Re: Tarski / Gödel and redefining the Foundation of Logic

On 2024-07-23 14:53:21 +0000, olcott said:

On 7/23/2024 3:07 AM, Mikko wrote:

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.
 

 L is the language of a formal mathematical system.
x is an expression of that language.
 When we understand that True(L,x) means that there is a finite
sequence of truth preserving operations in L from the semantic
meaning of x to x in L, then mathematical incompleteness is abolished.

No, it is not. From the meaning of "formal mathematical system" follows
that whether x is an expression of language L does not depend on semantics
or L is not a language of a formal mathiematical system. In addition,
the system is incomplete if there is a sentence that can be determined
to be true from the meaning of x but cannot be proven in the system.
--
Mikko