Re: Tarski / Gödel and redefining the Foundation of Logic

On 2024-07-22 14:40:41 +0000, olcott said:

On 7/22/2024 3:14 AM, Mikko wrote:

On 2024-07-21 13:20:04 +0000, olcott said:
 

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

On 2024-07-20 13:22:31 +0000, olcott said:
 

On 7/20/2024 3:42 AM, Mikko wrote:

On 2024-07-19 13:48:49 +0000, olcott said:
 

 Some undecidable expressions are only undecidable because
they are self contradictory. In other words they are undecidable
because there is something wrong with them.

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

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

 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.
 

 As Richard
Montague so aptly showed Semantics can be specified syntactically.
 

An arithmetic sentence is always about
numbers, not about sentences.

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

 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.
 

 You just directly contradicted yourself.

 I don't, and you cant show any contradiction.
 

 Gödel's proof had nothing what-so-ever to do with provability
except that he proved that g is unprovable in PA.

He also proved that its negation is unprovable in PA. He also proved
that every consistent extension of PA has a an sentence (different
from g) such that both it and its negation are unprovable.

A proof is about sentences, not about
numbers.
 

The Liar Paradox: "This sentence is not true"

 cannot be said in the language of Peano arithmetic.

 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.

 By Gödel's completeness theorem every consistent incomplete first order
theory has a model where at least one unprovable sentence is true.
 

https://liarparadox.org/Tarski_247_248.pdf // Tarski Liar Paradox basis
https://liarparadox.org/Tarski_275_276.pdf // Tarski proof

 It is very simple to redefine the foundation of logic to eliminate
incompleteness.

 Yes, as long as you don't care whether the resulting system is useful.
Classical logic has passed practical tests for thousands of years, so
it is hard to find a sysem with better empirical support.

 When we show how incompleteness is eliminated then this also shows
how undefinability is eliminated and this would have resulted in a
chatbot that eviscerated Fascist lies about election fraud long
before they could have taken hold in the minds of 45% of the electorate.

The simplest way to elimita incompleteness is to construct a theory
where everytihing is provable. Of course such theory is not useful.
The next simplest way is to construct a theory for a finite universe.
As the theory is complete it specifies the number of objects in the
universe. Then it is possible to evaluate every quantifier with a
simple finite loop or recursion, so the truth of every sentence is
computable.
This kind of theory may have some use but its applicability is very
limited. In particular, a complete theory cannot be used in situations
where somthing is not known.

Because people have been arguing against my correct system of reasoning
we will probably see the rise of the fourth Reich.

Trying something impossible does not prevent anything.
--
Mikko