _priori_ properties (among others)
we'll just have to take for granted are:

- Each string s would have a substring embedded within it, with the
string s itself being a trivial substring. (Note a substring itself
is just a string).

- There exists a string that has no substring but is a substring of
any given other string. (We'd call that the "Zero" string).

For example, the followings would be individuals that can be used
in constructing a language structure:

Ex 1: <0></0> (This is the "Zero" string).

Note: this string is syntactically isomorphic to
the familiar string 0.

Ex 2: <2><1><0></0></1></2> (This is a finite string).

Note: this string is syntactically isomorphic to
the familiar string SS0.

Ex 3: ...<2>...<1><0></0></1>...</2>.. (This could be am uncountably
infinite string).

-------------> Step 2 - Encoding the individuals.

In a structure-theoretical (but _finite_ ) formula mentioned earlier,
the symbols [, ] are reserved for encoding an individual. For example:

- [] would encode <0></0>, the "Zero" individual (string).

- If we let x be an individual variable symbol, then [x] would
encode a different individual.

-------------> Step 3 - Encoding a set individuals.

As mentioned earlier, a structure-theoretical set could be finite or
infinite, but all sets are encoded as a _finite_ string called a
_structure-theoretical formula_ . In such formula, the symbols:

- '{', '}' are reserved for indicating the delineation of a set.

- ',', is an auxiliary symbol, for (encoded) individual separation
(for clarity).

- 'this' symbol is used to encode the underlying (context-sensitive)

- Other set operation symbols and logical operator, as needed.

For instance, the following are valid formulas, encoding some sets.

Ex 1: {[]}.

Ex 2: U = { [] } U { x | (x e this.set) => ([x] e this.set) }.

Note: 'this.set' would encode the set delineated by the 2nd
pair of {, }.

-------------> Step 4 - Encoding a structure-theoretical set

A structure-theoretical set is encoded as a set, as Step 3 above, with
the following additional stipulations:

- The general format for an encoded structure-theoretical set M is:

M = { (Symbol, Set) }, where:

- There's exists one and only one (Symbol, Set) where
Symbol = 'U', and Set = (encoded) set of individuals
[which we'd call "the universe" of M].

- Other than above, Set = (encoded) set of n-ary tuples (n > 0)
[which we'd use for a predicate/function set].

- Note: A FOL language structure would be encoded as a structure-
theoretical set, but the reverse isn't necessarily true:
an encoded structure-theoretical set might _not_ encode
a language structure.

-------------> Step 5 - Structure-theoretical verifications

- Given an encoded structure-theoretical set M, and a formula
(statement) F, F is said to be verifiable to be true in M iff
the verification is done in a finite manner, or in an inductive

- A structure-theoretical set M is said to be not verifiable as
encoding a language structure, iff there exists a specific
statement F than can't be verified as true or false in M.

Otherwise, M is said to be encoding a language structure.


Are you with me so far?

There is no remainder in the mathematics of infinity.


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