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Re: Q on incompleteness proof
Posted:
Jun 29, 2013 2:46 PM


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 structuretheoretical (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 structuretheoretical set could be finite or infinite, but all sets are encoded as a _finite_ string called a _structuretheoretical 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 (contextsensitive) set.
 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 structuretheoretical set
A structuretheoretical set is encoded as a set, as Step 3 above, with the following additional stipulations:
 The general format for an encoded structuretheoretical 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 nary 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 structuretheoretical set might _not_ encode a language structure.
> Step 5  Structuretheoretical verifications
 Given an encoded structuretheoretical 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 manner.
 A structuretheoretical 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.
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Are you with me so far?
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 



