On Aug 27, 11:17 am, "LudovicoVan" <
ju...@diegidio.name> wrote:
>
> > Just to confirm my understanding: The statement "This statement is
> > false" cannot be formalized in number theory (the language of
> > arithmetic), which is the basis for pretty much all of mathematics. Is
> > that correct?
>
> Of course not! That statement can indeed be formalized, which is what makes
> it interesting... On the other hand, don't be surprised that arithmetic
> etc. are always involved: mathematical logic just is not logic, as it should
> be clear by now.
>
Not as a theory of arithmetic, which is what is generally meant by
formalized IN number theory.
T |- t1 ^ t2 ^ t3 ^ not(t3) ^ t4 ^ ...
T |- contradiction
ex contradictione sequitur quodlibet
from a contradiction, anything follows
{t3, !t3} |- W
A theorem and the negation of that theorem (even if it's one and the
same) forms a contradiction and an inconsistent theory.
Mind you TARSKI's ERROR is obvious
the same error as Godel's Proof.
A Godel Number implies a SYNTAX (Left Right GRAMMAR)
exists to construct the formula.
e.g. you could number all PREDICATES
A(x) P(x,y,z)
E(x) not( x = x )
These are both FORMULA's of Predicate Calculus.
But they are not both THEOREMS of Predicate Calculus.
There is nothing wrong with having a GODEL NUMBER FUNCTION within a
theory.
TRUE(#phi)
Banishing true(x) and false(x) predicates is just pruning the leaves
of your theory.
-------
Without a THEORY GRAMMAR (axioms)
You start with this
T |- W
a TRUE() predicate just gives you this!
{t3, !t3} |- W
-------
You guys don't know the difference between a SYNTAX GRAMMAR and THEORY
GRAMMAR.
http://www.microprolog.com/help.html
microPROLOG Syntax
Rule 1 LINE. --> FACT.
Rule 2 LINE. --> FACT IF TAIL.
Rule 3 LINE? --> TAIL?
Rule 4 TAIL --> FACT FACT ... FACT
Rule 5 FACT --> [term ARGS]
Rule 6 ARGS --> ARG ARG ... ARG
Rule 7 ARG --> term | VAR | FACT
e.g LINE.
--> FACT. ----------------------------- (Rule 1)
--> [term ARGS].
--> [term ARG].
--> [term term].
--> [lady gaga].
Following SYNTAX does not guarantee valid code!
Herc
--
http://tinyurl.com/TARSKI-PROOF
http://tinyurl.com/BLUEPRINTS-TARSKI