Re: How a True(X) predicate can be defined for the set of analytic knowledge

On 3/29/25 5:51 PM, olcott wrote:

On 3/29/2025 1:18 PM, Richard Damon wrote:

On 3/29/25 9:53 AM, olcott wrote:

On 3/29/2025 4:51 AM, joes wrote:

Am Fri, 28 Mar 2025 15:07:22 -0500 schrieb olcott:

On 3/28/2025 8:46 AM, Richard Damon wrote:

>

Ok, so therefore it includes all the "laws of mathematics" and the
"rules of inference" and thus, the system is capable of creating the
rules and properties of the Natural Numbers, so it supports the proofs
of Godel and Tarski, and thus there are statements in that sytstem that
are True but unprovable and no definition of the Truth Predicate can
handle those,

>
Yes it will showed the formal system can be defined that have all kinds
of issues because they were defined incoherently.

>
How is arithmetic (which is all it takes for Gödel's proof) incoherent?
>

>
To the best of my knowledge arithmetic itself cannot
be incomplete unless it can be shown that the sum of
two finite strings of digits cannot be derived.
>

>
Depends on what you consider "Arithmetic". If you just mean "sums" and the like, then maybe it can't be incomplete, because it can't ask questions that ask for proofs.
>

 It never could.

Then your version of arithmetic was never a logic system.
But, if your logic system creates arithmetic, then the logic system provides the logic that allows the arithmetic to create your problems.

 

Once you include first order logic with things like There exist a number such that ..., then you can perform Godel's proof and find that there is at least one statement that is true but can not be proven in the system.