Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of True(L,x)

On 2/22/25 1:27 PM, olcott wrote:

On 2/22/2025 3:18 AM, Mikko wrote:

On 2025-02-21 23:19:10 +0000, olcott said:
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On 2/20/2025 2:54 AM, Mikko wrote:

On 2025-02-18 03:59:08 +0000, olcott said:
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On 2/12/2025 4:21 AM, Mikko wrote:

On 2025-02-11 14:07:11 +0000, olcott said:
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On 2/11/2025 3:50 AM, Mikko wrote:

On 2025-02-10 11:48:16 +0000, olcott said:
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On 2/10/2025 2:55 AM, Mikko wrote:

On 2025-02-09 13:10:37 +0000, Richard Damon said:
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On 2/9/25 5:33 AM, Mikko wrote:

Of course, completness can be achieved if language is sufficiently
restricted so that sufficiently many arithemtic truths become inexpressible.
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It is far from clear that a theory of that kind can express all arithmetic
truths that Peano arithmetic can and avoid its incompletness.

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WHich, it seems, are the only type of logic system that Peter can understand.
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He can only think in primitive logic systems that can't reach the complexity needed for the proofs he talks about, but can't see the problem, as he just doesn't understand the needed concepts.

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That would be OK if he wouldn't try to solve problems that cannot even
exist in those systems.

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There are no problems than cannot be solved in a system
that can also reject semantically incorrect expressions.

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The topic of the discussion is completeness. Is there a complete system
that can solve all solvable problems?

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When the essence of the change is to simply reject expressions
that specify semantic nonsense there is no reduction in the
expressive power of such a system.

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The essence of the change is not sufficient to determine that.

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In the same way that 3 > 2 is stipulated the essence of the
change is that semantically incorrect expressions are rejected.
Disagreeing with this is the same as disagreeing that 3 > 2.

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That 3 > 2 need not be (and therefore usually isn't) stripualted.

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The defintion of the set of natural numbers stipulates this.

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 If NOTHING ever stipulates that 3 > 2 then NO ONE can
possibly know that 3 > 2 leaving the finite string
"3 > 2" merely random gibberish.
 The ultimate foundation of all truth that is made true
entirely by other expressions of language is these other
expressions of stipulated truth.
   

Right, and from the stipulated truths, we can show that in a logic system that meets the specified requiments (basically can handle the properties of the Natural Numbers) and which has a "True" predicate that returns true for all true statements, and false for all statements that are not true (either being false, or nonsense), we can show that the statement:
P defined as ~True(P) is a well formed expression that the predicate needs to handle, but there is no possible answer for it.
Thus, any logic system that meets those requirements, that of defining the properties of the Natural Numbers, can have a True predicate that meets that universal requirement and be a consistant logic system.
Unless you can find the error in that proof of the validity of the statement, you are just lying.
Note, finding the error is not done by saying that the results doesn't make sense, as that is the whole point of the proof, that the presumption of the existance of the True predicate is itself invalid.