Re: How a True(X) predicate can be defined for the set of analytic knowledge

On 2025-03-20 15:02:42 +0000, olcott said:

On 3/20/2025 8:09 AM, Mikko wrote:

On 2025-03-20 02:42:53 +0000, olcott said:
 

It is stipulated that analytic knowledge is limited to the
set of knowledge that can be expressed using language or
derived by applying truth preserving operations to elements
of this set.

 A simple example is the first order group theory.
 

When we begin with a set of basic facts and all inference
is limited to applying truth preserving operations to
elements of this set then a True(X) predicate cannot possibly
be thwarted.

 There is no computable predicate that tells whether a sentence
of the first order group theory can be proven.
 

 Likewise there currently does not exist any finite
proof that the Goldbach Conjecture is true or false
thus True(GC) is a type mismatch error.

However, it is possible that someone finds a proof of the conjecture
or its negation. Then the predicate True is no longer complete.

When we redefine logic systems such that they begin
with set of basic facts and are only allowed to
apply truth preserving operations to these basic
facts then every element of the system is provable
on the basis of these truth preserving operations.

However, it is possible (and, for sufficiently powerful sysems, certain)
that the provability is not computable.
--
Mikko