Re: Mathematical incompleteness has always been a misconception --- Tarski

On 2/25/2025 9:27 AM, Mikko wrote:

On 2025-02-24 21:31:26 +0000, olcott said:
 

On 2/24/2025 2:51 AM, Mikko wrote:

On 2025-02-22 17:24:59 +0000, olcott said:
 

On 2/22/2025 3:12 AM, Mikko wrote:

On 2025-02-21 23:22:23 +0000, olcott said:
 

On 2/20/2025 3:01 AM, Mikko wrote:

On 2025-02-18 13:50:22 +0000, olcott said:
 

There is nothing like that in the following concrete example:
LP := ~True(LP)
 In other words you are saying the Prolog is incorrect
to reject the Liar Paradox.
 Above translated to Prolog
 ?- LP = not(true(LP)).
LP = not(true(LP)).

 According to Prolog rules LP = not(true(LP)) is permitted to fail.
If it succeeds the operations using LP may misbehave. A memory
leak is also possible.
 

?- unify_with_occurs_check(LP, not(true(LP))).
false

 This merely means that the result of unification would be that LP conains
itself. It could be a selmantically valid result but is not in the scope
of Prolog language.
 

 It does not mean that. You are wrong.

 It does in the context where it was presented. More generally,
unify_with_occurs_check also fails if the arguments are not
unfiable. But this possibility is already excluded by their
successfull unification.

 IT CANNOT POSSIBLY BE SEMANTICALLY VALID

 Of course it is. Its semantics is well defined by the Prolog standard.

 Go freaking read the Clocksin and Mellish.
an "infinite term" means NOT SEMANTICALLY VALID.

 Prolog does not define any semantics other than the execution semantics
of a prolog program. Therefore no data structure has any own semantics.
 The result of the exectution of an instruction like LP == not(true(LP))
is not fully defined by the standard so we may say that that instruction
is semantically invalid.
 

 When we ask for Prolog to determine whether an expression
in Prolog is true according to its facts and rules and the
evaluation of the expression gets stuck in an infinite loop
then this expression IS SEMANTICALLY INCORRECT.