Re: Mathematical incompleteness has always been a misconception

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.
 The system is incomplete in the artificially contrivance way of
deliberately defined system to be insufficiently expressive.
 

And what about the fact that ther *IS* a contiguos sequence, infinite in length, that makes the statement true that you don't understand.
Part of your problem is you seem to never actually read the thing you are arguing about, but make a "simplifed" version of it, that might not be accurate, and argue against that strawman.
You are just proving that you don't understand how logic works, and that you are fundamentally ignorant of how logic actually works.
That there can be a more expressive system than the one in question, doesn't negate that the system itself exists. It seems you just don't undetstand that fundamental principle of logic systems, and don't understand the nature of "Formal Logic", only Abstract Philosophy, which can't actually prove anything as it doesn't have a defined base.