Re: Tarski / Gödel and redefining the Foundation of Logic

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.

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.

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).

That does not help much if you cannot determine whether a particular
string can be shown to be true.

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

However, every arithmetic sentence is either true or false.
--
Mikko