Re: Tarski / Gödel and redefining the Foundation of Logic

On 7/21/24 9:20 AM, olcott wrote:

On 7/21/2024 4:27 AM, Mikko wrote:

On 2024-07-20 13:22:31 +0000, olcott said:
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On 7/20/2024 3:42 AM, Mikko wrote:

On 2024-07-19 13:48:49 +0000, olcott said:
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Some undecidable expressions are only undecidable because
they are self contradictory. In other words they are undecidable
because there is something wrong with them.

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Being self-contradictory is a semantic property. Being uncdecidable is
independent of any semantics.

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Not it is not. When an expression is neither true nor false
that makes it neither provable nor refutable.

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There is no aithmetic sentence that is neither true or false. If the sentnece
contains both existentia and universal quantifiers it may be hard to find out
whether it is true or false but there is no sentence that is neither.
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 As Richard
Montague so aptly showed Semantics can be specified syntactically.
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An arithmetic sentence is always about
numbers, not about sentences.

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So when Gödel tried to show it could be about provability
he was wrong before he even started?

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Gödel did not try to show that an arithmetic sentence is about provability.
He constructed a sentence about numbers that is either true and provable
or false and unprovable in the theory that is an extension of Peano arithmetics.
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 You just directly contradicted yourself.

No, Godel didnt show that arithmetic is about provablity, but that provability can be reduced to arithmetic.
As with many things, you get your direction of implication reversed.

 

A proof is about sentences, not about
numbers.
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The Liar Paradox: "This sentence is not true"

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cannot be said in the language of Peano arithmetic.

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Since Tarski anchored his whole undefinability theorem in a self-contradictory sentence he only really showed that sentences that
are neither true nor false cannot be proven true.

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By Gödel's completeness theorem every consistent incomplete first order
theory has a model where at least one unprovable sentence is true.
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https://liarparadox.org/Tarski_247_248.pdf // Tarski Liar Paradox basis
https://liarparadox.org/Tarski_275_276.pdf // Tarski proof

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 It is very simple to redefine the foundation of logic to eliminate
incompleteness. Any expression x of language L that cannot be shown
to be true by some (possibly infinite) sequence of truth preserving operations in L is simply untrue in L: True(L, x).

And if you try that you find that your logic system can't handle very complex topics without becoming inconsistent. And that level turns out to be well below what we normally want out of logic.

 Tarski showed that True(Tarski_Theory, Liar_Paradox) cannot be defined
never understanding that Liar_Paradox is not a truth bearer.
 

But he didn't do that, that is just your misunderstanding of what he did. Your stupidity doesn't negate the work he did that was above your head.
If you think he made an error, show the exact step where he did something that violates the rules of logic that he was using.
Special note, that doesn't mean coming up with a result you find wrong, show the operation that he did that is incorrect.
Your problem is the point you like to point to isn't a point where he makes an assumption, but a point were he calls forward something he previously proved, so if you don't like that, you need to find the error with that proof.
But, since you don't seem to understand how formal logic and meta-theories work, I don't think you can understand that.
I think Formal Logic and Meta-Theories are just to abstract of a concept for you to understand.
Note, if you ACTUALLY want to "redefine" the Foundations of Logic, go ahead, just remember when you tear down a foundation, you also tear down everything built on it, so you need to rebuild them, especially since you GOAL seems to be to change some of the things above the foundation.
Go ahead and try that, but remember, you need to start AT THAT FOUNDATION, and to be honest, I don't think you understand how that sort of logic actually works well enough to make a foundation that would actual\ly hold anything. I don't think you have the time to do it eather since you wasted the last 20 years on your lies.