Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

On 5/5/2025 8:11 PM, Richard Damon wrote:

On 5/5/25 11:31 AM, olcott wrote:

On 5/5/2025 6:04 AM, Richard Damon wrote:

On 5/4/25 10:23 PM, olcott wrote:

When we define formal systems as a finite list of basic facts and allow semantic logical entailment as the only rule of inference we have systems that can express any truth that can be expressed in language.
>
Also with such systems Undecidability is impossible. The only incompleteness are things that are unknown or unknowable.

>
Can such a system include the mathematics of the natural numbers?
>
If so, your claim is false, as that is enough to create that undeciability.
>

>
It seems to me that the inferences steps that could
otherwise create undecidability cannot exist in the
system that I propose.'

 Only because it seems to create a trivially small system.
 

When I told you that the system comprises the entire
set of all general knowledge that can be expressed in
language many many times, you must have a mental defect
to to think that this system is very small.

>
For example: "This sentence is not true" cannot be
derived by applying semantic logical entailment to
basic facts. It is rejected as semantically unsound
on this basis.

 So?
 

>
Try to show any complete concrete example using
a system of basic facts and applying semantic logical
entailment where undecidability can be derived.

 That isn't what I said. I said that you system, to be decidable, couldn't include the mathematics of the Natural Numbers.
 

It does includes the mathematics of natural numbers
expressed as basic facts and truth preserving
operations applied to these basic facts.
When you start with truth and only apply truth
preserving operations you necessarily only end
up with truth. This means that you NEVER end
up with any undecidability.
The Liar Paradox: "this sentence is not true"
is rejected as untrue because it cannot be derived
by applying only truth preserving operations to
basic facts.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer