Re: Mathematical incompleteness has always been a misconception

On 2/4/25 11:30 PM, olcott wrote:

On 2/3/2025 6:39 PM, Richard Damon wrote:

On 2/3/25 12:00 PM, olcott wrote:

On 2/1/2025 12:23 PM, Richard Damon wrote:

On 2/1/25 1:10 PM, olcott wrote:

On 2/1/2025 7:56 AM, Richard Damon wrote:

On 1/31/25 10:43 PM, olcott wrote:

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.
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Within empirical truth this is possible.
Within analytical truth this is impossible.

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No, you only think it is impossible, becuase you don't know what you are talking about.
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Unless there is a semantic connection with
a truthmaker to what makes the expression
true IS IS NOT TRUE.

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Right, and that can be an INFINITE series of connection, which thus don't form a proof.

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It does make a {proof} within the foundational base meaning
of the term {proof} even though it may not meet the idiomatic
term-of-the-art meaning from math. The generic notion of {Truth}
itself is only defined in terms of base meanings. When math
diverges from this it is no longer talking about actual truth.
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The "foundational base meaning" of a proof in Formal Logic is a FINITE series.
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True[0] cannot possibly exist for any expression of language that
is only made true by a semantic connection to its truthmaker.

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Which can be a connection of infinite length.
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This makes the notion of provable[math] essentially a misnomer
because it attempts to override and supersede the most basic
foundation of the notion of truth itself.

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But provable is a statment about the existance of a FINITE sequence of connection
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 That IS NOT what Proof[0] means.
Proof[0] means that a connection to a truth-maker exists.
 

Show me an actual formal system defined that allows "Proof" to be an infinite connection to the truth-maker. All you are doing ios proving that you are just making up everything you say,
I think part of the problem is you just don't understand what a Formal System is, and since Incompleteness is a property of Formal System (not just general philosophy) that is an important part.
Of course, your big part for not understanding Formal Systens is you don't believe you need to follow the rules, and that is fundamental to Formal Logic, so its concepts are just foreign to you.
Sorry, you are just proving your total ignorance of what you talk about, and so ignorant that you can't see your ignorance.