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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic knowledge
Date: Fri, 21 Mar 2025 10:57:34 +0200
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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