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

On 3/21/2025 9:31 PM, Richard Damon wrote:

On 3/21/25 8:47 PM, olcott wrote:

On 3/21/2025 6:49 PM, Richard Damon wrote:

On 3/21/25 8:49 AM, olcott wrote:

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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And thus your concept of truth breaks.
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Truth, by its definition is an immutable thing, but you just defined it to be mutable.
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How often do we need to re-verify our truths?
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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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But you aren't begining with basic facts, but with what has been assumed to be the basic facts.

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That is not what I stipulated.
When we begin with what actual are the set of basic
facts and are only allowed to apply truth preserving
operations to these basic facts then it is self-evident
that True(X) must always be correct.

 But you can't stipulate that you cant' get to things that you can get to.
 If your system can define the Natural Numbers, then we get Godel and Tarski, and you can't stop it.
 

The entire semantics of G is defined in the body of human general
knowledge that can be expressed in language henceforth called (BOK).
The whole language metalanguage thing is already taken care of
in a hierarchy of types that expresses multiple levels of logic
in the same formal system and formal language.
https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer