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

On 3/28/25 4:05 PM, olcott wrote:

On 3/28/2025 7:16 AM, Mikko wrote:

On 2025-03-27 13:03:21 +0000, olcott said:
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On 3/27/2025 5:58 AM, Mikko wrote:

On 2025-03-26 18:01:14 +0000, olcott said:
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On 3/26/2025 3:36 AM, Mikko wrote:

On 2025-03-25 14:56:33 +0000, olcott said:
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On 3/25/2025 5:19 AM, Mikko wrote:

On 2025-03-22 17:53:28 +0000, olcott said:
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On 3/22/2025 11:43 AM, Mikko wrote:

On 2025-03-21 12:49:06 +0000, olcott said:
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On 3/21/2025 3:57 AM, Mikko wrote:

On 2025-03-20 15:02:42 +0000, olcott said:
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On 3/20/2025 8:09 AM, Mikko wrote:

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

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A simple example is the first order group theory.
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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.

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

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However, it is possible that someone finds a proof of the conjecture
or its negation. Then the predicate True is no longer complete.
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The set of all human general knowledge that can
be expressed using language gets updated.
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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.

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However, it is possible (and, for sufficiently powerful sysems, certain)
that the provability is not computable.
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When we begin with basic facts and only apply truth preserving
to the giant semantic tautology of the set of human knowledge
that can be expressed using language then every element in this
set is reachable by these same truth preserving operations.

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The set of human knowledge that can be expressed using language
is not a tautology.
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tautology, in logic, a statement so framed that
it cannot be denied without inconsistency.

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And human knowledge is not.

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What is taken to be knowledge might possibly be false.

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What actually <is> knowledge is impossibly false by
definition.

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What is presented as the body of human knowledge either is a very small
part of actual knowledge or contains false claims.
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I am NOT referring to what is merely presented as the body
of general knowledge, I am referring to the actual body of
general knowledge. Within this hypothesis it is easy to see
that True(X) would be infallible.

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In that case your True(X) is uncomputable and any theory that contains
it is incomplete.
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The body of general knowledge that can be expressed
using language is defined to be complete.

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That doesn't prevent us from presenting general knowledge that is not
in that "complete" body.
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 The problem of incompleteness is not inherent.
 

But is proven for any system that meets certain minimal requirements, which either you idea of "all general knowledge" isn't actually true by any stretch of the imagination, or your system WILL be incomplete.
Basically, If your idea of Knowledge includes the properties of the Natural Numbers, then your system has been proven by Godel to be incomplete.