Re: Mathematical incompleteness has always been a misconception

On 1/31/2025 7:52 PM, Richard Damon wrote:

On 1/31/25 12:42 PM, olcott wrote:

On 1/31/2025 10:08 AM, Richard Damon wrote:

On 1/31/25 10:20 AM, olcott wrote:

On 1/31/2025 8:49 AM, Richard Damon wrote:

On 1/30/25 8:24 PM, olcott wrote:

On 1/30/2025 7:06 PM, Richard Damon wrote:

On 1/30/25 6:10 PM, olcott wrote:

Within the entire body of analytical truth any expression of language that has no sequence of formalized semantic deductive inference steps from the formalized semantic foundational truths of this system are simply untrue in this system. (Isomorphic to provable from axioms).
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In other words when any expression of language of any system (formal or informal) has no semantic connection to its semantic meaning in this system then this expression is simply nonsense in this system. "This sentence is untrue" is Boolean nonsense.
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Copyright PL Olcott 2016 through 2025.
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Except that isn't what incompleteness says.
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Incompleteness is about the existance of statements which are TRUE, because there is a sequence of formal semantic deduction that reaches the statement, abet an infinite one, but there is no finite sequnce of formal semantic deduction to form a proof.
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That might be correct. If it is correct then all then
all that it is really saying is that math is incomplete
because some key pieces were intentionally left out.

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What was left out?
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If there exists no contiguous sequence of semantic deductive inference
steps from the basic facts of a system establishing that the semantic meaning of this expression has a value of Boolean true in this system then this expression is simply not true in this system even if it may be
true in other more expressive systems.
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The system is incomplete in the artificially contrivance way of
deliberately defined system to be insufficiently expressive.
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And what about the fact that ther *IS* a contiguos sequence, infinite in length, that makes the statement true that you don't understand.
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"Incomplete" means that there is no contiguous sequence of inference
steps within the expressiveness of this specific formal system.
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 No, "Incomplete" means that there is some true statement that can not be proven.
 

Within empirical truth this is possible.
Within analytical truth this is impossible.
Unless there is a semantic connection with
a truthmaker to what makes the expression
true IS IS NOT TRUE.
This is "proof" in the big sense of the term,
not the term-of-the-art thus idiomatic mathematical
meaning.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer